Pearson correlation coefficient

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class="vector-toc-numb">2</span> <span>Motivation/Intuition and Derivation</span> </div> </a> <ul id="toc-Motivation/Intuition_and_Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-For_a_population" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_a_population"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>For a population</span> </div> </a> <ul id="toc-For_a_population-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-For_a_sample" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_a_sample"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>For a sample</span> </div> </a> <ul id="toc-For_a_sample-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-For_jointly_gaussian_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_jointly_gaussian_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>For jointly gaussian distributions</span> </div> </a> <ul id="toc-For_jointly_gaussian_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Practical_issues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Practical_issues"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Practical issues</span> </div> </a> <ul id="toc-Practical_issues-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mathematical properties</span> </div> </a> <ul id="toc-Mathematical_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Interpretation</span> </div> </a> <button aria-controls="toc-Interpretation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Interpretation subsection</span> </button> <ul id="toc-Interpretation-sublist" class="vector-toc-list"> <li id="toc-Geometric_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Geometric interpretation</span> </div> </a> <ul id="toc-Geometric_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation_of_the_size_of_a_correlation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretation_of_the_size_of_a_correlation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Interpretation of the size of a correlation</span> </div> </a> <ul id="toc-Interpretation_of_the_size_of_a_correlation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Inference</span> </div> </a> <button aria-controls="toc-Inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Inference subsection</span> </button> <ul id="toc-Inference-sublist" class="vector-toc-list"> <li id="toc-Using_a_permutation_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_a_permutation_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Using a permutation test</span> </div> </a> <ul id="toc-Using_a_permutation_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_a_bootstrap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_a_bootstrap"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Using a bootstrap</span> </div> </a> <ul id="toc-Using_a_bootstrap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standard_error" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standard_error"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Standard error</span> </div> </a> <ul id="toc-Standard_error-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Testing_using_Student's_t-distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Testing_using_Student's_t-distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Testing using Student's <i>t</i>-distribution</span> </div> </a> <ul id="toc-Testing_using_Student's_t-distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_the_exact_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_the_exact_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Using the exact distribution</span> </div> </a> <ul id="toc-Using_the_exact_distribution-sublist" class="vector-toc-list"> <li id="toc-Using_the_exact_confidence_distribution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Using_the_exact_confidence_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.1</span> <span>Using the exact confidence distribution</span> </div> </a> <ul id="toc-Using_the_exact_confidence_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Using_the_Fisher_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_the_Fisher_transformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Using the Fisher transformation</span> </div> </a> <ul id="toc-Using_the_Fisher_transformation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_least_squares_regression_analysis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_least_squares_regression_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In least squares regression analysis</span> </div> </a> <ul id="toc-In_least_squares_regression_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sensitivity_to_the_data_distribution" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sensitivity_to_the_data_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Sensitivity to the data distribution</span> </div> </a> <button aria-controls="toc-Sensitivity_to_the_data_distribution-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Sensitivity to the data distribution subsection</span> </button> <ul id="toc-Sensitivity_to_the_data_distribution-sublist" class="vector-toc-list"> <li id="toc-Existence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Existence</span> </div> </a> <ul id="toc-Existence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sample_size" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sample_size"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Sample size</span> </div> </a> <ul id="toc-Sample_size-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Robustness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Robustness"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Robustness</span> </div> </a> <ul id="toc-Robustness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Variants" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Variants"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Variants</span> </div> </a> <button aria-controls="toc-Variants-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Variants subsection</span> </button> <ul id="toc-Variants-sublist" class="vector-toc-list"> <li id="toc-Adjusted_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adjusted_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Adjusted correlation coefficient</span> </div> </a> <ul id="toc-Adjusted_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weighted_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weighted_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Weighted correlation coefficient</span> </div> </a> <ul id="toc-Weighted_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reflective_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reflective_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Reflective correlation coefficient</span> </div> </a> <ul id="toc-Reflective_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaled_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaled_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Scaled correlation coefficient</span> </div> </a> <ul id="toc-Scaled_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pearson's_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pearson's_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Pearson's distance</span> </div> </a> <ul id="toc-Pearson's_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Circular_correlation_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circular_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.6</span> <span>Circular correlation coefficient</span> </div> </a> <ul id="toc-Circular_correlation_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_correlation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_correlation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.7</span> <span>Partial correlation</span> </div> </a> <ul id="toc-Partial_correlation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pearson_correlation_coefficient_in_quantum_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pearson_correlation_coefficient_in_quantum_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.8</span> <span>Pearson correlation coefficient in quantum systems</span> </div> </a> <ul id="toc-Pearson_correlation_coefficient_in_quantum_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decorrelation_of_n_random_variables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Decorrelation_of_n_random_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Decorrelation of <i>n</i> random variables</span> </div> </a> <ul id="toc-Decorrelation_of_n_random_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software_implementations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Software_implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Software implementations</span> </div> </a> <ul id="toc-Software_implementations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Pearson correlation coefficient</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go 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Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D9%85%D9%84_%D8%A7%D9%84%D8%A7%D8%B1%D8%AA%D8%A8%D8%A7%D8%B7_%D9%84%D8%A8%D9%8A%D8%B1%D8%B3%D9%88%D9%86" title="معامل الارتباط لبيرسون – Arabic" lang="ar" hreflang="ar" data-title="معامل الارتباط لبيرسون" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coeficient_de_correlaci%C3%B3_de_Pearson" title="Coeficient de correlació de Pearson – Catalan" lang="ca" hreflang="ca" data-title="Coeficient de correlació de Pearson" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Korrelationskoeffizient_nach_Bravais-Pearson" title="Korrelationskoeffizient nach Bravais-Pearson – German" lang="de" hreflang="de" data-title="Korrelationskoeffizient nach Bravais-Pearson" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaarne_korrelatsioonikordaja" title="Lineaarne korrelatsioonikordaja – Estonian" lang="et" hreflang="et" data-title="Lineaarne korrelatsioonikordaja" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coeficiente_de_correlaci%C3%B3n_de_Pearson" title="Coeficiente de correlación de Pearson – Spanish" lang="es" hreflang="es" data-title="Coeficiente de correlación de Pearson" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Korrelazio-koefiziente" title="Korrelazio-koefiziente – Basque" lang="eu" hreflang="eu" data-title="Korrelazio-koefiziente" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B6%D8%B1%DB%8C%D8%A8_%D9%87%D9%85%D8%A8%D8%B3%D8%AA%DA%AF%DB%8C_%D9%BE%DB%8C%D8%B1%D8%B3%D9%88%D9%86" title="ضریب همبستگی پیرسون – Persian" lang="fa" hreflang="fa" data-title="ضریب همبستگی پیرسون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%BC%EC%96%B4%EC%8A%A8_%EC%83%81%EA%B4%80_%EA%B3%84%EC%88%98" title="피어슨 상관 계수 – Korean" lang="ko" hreflang="ko" data-title="피어슨 상관 계수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Indice_di_correlazione_di_Pearson" title="Indice di correlazione di Pearson – Italian" lang="it" hreflang="it" data-title="Indice di correlazione di Pearson" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%90%D7%9D_%D7%A4%D7%99%D7%A8%D7%A1%D7%95%D7%9F" title="מתאם פירסון – Hebrew" lang="he" hreflang="he" data-title="מתאם פירסון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/P%C4%ABrsona_korel%C4%81cijas_koeficients" title="Pīrsona korelācijas koeficients – Latvian" lang="lv" hreflang="lv" data-title="Pīrsona korelācijas koeficients" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Correlatieco%C3%ABffici%C3%ABnt" title="Correlatiecoëfficiënt – Dutch" lang="nl" hreflang="nl" data-title="Correlatiecoëfficiënt" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%94%E3%82%A2%E3%82%BD%E3%83%B3%E3%81%AE%E7%A9%8D%E7%8E%87%E7%9B%B8%E9%96%A2%E4%BF%82%E6%95%B0" title="ピアソンの積率相関係数 – Japanese" lang="ja" hreflang="ja" data-title="ピアソンの積率相関係数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Pearsons_produkt-moment_korrelasjonskoeffisient" title="Pearsons produkt-moment korrelasjonskoeffisient – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Pearsons produkt-moment 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href="/w/index.php?title=Pearson_product-moment_correlation_coefficient&redirect=no" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment correlation coefficient</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Measure of linear correlation</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Correlation_coefficient.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Correlation_coefficient.png/400px-Correlation_coefficient.png" decoding="async" width="400" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Correlation_coefficient.png/600px-Correlation_coefficient.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Correlation_coefficient.png/800px-Correlation_coefficient.png 2x" data-file-width="960" data-file-height="530" /></a><figcaption>Examples of scatter diagrams with different values of correlation coefficient (<i>ρ</i>)</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Correlation_examples2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Correlation_examples2.svg/400px-Correlation_examples2.svg.png" decoding="async" width="400" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Correlation_examples2.svg/600px-Correlation_examples2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Correlation_examples2.svg/800px-Correlation_examples2.svg.png 2x" data-file-width="506" data-file-height="231" /></a><figcaption>Several sets of (<i>x</i>, <i>y</i>) points, with the correlation coefficient of <i>x</i> and <i>y</i> for each set. The correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of <i>Y</i> is zero.</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>Pearson correlation coefficient</b> (<b>PCC</b>)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Correlation_coefficient" title="Correlation coefficient">correlation coefficient</a> that measures <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> correlation between two sets of data. It is the ratio between the <a href="/wiki/Covariance" title="Covariance">covariance</a> of two variables and the product of their <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a>; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear <a href="/wiki/Correlation" title="Correlation">correlation</a> of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of children from a primary school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Naming_and_history">Naming and history</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=1" title="Edit section: Naming and history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It was developed by <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> from a related idea introduced by <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a> in the 1880s, and for which the mathematical formula was derived and published by <a href="/wiki/Auguste_Bravais" title="Auguste Bravais">Auguste Bravais</a> in 1844.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The naming of the coefficient is thus an example of <a href="/wiki/Stigler%27s_Law" class="mw-redirect" title="Stigler's Law">Stigler's Law</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Motivation/Intuition_and_Derivation"><span id="Motivation.2FIntuition_and_Derivation"></span>Motivation/Intuition and Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=2" title="Edit section: Motivation/Intuition and Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The correlation coefficient can be derived by considering the cosine of the angle between two points representing the two sets of x and y co-ordinate data.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This expression is therefore a number between -1 and 1 and is equal to unity when all the points lie on a straight line. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pearson's correlation coefficient is the <a href="/wiki/Covariance" title="Covariance">covariance</a> of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moment</a> about the origin) of the product of the mean-adjusted random variables; hence the modifier <i>product-moment</i> in the name.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="The material near this tag needs to be fact-checked with the cited source(s). (February 2024)">verification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="For_a_population">For a population</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=4" title="Edit section: For a population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pearson's correlation coefficient, when applied to a <a href="/wiki/Statistical_population" title="Statistical population">population</a>, is commonly represented by the Greek letter <i>ρ</i> (rho) and may be referred to as the <i>population correlation coefficient</i> or the <i>population Pearson correlation coefficient</i>. Given a pair of random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f29b9537685f499713112d6802e811cbf51bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}"></span> (for example, Height and Weight), the formula for <i>ρ</i><sup id="cite_ref-RealCorBasic_13-0" class="reference"><a href="#cite_note-RealCorBasic-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> is<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76ccfa7c2ed7f5b085115086107bbe25d329cec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.499ex; height:6.009ex;" alt="{\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b35a5e29cef19ef8b54ae74d92322f3df6dbbea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.422ex; height:1.676ex;" alt="{\displaystyle \operatorname {cov} }"></span> is the <a href="/wiki/Covariance" title="Covariance">covariance</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380c53c60c8301a5c80924b66363d831dfa80b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \sigma _{X}}"></span> is the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9fb889441f514e155f65e77dc5b7c7a5a84f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.814ex; height:2.009ex;" alt="{\displaystyle \sigma _{Y}}"></span> is the standard deviation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>.</li></ul> <p>The formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61e8a148f562999bb9c835b5b5568955a85cdae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.019ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (X,Y)}"></span> can be expressed in terms of <a href="/wiki/Mean" title="Mean">mean</a> and <a href="/wiki/Expected_Value" class="mw-redirect" title="Expected Value">expectation</a>. Since<sup id="cite_ref-RealCorBasic_13-1" class="reference"><a href="#cite_note-RealCorBasic-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X,Y)=\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X,Y)=\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e88bc4ba085b98d5cca09b958ad378d50127308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.583ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (X,Y)=\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})],}"></span></dd></dl> <p>the formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> can also be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042c646e848d2dc6e15d7b5c7a5b891941b2eab6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.299ex; height:6.009ex;" alt="{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9fb889441f514e155f65e77dc5b7c7a5a84f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.814ex; height:2.009ex;" alt="{\displaystyle \sigma _{Y}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380c53c60c8301a5c80924b66363d831dfa80b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \sigma _{X}}"></span> are defined as above</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfe6d3f115b8d6cb595119ea9bc7962a11db65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \mu _{X}}"></span> is the mean of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d61d6c2c7513bb6589518e6c034e7988aadf91c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.888ex; height:2.176ex;" alt="{\displaystyle \mu _{Y}}"></span> is the mean of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\mathbb {E} } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathbb {E} } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e2b171416452bdde30128e279b76ba4d872773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \operatorname {\mathbb {E} } }"></span> is the expectation.</li></ul> <p>The formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> can be expressed in terms of uncentered moments. Since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu _{X}={}&\operatorname {\mathbb {E} } [\,X\,]\\\mu _{Y}={}&\operatorname {\mathbb {E} } [\,Y\,]\\\sigma _{X}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(X-\operatorname {\mathbb {E} } [X]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}\\\sigma _{Y}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(Y-\operatorname {\mathbb {E} } [Y]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\,\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}\\&\operatorname {\mathbb {E} } [\,\left(X-\mu _{X}\right)\left(Y-\mu _{Y}\right)\,]=\operatorname {\mathbb {E} } [\,\left(X-\operatorname {\mathbb {E} } [\,X\,]\right)\left(Y-\operatorname {\mathbb {E} } [\,Y\,]\right)\,]=\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{X}={}&\operatorname {\mathbb {E} } [\,X\,]\\\mu _{Y}={}&\operatorname {\mathbb {E} } [\,Y\,]\\\sigma _{X}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(X-\operatorname {\mathbb {E} } [X]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}\\\sigma _{Y}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(Y-\operatorname {\mathbb {E} } [Y]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\,\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}\\&\operatorname {\mathbb {E} } [\,\left(X-\mu _{X}\right)\left(Y-\mu _{Y}\right)\,]=\operatorname {\mathbb {E} } [\,\left(X-\operatorname {\mathbb {E} } [\,X\,]\right)\left(Y-\operatorname {\mathbb {E} } [\,Y\,]\right)\,]=\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2469cdb397ef7d50c200b03c9e9f7311f0ab2b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:87.118ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\mu _{X}={}&\operatorname {\mathbb {E} } [\,X\,]\\\mu _{Y}={}&\operatorname {\mathbb {E} } [\,Y\,]\\\sigma _{X}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(X-\operatorname {\mathbb {E} } [X]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}\\\sigma _{Y}^{2}={}&\operatorname {\mathbb {E} } \left[\,\left(Y-\operatorname {\mathbb {E} } [Y]\right)^{2}\,\right]=\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\,\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}\\&\operatorname {\mathbb {E} } [\,\left(X-\mu _{X}\right)\left(Y-\mu _{Y}\right)\,]=\operatorname {\mathbb {E} } [\,\left(X-\operatorname {\mathbb {E} } [\,X\,]\right)\left(Y-\operatorname {\mathbb {E} } [\,Y\,]\right)\,]=\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]\,,\end{aligned}}}"></span></dd></dl> <p>the formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> can also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]}{{\sqrt {\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}}}~{\sqrt {\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]}{{\sqrt {\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}}}~{\sqrt {\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5984dfb290912b0e0b92a984bf49cdd628c38b2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:49.995ex; height:8.509ex;" alt="{\displaystyle \rho _{X,Y}={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]-\operatorname {\mathbb {E} } [\,X\,]\operatorname {\mathbb {E} } [\,Y\,]}{{\sqrt {\operatorname {\mathbb {E} } \left[\,X^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,X\,]\right)^{2}}}~{\sqrt {\operatorname {\mathbb {E} } \left[\,Y^{2}\,\right]-\left(\operatorname {\mathbb {E} } [\,Y\,]\right)^{2}}}}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="For_a_sample">For a sample</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=5" title="Edit section: For a sample"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pearson's correlation coefficient, when applied to a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a>, is commonly represented by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span> and may be referred to as the <i>sample correlation coefficient</i> or the <i>sample Pearson correlation coefficient</i>. We can obtain a formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span> by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{(x_{1},y_{1}),\ldots ,(x_{n},y_{n})\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},y_{1}),\ldots ,(x_{n},y_{n})\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594dca31c11c4148e945d30efde2027812bce8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.673ex; height:2.843ex;" alt="{\displaystyle \left\{(x_{1},y_{1}),\ldots ,(x_{n},y_{n})\right\}}"></span> consisting of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> pairs, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span> is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9c2079a3ffc1aacd36201ea0a3fb2460dc226f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:40.436ex; height:8.676ex;" alt="{\displaystyle r_{xy}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is sample size</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i},y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i},y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dccea2bb6a826b389fd2d042508c3d30443dd77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.102ex; height:2.009ex;" alt="{\displaystyle x_{i},y_{i}}"></span> are the individual sample points indexed with <i>i</i></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5792e9289b8786ab64a5ef4e0cd083f9c151062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.508ex; height:3.343ex;" alt="{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}"></span> (the sample mean); and analogously for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span>.</li></ul> <p>Rearranging gives us this<sup id="cite_ref-RealCorBasic_13-2" class="reference"><a href="#cite_note-RealCorBasic-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {\sum _{i}x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{{\sqrt {\sum _{i}x_{i}^{2}-n{\bar {x}}^{2}}}~{\sqrt {\sum _{i}y_{i}^{2}-n{\bar {y}}^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>n</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>n</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {\sum _{i}x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{{\sqrt {\sum _{i}x_{i}^{2}-n{\bar {x}}^{2}}}~{\sqrt {\sum _{i}y_{i}^{2}-n{\bar {y}}^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6805059197ea181c016eee1c6fff42b6215a9940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:37.938ex; height:8.509ex;" alt="{\displaystyle r_{xy}={\frac {\sum _{i}x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{{\sqrt {\sum _{i}x_{i}^{2}-n{\bar {x}}^{2}}}~{\sqrt {\sum _{i}y_{i}^{2}-n{\bar {y}}^{2}}}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f83533d3ab105f9972137dcc5a47411fc2ba5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.231ex; height:2.343ex;" alt="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"></span> are defined as above. </p><p>Rearranging again gives us this formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2}}}~{\sqrt {n\sum y_{i}^{2}-\left(\sum y_{i}\right)^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∑</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>∑</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>∑</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∑</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2}}}~{\sqrt {n\sum y_{i}^{2}-\left(\sum y_{i}\right)^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97e4e80367fa844f60007ef8d43f683edaea9bb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:47.85ex; height:8.509ex;" alt="{\displaystyle r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2}}}~{\sqrt {n\sum y_{i}^{2}-\left(\sum y_{i}\right)^{2}}}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,x_{i},y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,x_{i},y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493671619244268bb3ee6a7adca3a28521296a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.531ex; height:2.009ex;" alt="{\displaystyle n,x_{i},y_{i}}"></span> are defined as above. </p><p>This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be <a href="/wiki/Numerical_stability" title="Numerical stability">numerically unstable</a>. </p><p>An equivalent expression gives the formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span> as the mean of the products of the <a href="/wiki/Standard_score" title="Standard score">standard scores</a> as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9363d4a765bda05563bf32c9216e3cf250ac387d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.782ex; height:6.843ex;" alt="{\displaystyle r_{xy}={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f83533d3ab105f9972137dcc5a47411fc2ba5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.231ex; height:2.343ex;" alt="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"></span> are defined as above, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x},s_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{x},s_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a094967e19f2a2fe7f6e0d75d1b33acef8c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.437ex; height:2.343ex;" alt="{\displaystyle s_{x},s_{y}}"></span> are defined below</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddffcae5dec630f0ea0149d90c7d23e75b363a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.401ex; height:4.843ex;" alt="{\textstyle \left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)}"></span> is the standard score (and analogously for the standard score of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>).</li></ul> <p>Alternative formulae for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span> are also available. For example, one can use the following formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3bfa846f27e4f6c993bd259805d3ce59f4882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle r_{xy}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea4ff80b5f62cbad42cd98edef63a4e5dcfe930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.749ex; height:6.509ex;" alt="{\displaystyle r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f83533d3ab105f9972137dcc5a47411fc2ba5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.231ex; height:2.343ex;" alt="{\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}"></span> are defined as above and:</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{x}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s_{x}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3771178992d5cc958247dd962f4c159c2b019814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.512ex; height:4.843ex;" alt="{\textstyle s_{x}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}"></span> (the <a href="/wiki/Sample_standard_deviation" class="mw-redirect" title="Sample standard deviation">sample standard deviation</a>); and analogously for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec93eb58677e5642c86ecdb2430703a8ee57da2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle s_{y}}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="For_jointly_gaussian_distributions">For jointly gaussian distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=6" title="Edit section: For jointly gaussian distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f29b9537685f499713112d6802e811cbf51bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}"></span> is <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">jointly</a> <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">gaussian</a>, with mean zero and <a href="/wiki/Variance" title="Variance">variance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma ={\begin{bmatrix}\sigma _{X}^{2}&\rho _{X,Y}\sigma _{X}\sigma _{Y}\\\rho _{X,Y}\sigma _{X}\sigma _{Y}&\sigma _{Y}^{2}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ={\begin{bmatrix}\sigma _{X}^{2}&\rho _{X,Y}\sigma _{X}\sigma _{Y}\\\rho _{X,Y}\sigma _{X}\sigma _{Y}&\sigma _{Y}^{2}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69812c4687924ff0548609e3deee8d702d931ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.523ex; margin-bottom: -0.315ex; width:30.943ex; height:6.843ex;" alt="{\displaystyle \Sigma ={\begin{bmatrix}\sigma _{X}^{2}&\rho _{X,Y}\sigma _{X}\sigma _{Y}\\\rho _{X,Y}\sigma _{X}\sigma _{Y}&\sigma _{Y}^{2}\\\end{bmatrix}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Practical_issues">Practical issues</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=7" title="Edit section: Practical issues"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under heavy noise conditions, extracting the correlation coefficient between two sets of <a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">stochastic variables</a> is nontrivial, in particular where <a href="/wiki/Canonical_Correlation_Analysis" class="mw-redirect" title="Canonical Correlation Analysis">Canonical Correlation Analysis</a> reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In case of missing data, Garren derived the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimator.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Some distributions (e.g., <a href="/wiki/Stable_distribution" title="Stable distribution">stable distributions</a> other than a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>) do not have a defined variance. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=8" title="Edit section: Mathematical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The values of both the sample and population Pearson correlation coefficients are on or between −1 and 1. Correlations equal to +1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely <a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">supported</a> on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(<i>X</i>,<i>Y</i>) = corr(<i>Y</i>,<i>X</i>). </p><p>A key mathematical property of the Pearson correlation coefficient is that it is <a href="/wiki/Invariant_estimator" title="Invariant estimator">invariant</a> under separate changes in location and scale in the two variables. That is, we may transform <i>X</i> to <span class="texhtml"><i>a</i> + <i>bX</i></span> and transform <i>Y</i> to <span class="texhtml"><i>c</i> + <i>dY</i></span>, where <i>a</i>, <i>b</i>, <i>c</i>, and <i>d</i> are constants with <span class="texhtml"><i>b</i>, <i>d</i> > 0</span>, without changing the correlation coefficient. (This holds for both the population and sample Pearson correlation coefficients.) More general linear transformations do change the correlation: see <i><a href="#Decorrelation_of_n_random_variables">§ Decorrelation of n random variables</a></i> for an application of this. </p> <div class="mw-heading mw-heading2"><h2 id="Interpretation">Interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=9" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The correlation coefficient ranges from −1 to 1. An absolute value of exactly 1 implies that a linear equation describes the relationship between <i>X</i> and <i>Y</i> perfectly, with all data points lying on a <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a>. The correlation sign is determined by the <a href="/wiki/Regression_slope" class="mw-redirect" title="Regression slope">regression slope</a>: a value of +1 implies that all data points lie on a line for which <i>Y</i> increases as <i>X</i> increases, whereas a value of -1 implies a line where <i>Y</i> increases while <i>X</i> decreases.<sup id="cite_ref-STAT_462_17-0" class="reference"><a href="#cite_note-STAT_462-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> A value of 0 implies that there is no linear dependency between the variables.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>More generally, <span class="texhtml">(<i>X</i><sub><i>i</i></sub> − <span style="text-decoration:overline;"><i>X</i></span>)(<i>Y</i><sub><i>i</i></sub> − <span style="text-decoration:overline;"><i>Y</i></span>)</span> is positive if and only if <i>X</i><sub><i>i</i></sub> and <i>Y</i><sub><i>i</i></sub> lie on the same side of their respective means. Thus the correlation coefficient is positive if <i>X</i><sub><i>i</i></sub> and <i>Y</i><sub><i>i</i></sub> tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative (<a href="/wiki/Anti-correlation" class="mw-redirect" title="Anti-correlation">anti-correlation</a>) if <i>X</i><sub><i>i</i></sub> and <i>Y</i><sub><i>i</i></sub> tend to lie on opposite sides of their respective means. Moreover, the stronger either tendency is, the larger is the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the correlation coefficient. </p><p>Rodgers and Nicewander<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> cataloged thirteen ways of interpreting correlation or simple functions of it: </p> <ul><li>Function of raw scores and means</li> <li>Standardized covariance</li> <li>Standardized slope of the regression line</li> <li>Geometric mean of the two regression slopes</li> <li>Square root of the ratio of two variances</li> <li>Mean cross-product of standardized variables</li> <li>Function of the angle between two standardized regression lines</li> <li>Function of the angle between two variable vectors</li> <li>Rescaled variance of the difference between standardized scores</li> <li>Estimated from the balloon rule</li> <li>Related to the bivariate ellipses of isoconcentration</li> <li>Function of test statistics from designed experiments</li> <li>Ratio of two means</li></ul> <div class="mw-heading mw-heading3"><h3 id="Geometric_interpretation">Geometric interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=10" title="Edit section: Geometric interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Regression_lines.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Regression_lines.png/330px-Regression_lines.png" decoding="async" width="330" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Regression_lines.png/495px-Regression_lines.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d1/Regression_lines.png 2x" data-file-width="603" data-file-height="525" /></a><figcaption>Regression lines for <span class="texhtml"><i>y</i> = <i>g</i><sub><i>X</i></sub>(<i>x</i>)</span> [<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:red">red</span>] and <span class="texhtml"><i>x</i> = <i>g</i><sub><i>Y</i></sub>(<i>y</i>)</span> [<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:blue">blue</span>]</figcaption></figure> <p>For uncentered data, there is a relation between the correlation coefficient and the angle <i>φ</i> between the two regression lines, <span class="nowrap"><i>y</i> = <i>g</i><sub><i>X</i></sub>(<i>x</i>)</span> and <span class="nowrap"><i>x</i> = <i>g</i><sub><i>Y</i></sub>(<i>y</i>)</span>, obtained by regressing <i>y</i> on <i>x</i> and <i>x</i> on <i>y</i> respectively. (Here, <i>φ</i> is measured counterclockwise within the first quadrant formed around the lines' intersection point if <span class="texhtml"><i>r</i> > 0</span>, or counterclockwise from the fourth to the second quadrant if <span class="nowrap"><i>r</i> < 0</span>.) One can show<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> that if the standard deviations are equal, then <span class="nowrap"><i>r</i> = sec <i>φ</i> − tan <i>φ</i></span>, where sec and tan are <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>. </p><p>For centered data (i.e., data which have been shifted by the sample means of their respective variables so as to have an average of zero for each variable), the correlation coefficient can also be viewed as the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> of the <a href="/wiki/Angle" title="Angle">angle</a> <i>θ</i> between the two observed <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vectors</a> in <i>N</i>-dimensional space (for <i>N</i> observations of each variable).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let <b>x</b> and <b>y</b> be ordered 5-element vectors containing the above data: <span class="nowrap"><b>x</b> = (1, 2, 3, 5, 8)</span> and <span class="nowrap"><b>y</b> = (0.11, 0.12, 0.13, 0.15, 0.18)</span>. </p><p>By the usual procedure for finding the angle <i>θ</i> between two vectors (see <a href="/wiki/Dot_product" title="Dot product">dot product</a>), the <i>uncentered</i> correlation coefficient is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {2.93}{{\sqrt {103}}{\sqrt {0.0983}}}}=0.920814711.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.93</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>103</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0.0983</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.920814711.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {2.93}{{\sqrt {103}}{\sqrt {0.0983}}}}=0.920814711.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9246ea010f8aa2a5c7388806a81621a602b241f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:50.152ex; height:6.176ex;" alt="{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {2.93}{{\sqrt {103}}{\sqrt {0.0983}}}}=0.920814711.}"></span></dd></dl> <p>This uncentered correlation coefficient is identical with the <a href="/wiki/Cosine_similarity" title="Cosine similarity">cosine similarity</a>. The above data were deliberately chosen to be perfectly correlated: <span class="texhtml"><i>y</i> = 0.10 + 0.01 <i>x</i></span>. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting <b>x</b> by <span class="texhtml">ℰ(<b>x</b>) = 3.8</span> and <b>y</b> by <span class="texhtml">ℰ(<b>y</b>) = 0.138</span>) yields <span class="texhtml"><b>x</b> = (−2.8, −1.8, −0.8, 1.2, 4.2)</span> and <span class="texhtml"><b>y</b> = (−0.028, −0.018, −0.008, 0.012, 0.042)</span>, from which </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {0.308}{{\sqrt {30.8}}{\sqrt {0.00308}}}}=1=\rho _{xy},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.308</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>30.8</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0.00308</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {0.308}{{\sqrt {30.8}}{\sqrt {0.00308}}}}=1=\rho _{xy},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5c7f0af0f3cdc5fa45784605b88c9e085b62c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:47.142ex; height:6.176ex;" alt="{\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {0.308}{{\sqrt {30.8}}{\sqrt {0.00308}}}}=1=\rho _{xy},}"></span></dd></dl> <p>as expected. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation_of_the_size_of_a_correlation">Interpretation of the size of a correlation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=11" title="Edit section: Interpretation of the size of a correlation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pearson_correlation_and_prediction_intervals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Pearson_correlation_and_prediction_intervals.svg/200px-Pearson_correlation_and_prediction_intervals.svg.png" decoding="async" width="200" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Pearson_correlation_and_prediction_intervals.svg/300px-Pearson_correlation_and_prediction_intervals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Pearson_correlation_and_prediction_intervals.svg/400px-Pearson_correlation_and_prediction_intervals.svg.png 2x" data-file-width="956" data-file-height="670" /></a><figcaption>This figure gives a sense of how the usefulness of a Pearson correlation for predicting values varies with its magnitude. Given jointly normal <i>X</i>, <i>Y</i> with correlation <i>ρ</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\sqrt {1-\rho ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\sqrt {1-\rho ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2503071eb5432a487fd7f300442f2cf5fc6073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.586ex; height:4.843ex;" alt="{\displaystyle 1-{\sqrt {1-\rho ^{2}}}}"></span> (plotted here as a function of <i>ρ</i>) is the factor by which a given <a href="/wiki/Prediction_interval" title="Prediction interval">prediction interval</a> for <i>Y</i> may be reduced given the corresponding value of <i>X</i>. For example, if <i>ρ</i> = 0.5, then the 95% prediction interval of <i>Y</i>|<i>X</i> will be about 13% smaller than the 95% prediction interval of <i>Y</i>.</figcaption></figure> <p>Several authors have offered guidelines for the interpretation of a correlation coefficient.<sup id="cite_ref-Buda_22-0" class="reference"><a href="#cite_note-Buda-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cohen88_23-0" class="reference"><a href="#cite_note-Cohen88-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> However, all such criteria are in some ways arbitrary.<sup id="cite_ref-Cohen88_23-1" class="reference"><a href="#cite_note-Cohen88-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.8 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences, where there may be a greater contribution from complicating factors. </p> <div class="mw-heading mw-heading2"><h2 id="Inference">Inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=12" title="Edit section: Inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Statistical inference based on Pearson's correlation coefficient often focuses on one of the following two aims: </p> <ul><li>One aim is to test the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> that the true correlation coefficient <i>ρ</i> is equal to 0, based on the value of the sample correlation coefficient <i>r</i>.</li> <li>The other aim is to derive a <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> that, on repeated sampling, has a given probability of containing <i>ρ</i>.</li></ul> <p>Methods of achieving one or both of these aims are discussed below. </p> <div class="mw-heading mw-heading3"><h3 id="Using_a_permutation_test">Using a permutation test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=13" title="Edit section: Using a permutation test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Permutation_test" title="Permutation test">Permutation tests</a> provide a direct approach to performing hypothesis tests and constructing confidence intervals. A permutation test for Pearson's correlation coefficient involves the following two steps: </p> <ol><li>Using the original paired data (<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i</i></sub>), randomly redefine the pairs to create a new data set (<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i′</i></sub>), where the <i>i′</i> are a <a href="/wiki/Permutation" title="Permutation">permutation</a> of the set {1,...,<i>n</i>}. The permutation <i>i′</i> is selected randomly, with equal probabilities placed on all <i>n</i>! possible permutations. This is equivalent to drawing the <i>i′</i> randomly without replacement from the set {1, ..., <i>n</i>}. In <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrapping</a>, a closely related approach, the <i>i</i> and the <i>i′</i> are equal and drawn with replacement from {1, ..., <i>n</i>};</li> <li>Construct a correlation coefficient <i>r</i> from the randomized data.</li></ol> <p>To perform the permutation test, repeat steps (1) and (2) a large number of times. The <a href="/wiki/P-value" title="P-value">p-value</a> for the permutation test is the proportion of the <i>r</i> values generated in step (2) that are larger than the Pearson correlation coefficient that was calculated from the original data. Here "larger" can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a <a href="/wiki/Two-tailed_test" class="mw-redirect" title="Two-tailed test">two-sided</a> or <a href="/wiki/One-sided_test" class="mw-redirect" title="One-sided test">one-sided</a> test is desired. </p> <div class="mw-heading mw-heading3"><h3 id="Using_a_bootstrap">Using a bootstrap</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=14" title="Edit section: Using a bootstrap"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrap</a> can be used to construct confidence intervals for Pearson's correlation coefficient. In the "non-parametric" bootstrap, <i>n</i> pairs (<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i</i></sub>) are resampled "with replacement" from the observed set of <i>n</i> pairs, and the correlation coefficient <i>r</i> is calculated based on the resampled data. This process is repeated a large number of times, and the empirical distribution of the resampled <i>r</i> values are used to approximate the <a href="/wiki/Sampling_distribution" title="Sampling distribution">sampling distribution</a> of the statistic. A 95% <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for <i>ρ</i> can be defined as the interval spanning from the 2.5th to the 97.5th <a href="/wiki/Percentile" title="Percentile">percentile</a> of the resampled <i>r</i> values. </p> <div class="mw-heading mw-heading3"><h3 id="Standard_error">Standard error</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=15" title="Edit section: Standard error"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are random variables, with a simple linear relationship between them with an additive normal noise (i.e., y= a + bx + e), then a <a href="/wiki/Standard_error" title="Standard error">standard error</a> associated to the correlation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{r}={\sqrt {\frac {1-r^{2}}{n-2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{r}={\sqrt {\frac {1-r^{2}}{n-2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9220ab72ee5018aee2942b3cf906fc776c2bec0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.665ex; height:7.676ex;" alt="{\displaystyle \sigma _{r}={\sqrt {\frac {1-r^{2}}{n-2}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the correlation and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> the sample size.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Testing_using_Student's_t-distribution"><span id="Testing_using_Student.27s_t-distribution"></span>Testing using Student's <i>t</i>-distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=16" title="Edit section: Testing using Student's t-distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Critical_correlation_vs._sample_size.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Critical_correlation_vs._sample_size.svg/324px-Critical_correlation_vs._sample_size.svg.png" decoding="async" width="324" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Critical_correlation_vs._sample_size.svg/486px-Critical_correlation_vs._sample_size.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Critical_correlation_vs._sample_size.svg/648px-Critical_correlation_vs._sample_size.svg.png 2x" data-file-width="723" data-file-height="405" /></a><figcaption>Critical values of Pearson's correlation coefficient that must be exceeded to be considered significantly nonzero at the 0.05 level</figcaption></figure><p>For pairs from an uncorrelated <a href="/wiki/Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a>, the <a href="/wiki/Sampling_distribution" title="Sampling distribution">sampling distribution</a> of the <a href="/wiki/Studentized" class="mw-redirect" title="Studentized">studentized</a> Pearson's correlation coefficient follows <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i>-distribution</a> with degrees of freedom <i>n</i> − 2. Specifically, if the underlying variables have a bivariate normal distribution, the variable </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {r}{\sigma _{r}}}=r{\sqrt {\frac {n-2}{1-r^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\frac {r}{\sigma _{r}}}=r{\sqrt {\frac {n-2}{1-r^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e397cde0effd45d5eb033aa678ead227929ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.488ex; height:7.509ex;" alt="{\displaystyle t={\frac {r}{\sigma _{r}}}=r{\sqrt {\frac {n-2}{1-r^{2}}}}}"></span></dd></dl> <p>has a student's <i>t</i>-distribution in the null case (zero correlation).<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> This holds approximately in case of non-normal observed values if sample sizes are large enough.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For determining the critical values for <i>r</i> the inverse function is needed: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {t}{\sqrt {n-2+t^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {t}{\sqrt {n-2+t^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f585a954feeb53a614741b1b137cb07a57384d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.086ex; height:6.509ex;" alt="{\displaystyle r={\frac {t}{\sqrt {n-2+t^{2}}}}.}"></span></dd></dl> <p>Alternatively, large sample, asymptotic approaches can be used. </p><p>Another early paper<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> provides graphs and tables for general values of <i>ρ</i>, for small sample sizes, and discusses computational approaches. </p><p>In the case where the underlying variables are not normal, the sampling distribution of Pearson's correlation coefficient follows a Student's <i>t</i>-distribution, but the degrees of freedom are reduced.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Using_the_exact_distribution">Using the exact distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=17" title="Edit section: Using the exact distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For data that follow a <a href="/wiki/Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a>, the exact density function <i>f</i>(<i>r</i>) for the sample correlation coefficient <i>r</i> of a normal bivariate is<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(r)={\frac {(n-2)\,\mathrm {\Gamma } (n-1)\left(1-\rho ^{2}\right)^{\frac {n-1}{2}}\left(1-r^{2}\right)^{\frac {n-4}{2}}}{{\sqrt {2\pi }}\,\operatorname {\Gamma } {\mathord {\left(n-{\tfrac {1}{2}}\right)}}(1-\rho r)^{n-{\frac {3}{2}}}}}{}_{2}\mathrm {F} _{1}{\mathord {\left({\tfrac {1}{2}},{\tfrac {1}{2}};{\tfrac {1}{2}}(2n-1);{\tfrac {1}{2}}(\rho r+1)\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(r)={\frac {(n-2)\,\mathrm {\Gamma } (n-1)\left(1-\rho ^{2}\right)^{\frac {n-1}{2}}\left(1-r^{2}\right)^{\frac {n-4}{2}}}{{\sqrt {2\pi }}\,\operatorname {\Gamma } {\mathord {\left(n-{\tfrac {1}{2}}\right)}}(1-\rho r)^{n-{\frac {3}{2}}}}}{}_{2}\mathrm {F} _{1}{\mathord {\left({\tfrac {1}{2}},{\tfrac {1}{2}};{\tfrac {1}{2}}(2n-1);{\tfrac {1}{2}}(\rho r+1)\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4812ab82090e8ce42304c8adeb72bb8478aede05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:80.02ex; height:9.676ex;" alt="{\displaystyle f(r)={\frac {(n-2)\,\mathrm {\Gamma } (n-1)\left(1-\rho ^{2}\right)^{\frac {n-1}{2}}\left(1-r^{2}\right)^{\frac {n-4}{2}}}{{\sqrt {2\pi }}\,\operatorname {\Gamma } {\mathord {\left(n-{\tfrac {1}{2}}\right)}}(1-\rho r)^{n-{\frac {3}{2}}}}}{}_{2}\mathrm {F} _{1}{\mathord {\left({\tfrac {1}{2}},{\tfrac {1}{2}};{\tfrac {1}{2}}(2n-1);{\tfrac {1}{2}}(\rho r+1)\right)}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>c</mi> <mo>;</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98109d3d8cc5865398696e396bc469aff388b963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.86ex; height:2.843ex;" alt="{\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)}"></span> is the <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Gaussian hypergeometric function</a>. </p><p>In the special case when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}"></span> (zero population correlation), the exact density function <i>f</i>(<i>r</i>) can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(r)={\frac {\left(1-r^{2}\right)^{\frac {n-4}{2}}}{\operatorname {\mathrm {B} } {\mathord {\left({\tfrac {1}{2}},{\tfrac {n-2}{2}}\right)}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(r)={\frac {\left(1-r^{2}\right)^{\frac {n-4}{2}}}{\operatorname {\mathrm {B} } {\mathord {\left({\tfrac {1}{2}},{\tfrac {n-2}{2}}\right)}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2524deaafdb59c5e9d49b117132b1442128c4bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.528ex; height:10.009ex;" alt="{\displaystyle f(r)={\frac {\left(1-r^{2}\right)^{\frac {n-4}{2}}}{\operatorname {\mathrm {B} } {\mathord {\left({\tfrac {1}{2}},{\tfrac {n-2}{2}}\right)}}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93003d072991ba424a73ed1e081afe55c124b6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle \mathrm {B} }"></span> is the <a href="/wiki/Beta_function" title="Beta function">beta function</a>, which is one way of writing the density of a Student's t-distribution for a <a href="/wiki/Studentized" class="mw-redirect" title="Studentized">studentized</a> sample correlation coefficient, as above. </p> <div class="mw-heading mw-heading4"><h4 id="Using_the_exact_confidence_distribution">Using the exact confidence distribution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=18" title="Edit section: Using the exact confidence distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Confidence intervals and tests can be calculated from a <a href="/wiki/Confidence_distribution" title="Confidence distribution">confidence distribution</a>. An exact confidence density for <i>ρ</i> is<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (\rho \mid r)={\frac {\nu (\nu -1)\Gamma (\nu -1)}{{\sqrt {2\pi }}\Gamma \left(\nu +{\frac {1}{2}}\right)}}\left(1-r^{2}\right)^{\frac {\nu -1}{2}}\cdot \left(1-\rho ^{2}\right)^{\frac {\nu -2}{2}}\cdot \left(1-r\rho \right)^{\frac {1-2\nu }{2}}\operatorname {F} \left({\tfrac {3}{2}},-{\tfrac {1}{2}};\nu +{\tfrac {1}{2}};{\tfrac {1+r\rho }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>∣</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>ν</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ν</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi mathvariant="normal">F</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mi>ν</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>r</mi> <mi>ρ</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (\rho \mid r)={\frac {\nu (\nu -1)\Gamma (\nu -1)}{{\sqrt {2\pi }}\Gamma \left(\nu +{\frac {1}{2}}\right)}}\left(1-r^{2}\right)^{\frac {\nu -1}{2}}\cdot \left(1-\rho ^{2}\right)^{\frac {\nu -2}{2}}\cdot \left(1-r\rho \right)^{\frac {1-2\nu }{2}}\operatorname {F} \left({\tfrac {3}{2}},-{\tfrac {1}{2}};\nu +{\tfrac {1}{2}};{\tfrac {1+r\rho }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13a5edb898895aa32091a9613ec6c68e368c3e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:90.358ex; height:7.176ex;" alt="{\displaystyle \pi (\rho \mid r)={\frac {\nu (\nu -1)\Gamma (\nu -1)}{{\sqrt {2\pi }}\Gamma \left(\nu +{\frac {1}{2}}\right)}}\left(1-r^{2}\right)^{\frac {\nu -1}{2}}\cdot \left(1-\rho ^{2}\right)^{\frac {\nu -2}{2}}\cdot \left(1-r\rho \right)^{\frac {1-2\nu }{2}}\operatorname {F} \left({\tfrac {3}{2}},-{\tfrac {1}{2}};\nu +{\tfrac {1}{2}};{\tfrac {1+r\rho }{2}}\right)}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082e1af95112d5663f599556b8f59fb816c237b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.518ex; height:2.176ex;" alt="{\displaystyle \operatorname {F} }"></span> is the Gaussian hypergeometric function and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =n-1>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =n-1>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90621e9d8a7d137425c6871c269cc3dc8acddfe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.989ex; height:2.343ex;" alt="{\displaystyle \nu =n-1>1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Using_the_Fisher_transformation">Using the Fisher transformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=19" title="Edit section: Using the Fisher transformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fisher_transformation" title="Fisher transformation">Fisher transformation</a></div> <p>In practice, <a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> and <a href="/wiki/Hypothesis_test" class="mw-redirect" title="Hypothesis test">hypothesis tests</a> relating to <i>ρ</i> are usually carried out using the, <a href="/wiki/Variance-stabilizing_transformation" title="Variance-stabilizing transformation">Variance-stabilizing transformation</a>, <a href="/wiki/Fisher_transformation" title="Fisher transformation">Fisher transformation</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span><i>:</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(r)\equiv {\tfrac {1}{2}}\,\ln \left({\frac {1+r}{1-r}}\right)=\operatorname {artanh} (r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>r</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(r)\equiv {\tfrac {1}{2}}\,\ln \left({\frac {1+r}{1-r}}\right)=\operatorname {artanh} (r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d292b07d038e9969434f745faa99f170296f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.06ex; height:6.176ex;" alt="{\displaystyle F(r)\equiv {\tfrac {1}{2}}\,\ln \left({\frac {1+r}{1-r}}\right)=\operatorname {artanh} (r)}"></span></dd></dl> <p><i>F</i>(<i>r</i>) approximately follows a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{mean}}=F(\rho )=\operatorname {artanh} (\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mean</mtext> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{mean}}=F(\rho )=\operatorname {artanh} (\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654763efc6d55f537bffdfe1bdc33599e1f3d2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.11ex; height:2.843ex;" alt="{\displaystyle {\text{mean}}=F(\rho )=\operatorname {artanh} (\rho )}"></span><span class="nowrap">    </span>and <a href="/wiki/Standard_error" title="Standard error">standard error</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\text{SE}}={\frac {1}{\sqrt {n-3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>SE</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> <mo>−</mo> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\text{SE}}={\frac {1}{\sqrt {n-3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f040350f3f8007ca5b83524778904a0cf56e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.244ex; height:6.176ex;" alt="{\displaystyle ={\text{SE}}={\frac {1}{\sqrt {n-3}}},}"></span></dd></dl> <p>where <i>n</i> is the sample size. The approximation error is lowest for a large sample size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c04a9d26b86af8c6205ba2a6287fd655b6b714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}"></span> and increases otherwise. </p><p>Using the approximation, a <a href="/wiki/Standard_score" title="Standard score">z-score</a> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\frac {x-{\text{mean}}}{\text{SE}}}=[F(r)-F(\rho _{0})]{\sqrt {n-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mean</mtext> </mrow> </mrow> <mtext>SE</mtext> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\frac {x-{\text{mean}}}{\text{SE}}}=[F(r)-F(\rho _{0})]{\sqrt {n-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7a3d54a70f9005e3bf9a2accf62cbf0fa0ea71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.587ex; height:5.176ex;" alt="{\displaystyle z={\frac {x-{\text{mean}}}{\text{SE}}}=[F(r)-F(\rho _{0})]{\sqrt {n-3}}}"></span></dd></dl> <p>under the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\rho _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f238f1650226a8b22c2e7daea44054d5616bf57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.557ex; height:2.176ex;" alt="{\displaystyle \rho =\rho _{0}}"></span>, given the assumption that the sample pairs are <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> and follow a <a href="/wiki/Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a>. Thus an approximate <a href="/wiki/P-value" title="P-value">p-value</a> can be obtained from a normal probability table. For example, if <i>z</i> = 2.2 is observed and a two-sided p-value is desired to test the null hypothesis that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}"></span>, the p-value is <span class="nowrap">2 Φ(−2.2) = 0.028</span>, where Φ is the standard normal <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>. </p><p>To obtain a confidence interval for ρ, we first compute a confidence interval for <i>F</i>(<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span></i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100(1-\alpha )\%{\text{CI}}:\operatorname {artanh} (\rho )\in [\operatorname {artanh} (r)\pm z_{\alpha /2}{\text{SE}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">%</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>CI</mtext> </mrow> <mo>:</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SE</mtext> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100(1-\alpha )\%{\text{CI}}:\operatorname {artanh} (\rho )\in [\operatorname {artanh} (r)\pm z_{\alpha /2}{\text{SE}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/affc3f0ee39499c97bb851229113f49d83100bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.359ex; height:3.176ex;" alt="{\displaystyle 100(1-\alpha )\%{\text{CI}}:\operatorname {artanh} (\rho )\in [\operatorname {artanh} (r)\pm z_{\alpha /2}{\text{SE}}]}"></span></dd></dl> <p>The inverse Fisher transformation brings the interval back to the correlation scale. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100(1-\alpha )\%{\text{CI}}:\rho \in [\tanh(\operatorname {artanh} (r)-z_{\alpha /2}{\text{SE}}),\tanh(\operatorname {artanh} (r)+z_{\alpha /2}{\text{SE}})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">%</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>CI</mtext> </mrow> <mo>:</mo> <mi>ρ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SE</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SE</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100(1-\alpha )\%{\text{CI}}:\rho \in [\tanh(\operatorname {artanh} (r)-z_{\alpha /2}{\text{SE}}),\tanh(\operatorname {artanh} (r)+z_{\alpha /2}{\text{SE}})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf658969d39ea848505750b5cd76db21da78dd5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:75.089ex; height:3.176ex;" alt="{\displaystyle 100(1-\alpha )\%{\text{CI}}:\rho \in [\tanh(\operatorname {artanh} (r)-z_{\alpha /2}{\text{SE}}),\tanh(\operatorname {artanh} (r)+z_{\alpha /2}{\text{SE}})]}"></span></dd></dl> <p>For example, suppose we observe <i>r</i> = 0.7 with a sample size of <i>n</i>=50, and we wish to obtain a 95% confidence interval for <i>ρ</i>. The transformed value is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {arctanh} \left(r\right)=0.8673}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>arctanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.8673</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {arctanh} \left(r\right)=0.8673}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4516b17b353dbe2318e7dce8860da655066da54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.174ex; height:2.843ex;" alt="{\textstyle \operatorname {arctanh} \left(r\right)=0.8673}"></span>, so the confidence interval on the transformed scale is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.8673\pm {\frac {1.96}{\sqrt {47}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.8673</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.96</mn> <msqrt> <mn>47</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.8673\pm {\frac {1.96}{\sqrt {47}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1b7c305b535d0a677a532120f2cb5a6c798f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.396ex; height:6.176ex;" alt="{\displaystyle 0.8673\pm {\frac {1.96}{\sqrt {47}}}}"></span>, or (0.5814, 1.1532). Converting back to the correlation scale yields (0.5237, 0.8188). </p> <div class="mw-heading mw-heading2"><h2 id="In_least_squares_regression_analysis">In least squares regression analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=20" title="Edit section: In least squares regression analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For more general, non-linear dependency, see <a href="/wiki/Coefficient_of_determination#In_a_multiple_linear_model" title="Coefficient of determination">Coefficient of determination § In a multiple linear model</a>.</div> <p>The square of the sample correlation coefficient is typically denoted <i>r</i><sup>2</sup> and is a special case of the <a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a>. In this case, it estimates the fraction of the variance in <i>Y</i> that is explained by <i>X</i> in a <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple linear regression</a>. So if we have the observed dataset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1},\dots ,Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{1},\dots ,Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010c105dda336a4624a635ea54886fb040034d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.152ex; height:2.509ex;" alt="{\displaystyle Y_{1},\dots ,Y_{n}}"></span> and the fitted dataset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Y}}_{1},\dots ,{\hat {Y}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Y}}_{1},\dots ,{\hat {Y}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b92b6ed682745ccf89687f13a8d9babd6b98b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.998ex; height:3.176ex;" alt="{\displaystyle {\hat {Y}}_{1},\dots ,{\hat {Y}}_{n}}"></span> then as a starting point the total variation in the <i>Y</i><sub><i>i</i></sub> around their average value can be decomposed as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}(Y_{i}-{\bar {Y}})^{2}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}+\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}(Y_{i}-{\bar {Y}})^{2}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}+\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9118c1be467c66001f725256173b44a232ff3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.756ex; height:5.509ex;" alt="{\displaystyle \sum _{i}(Y_{i}-{\bar {Y}})^{2}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}+\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2},}"></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Y}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Y}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3f64a5026a18a1d7ff1b9991ad98465260fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:3.176ex;" alt="{\displaystyle {\hat {Y}}_{i}}"></span> are the fitted values from the regression analysis. This can be rearranged to give </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}+{\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}+{\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a66f944f825488f551eee242c6ee5d89172501c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.405ex; height:7.343ex;" alt="{\displaystyle 1={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}+{\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}.}"></span></dd></dl> <p>The two summands above are the fraction of variance in <i>Y</i> that is explained by <i>X</i> (right) and that is unexplained by <i>X</i> (left). </p><p>Next, we apply a property of <a href="/wiki/Least_squares" title="Least squares">least squares</a> regression models, that the sample covariance between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {Y}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {Y}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3f64a5026a18a1d7ff1b9991ad98465260fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:3.176ex;" alt="{\displaystyle {\hat {Y}}_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}-{\hat {Y}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}-{\hat {Y}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd03d5a74128859466ac10cc4561d4aeb8d72d49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.564ex; height:3.176ex;" alt="{\displaystyle Y_{i}-{\hat {Y}}_{i}}"></span> is zero. Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written (calculation is under expectation, assumes Gaussian statistics) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r(Y,{\hat {Y}})&={\frac {\sum _{i}(Y_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i}+{\hat {Y}}_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}[(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})+({\hat {Y}}_{i}-{\bar {Y}})^{2}]}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\sqrt {\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r(Y,{\hat {Y}})&={\frac {\sum _{i}(Y_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i}+{\hat {Y}}_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}[(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})+({\hat {Y}}_{i}-{\bar {Y}})^{2}]}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\sqrt {\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86595f3f77e8ee96952760d9176a5fa140cc562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.505ex; width:47.931ex; height:50.176ex;" alt="{\displaystyle {\begin{aligned}r(Y,{\hat {Y}})&={\frac {\sum _{i}(Y_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i}+{\hat {Y}}_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}[(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})+({\hat {Y}}_{i}-{\bar {Y}})^{2}]}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\[6pt]&={\sqrt {\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}.\end{aligned}}}"></span></dd></dl> <p>Thus </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b69ee444a1cf744688c7629e8379f8365436bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.731ex; height:7.343ex;" alt="{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(Y,{\hat {Y}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(Y,{\hat {Y}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96eb90dcf4a4f321c44761777ca0b5a3b557dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.493ex; height:3.343ex;" alt="{\displaystyle r(Y,{\hat {Y}})^{2}}"></span> is the proportion of variance in <i>Y</i> explained by a linear function of <i>X</i>. </p><p>In the derivation above, the fact that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c05d40d242e53a1773c0b3d16091f6038300d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.985ex; height:5.509ex;" alt="{\displaystyle \sum _{i}(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})=0}"></span></dd></dl> <p>can be proved by noticing that the partial derivatives of the <a href="/wiki/Residual_sum_of_squares" title="Residual sum of squares">residual sum of squares</a> (<span class="texhtml">RSS</span>) over <i>β</i><sub>0</sub> and <i>β</i><sub>1</sub> are equal to 0 in the least squares model, where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{RSS}}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>RSS</mtext> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{RSS}}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0767dfb81d4b557639ff4a261ce799dd180c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.176ex; height:5.509ex;" alt="{\displaystyle {\text{RSS}}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}"></span>.</dd></dl> <p>In the end, the equation can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {{\text{SS}}_{\text{reg}}}{{\text{SS}}_{\text{tot}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {{\text{SS}}_{\text{reg}}}{{\text{SS}}_{\text{tot}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3a38b66d8c711fed7d3ce42bbf4b8527fa55a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.441ex; height:6.176ex;" alt="{\displaystyle r(Y,{\hat {Y}})^{2}={\frac {{\text{SS}}_{\text{reg}}}{{\text{SS}}_{\text{tot}}}}}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SS}}_{\text{reg}}=\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SS}}_{\text{reg}}=\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2326f7ddf5f4b56f75e373683b5c990d91917b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.518ex; height:5.509ex;" alt="{\displaystyle {\text{SS}}_{\text{reg}}=\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SS}}_{\text{tot}}=\sum _{i}(Y_{i}-{\bar {Y}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SS}}_{\text{tot}}=\sum _{i}(Y_{i}-{\bar {Y}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185d77782aeb203c935c650891e8a9c70f38ceb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21ex; height:5.509ex;" alt="{\displaystyle {\text{SS}}_{\text{tot}}=\sum _{i}(Y_{i}-{\bar {Y}})^{2}}"></span>.</li></ul> <p>The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SS}}_{\text{reg}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>reg</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SS}}_{\text{reg}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bb9bbb9357f5fc855305e0a69f112bddb99cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle {\text{SS}}_{\text{reg}}}"></span> is called the regression sum of squares, also called the <a href="/wiki/Explained_sum_of_squares" title="Explained sum of squares">explained sum of squares</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SS}}_{\text{tot}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SS}}_{\text{tot}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f89b1b62a6c558284ef336502630d606a8c1cf4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.919ex; height:2.509ex;" alt="{\displaystyle {\text{SS}}_{\text{tot}}}"></span> is the <a href="/wiki/Total_sum_of_squares" title="Total sum of squares">total sum of squares</a> (proportional to the <a href="/wiki/Variance" title="Variance">variance</a> of the data). </p> <div class="mw-heading mw-heading2"><h2 id="Sensitivity_to_the_data_distribution">Sensitivity to the data distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=21" title="Edit section: Sensitivity to the data distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Correlation_and_dependence#Sensitivity_to_the_data_distribution" class="mw-redirect" title="Correlation and dependence">Correlation and dependence § Sensitivity to the data distribution</a></div> <div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=22" title="Edit section: Existence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The population Pearson correlation coefficient is defined in terms of <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a>, and therefore exists for any bivariate <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> for which the <a href="/wiki/Statistical_population" title="Statistical population">population</a> <a href="/wiki/Covariance" title="Covariance">covariance</a> is defined and the <a href="/wiki/Marginal_distribution" title="Marginal distribution">marginal</a> <a href="/wiki/Population_variance" class="mw-redirect" title="Population variance">population variances</a> are defined and are non-zero. Some probability distributions, such as the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>, have undefined variance and hence ρ is not defined if <i>X</i> or <i>Y</i> follows such a distribution. In some practical applications, such as those involving data suspected to follow a <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy-tailed distribution</a>, this is an important consideration. However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined. </p> <div class="mw-heading mw-heading3"><h3 id="Sample_size">Sample size</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=23" title="Edit section: Sample size"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If the sample size is moderate or large and the population is normal, then, in the case of the bivariate <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the sample correlation coefficient is the <a href="/wiki/Maximum_likelihood_estimate" class="mw-redirect" title="Maximum likelihood estimate">maximum likelihood estimate</a> of the population correlation coefficient, and is <a href="/wiki/Asymptotic_distribution" title="Asymptotic distribution">asymptotically</a> <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> and <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficient</a>, which roughly means that it is impossible to construct a more accurate estimate than the sample correlation coefficient.</li> <li>If the sample size is large and the population is not normal, then the sample correlation coefficient remains approximately unbiased, but may not be efficient.</li> <li>If the sample size is large, then the sample correlation coefficient is a <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent estimator</a> of the population correlation coefficient as long as the sample means, variances, and covariance are consistent (which is guaranteed when the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> can be applied).</li> <li>If the sample size is small, then the sample correlation coefficient <i>r</i> is not an unbiased estimate of <i>ρ</i>.<sup id="cite_ref-RealCorBasic_13-3" class="reference"><a href="#cite_note-RealCorBasic-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The adjusted correlation coefficient must be used instead: see elsewhere in this article for the definition.</li> <li>Correlations can be different for imbalanced <a href="/wiki/Dichotomous_variable" class="mw-redirect" title="Dichotomous variable">dichotomous</a> data when there is variance error in sample.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Robustness">Robustness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=24" title="Edit section: Robustness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like many commonly used statistics, the sample <a href="/wiki/Statistic" title="Statistic">statistic</a> <i>r</i> is not <a href="/wiki/Robust_statistics" title="Robust statistics">robust</a>,<sup id="cite_ref-wilcox_35-0" class="reference"><a href="#cite_note-wilcox-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> so its value can be misleading if <a href="/wiki/Outlier" title="Outlier">outliers</a> are present.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Specifically, the PMCC is neither distributionally robust,<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> nor outlier resistant<sup id="cite_ref-wilcox_35-1" class="reference"><a href="#cite_note-wilcox-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> (see <i><a href="/wiki/Robust_statistics#Definition" title="Robust statistics">Robust statistics § Definition</a></i>). Inspection of the <a href="/wiki/Scatterplot" class="mw-redirect" title="Scatterplot">scatterplot</a> between <i>X</i> and <i>Y</i> will typically reveal a situation where lack of robustness might be an issue, and in such cases it may be advisable to use a robust measure of association. Note however that while most robust estimators of association measure <a href="/wiki/Statistical_dependence" class="mw-redirect" title="Statistical dependence">statistical dependence</a> in some way, they are generally not interpretable on the same scale as the Pearson correlation coefficient. </p><p>Statistical inference for Pearson's correlation coefficient is sensitive to the data distribution. Exact tests, and asymptotic tests based on the <a href="/wiki/Fisher_transformation" title="Fisher transformation">Fisher transformation</a> can be applied if the data are approximately normally distributed, but may be misleading otherwise. In some situations, the <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrap</a> can be applied to construct confidence intervals, and <a href="/wiki/Permutation_test" title="Permutation test">permutation tests</a> can be applied to carry out hypothesis tests. These <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric</a> approaches may give more meaningful results in some situations where bivariate normality does not hold. However the standard versions of these approaches rely on <a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">exchangeability</a> of the data, meaning that there is no ordering or grouping of the data pairs being analyzed that might affect the behavior of the correlation estimate. </p><p>A stratified analysis is one way to either accommodate a lack of bivariate normality, or to isolate the correlation resulting from one factor while controlling for another. If <i>W</i> represents cluster membership or another factor that it is desirable to control, we can <a href="/wiki/Stratified_sampling" title="Stratified sampling">stratify</a> the data based on the value of <i>W</i>, then calculate a correlation coefficient within each stratum. The stratum-level estimates can then be combined to estimate the overall correlation while controlling for <i>W</i>.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Variants">Variants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=25" title="Edit section: Variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Correlation_and_dependence#Other_measures_of_dependence_among_random_variables" class="mw-redirect" title="Correlation and dependence">Correlation and dependence § Other measures of dependence among random variables</a></div> <p>Variations of the correlation coefficient can be calculated for different purposes. Here are some examples. </p> <div class="mw-heading mw-heading3"><h3 id="Adjusted_correlation_coefficient">Adjusted correlation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=26" title="Edit section: Adjusted correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sample correlation coefficient <span class="texhtml mvar" style="font-style:italic;">r</span> is not an unbiased estimate of <span class="texhtml mvar" style="font-style:italic;">ρ</span>. For data that follows a <a href="/wiki/Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a>, the expectation <span class="texhtml">E[<i>r</i>]</span> for the sample correlation coefficient <span class="texhtml mvar" style="font-style:italic;">r</span> of a normal bivariate is<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\mathbb {E} } \left[r\right]=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2n}}+\cdots ,\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>[</mo> <mi>r</mi> <mo>]</mo> </mrow> <mo>=</mo> <mi>ρ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathbb {E} } \left[r\right]=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2n}}+\cdots ,\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683b838e709e3b32a3c22dfec4fa665a593f42ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.768ex; height:6.009ex;" alt="{\displaystyle \operatorname {\mathbb {E} } \left[r\right]=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2n}}+\cdots ,\quad }"></span> therefore <span class="texhtml mvar" style="font-style:italic;">r</span> is a biased estimator of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae3f23f76f614ab4dc47bfc296699c2be740666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.849ex; height:2.176ex;" alt="{\displaystyle \rho .}"></span></dd></dl> <p>The unique minimum variance unbiased estimator <span class="texhtml"><i>r</i><sub>adj</sub></span> is given by<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{adj}}=r\,\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {n-1}{2}};1-r^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> <msub> <mi mathvariant="bold">F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{adj}}=r\,\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {n-1}{2}};1-r^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d4b4792e418cc86e6d5f37c095157474d27d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.373ex; height:6.176ex;" alt="{\displaystyle r_{\text{adj}}=r\,\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {n-1}{2}};1-r^{2}\right),}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cab944a623d7a357cbef615f6ca81d32206107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.477ex; height:2.009ex;" alt="{\displaystyle r,n}"></span> are defined as above,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {_{2}F_{1}} (a,b;c;z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> <msub> <mi mathvariant="bold">F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>c</mi> <mo>;</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {_{2}F_{1}} (a,b;c;z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2c006415714efae128b57c0fad2b0c37257065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.271ex; height:2.843ex;" alt="{\displaystyle \mathbf {_{2}F_{1}} (a,b;c;z)}"></span> is the <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Gaussian hypergeometric function</a>.</li></ul> <p>An approximately unbiased estimator <span class="texhtml"><i>r</i><sub>adj</sub></span> can be obtained<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2012)">citation needed</span></a></i>]</sup> by truncating <span class="texhtml">E[<i>r</i>]</span> and solving this truncated equation: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\operatorname {\mathbb {E} } [r]\approx r_{\text{adj}}-{\frac {r_{\text{adj}}\left(1-r_{\text{adj}}^{2}\right)}{2n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>r</mi> <mo stretchy="false">]</mo> <mo>≈</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\operatorname {\mathbb {E} } [r]\approx r_{\text{adj}}-{\frac {r_{\text{adj}}\left(1-r_{\text{adj}}^{2}\right)}{2n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e094c3fdfcb0bfd127f4be74e582f22a407201c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.188ex; height:7.509ex;" alt="{\displaystyle r=\operatorname {\mathbb {E} } [r]\approx r_{\text{adj}}-{\frac {r_{\text{adj}}\left(1-r_{\text{adj}}^{2}\right)}{2n}}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>An approximate solution<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2012)">citation needed</span></a></i>]</sup> to equation (<b><a href="#math_2">2</a></b>) is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{adj}}\approx r\left[1+{\frac {1-r^{2}}{2n}}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> </msub> <mo>≈</mo> <mi>r</mi> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{adj}}\approx r\left[1+{\frac {1-r^{2}}{2n}}\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf3f71f2cfe17f8f0d422d5ac0d482cc429a925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.488ex; height:6.343ex;" alt="{\displaystyle r_{\text{adj}}\approx r\left[1+{\frac {1-r^{2}}{2n}}\right],}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>where in (<b><a href="#math_3">3</a></b>) </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cab944a623d7a357cbef615f6ca81d32206107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.477ex; height:2.009ex;" alt="{\displaystyle r,n}"></span> are defined as above,</li> <li><span class="texhtml"><i>r</i><sub>adj</sub></span> is a suboptimal estimator,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2012)">citation needed</span></a></i>]</sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="suboptimal in what sense? (February 2015)">clarification needed</span></a></i>]</sup></li> <li><span class="texhtml"><i>r</i><sub>adj</sub></span> can also be obtained by maximizing log(<i>f</i>(<i>r</i>)),</li> <li><span class="texhtml"><i>r</i><sub>adj</sub></span> has minimum variance for large values of <span class="texhtml mvar" style="font-style:italic;">n</span>,</li> <li><span class="texhtml"><i>r</i><sub>adj</sub></span> has a bias of order <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">(<i>n</i> − 1)</span></span></span>.</li></ul> <p>Another proposed<sup id="cite_ref-RealCorBasic_13-4" class="reference"><a href="#cite_note-RealCorBasic-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> adjusted correlation coefficient is<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="is this in a published article? (February 2015)">citation needed</span></a></i>]</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{adj}}={\sqrt {1-{\frac {(1-r^{2})(n-1)}{(n-2)}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>adj</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{adj}}={\sqrt {1-{\frac {(1-r^{2})(n-1)}{(n-2)}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c19e17c9a6c528dbf5fde1b5577d9d65bd4e85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.55ex; height:7.509ex;" alt="{\displaystyle r_{\text{adj}}={\sqrt {1-{\frac {(1-r^{2})(n-1)}{(n-2)}}}}.}"></span></dd></dl> <p><span class="texhtml"><i>r</i><sub>adj</sub> ≈ <i>r</i></span> for large values of <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Weighted_correlation_coefficient">Weighted correlation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=27" title="Edit section: Weighted correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose observations to be correlated have differing degrees of importance that can be expressed with a weight vector <i>w</i>. To calculate the correlation between vectors <i>x</i> and <i>y</i> with the weight vector <i>w</i> (all of length <i>n</i>),<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Weighted mean: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {m} (x;w)={\frac {\sum _{i}w_{i}x_{i}}{\sum _{i}w_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">m</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {m} (x;w)={\frac {\sum _{i}w_{i}x_{i}}{\sum _{i}w_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c9277e02db423861aa4ad28c3a66dbdfa8850d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.588ex; height:6.509ex;" alt="{\displaystyle \operatorname {m} (x;w)={\frac {\sum _{i}w_{i}x_{i}}{\sum _{i}w_{i}}}.}"></span></li> <li>Weighted covariance <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (x,y;w)={\frac {\sum _{i}w_{i}\cdot (x_{i}-\operatorname {m} (x;w))(y_{i}-\operatorname {m} (y;w))}{\sum _{i}w_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">m</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">m</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (x,y;w)={\frac {\sum _{i}w_{i}\cdot (x_{i}-\operatorname {m} (x;w))(y_{i}-\operatorname {m} (y;w))}{\sum _{i}w_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc4f106a16942bd22b5c8721a0a567816dce50d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.553ex; height:6.509ex;" alt="{\displaystyle \operatorname {cov} (x,y;w)={\frac {\sum _{i}w_{i}\cdot (x_{i}-\operatorname {m} (x;w))(y_{i}-\operatorname {m} (y;w))}{\sum _{i}w_{i}}}.}"></span></li> <li>Weighted correlation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {corr} (x,y;w)={\frac {\operatorname {cov} (x,y;w)}{\sqrt {\operatorname {cov} (x,x;w)\operatorname {cov} (y,y;w)}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>corr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {corr} (x,y;w)={\frac {\operatorname {cov} (x,y;w)}{\sqrt {\operatorname {cov} (x,x;w)\operatorname {cov} (y,y;w)}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d703455daf32e17a67de117ef93433125980424" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.234ex; height:7.009ex;" alt="{\displaystyle \operatorname {corr} (x,y;w)={\frac {\operatorname {cov} (x,y;w)}{\sqrt {\operatorname {cov} (x,x;w)\operatorname {cov} (y,y;w)}}}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Reflective_correlation_coefficient">Reflective correlation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=28" title="Edit section: Reflective correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2011)">citation needed</span></a></i>]</sup> The population reflective correlation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {corr} _{r}(X,Y)={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]}{\sqrt {\operatorname {\mathbb {E} } [\,X^{2}\,]\cdot \operatorname {\mathbb {E} } [\,Y^{2}\,]}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>corr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> <mi>Y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mrow> <msqrt> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {corr} _{r}(X,Y)={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]}{\sqrt {\operatorname {\mathbb {E} } [\,X^{2}\,]\cdot \operatorname {\mathbb {E} } [\,Y^{2}\,]}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d897e4b303a062ed14cc9f88f35f5c8ffc91f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.415ex; height:7.009ex;" alt="{\displaystyle \operatorname {corr} _{r}(X,Y)={\frac {\operatorname {\mathbb {E} } [\,X\,Y\,]}{\sqrt {\operatorname {\mathbb {E} } [\,X^{2}\,]\cdot \operatorname {\mathbb {E} } [\,Y^{2}\,]}}}.}"></span></dd></dl> <p>The reflective correlation is symmetric, but it is not invariant under translation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {corr} _{r}(X,Y)=\operatorname {corr} _{r}(Y,X)=\operatorname {corr} _{r}(X,bY)\neq \operatorname {corr} _{r}(X,a+bY),\quad a\neq 0,b>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>corr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>corr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>corr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>b</mi> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <msub> <mi>corr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {corr} _{r}(X,Y)=\operatorname {corr} _{r}(Y,X)=\operatorname {corr} _{r}(X,bY)\neq \operatorname {corr} _{r}(X,a+bY),\quad a\neq 0,b>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45be7f276b0b408d286c9d49ed7dff78e2a0aa69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:77.501ex; height:2.843ex;" alt="{\displaystyle \operatorname {corr} _{r}(X,Y)=\operatorname {corr} _{r}(Y,X)=\operatorname {corr} _{r}(X,bY)\neq \operatorname {corr} _{r}(X,a+bY),\quad a\neq 0,b>0.}"></span></dd></dl> <p>The sample reflective correlation is equivalent to <a href="/wiki/Cosine_similarity" title="Cosine similarity">cosine similarity</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rr_{xy}={\frac {\sum x_{i}y_{i}}{\sqrt {(\sum x_{i}^{2})(\sum y_{i}^{2})}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msqrt> <mo stretchy="false">(</mo> <mo>∑</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>∑</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rr_{xy}={\frac {\sum x_{i}y_{i}}{\sqrt {(\sum x_{i}^{2})(\sum y_{i}^{2})}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f362146fa321cdb0444e6b66603bd9d1ea02a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:24.891ex; height:8.509ex;" alt="{\displaystyle rr_{xy}={\frac {\sum x_{i}y_{i}}{\sqrt {(\sum x_{i}^{2})(\sum y_{i}^{2})}}}.}"></span></dd></dl> <p>The weighted version of the sample reflective correlation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rr_{xy,w}={\frac {\sum w_{i}x_{i}y_{i}}{\sqrt {(\sum w_{i}x_{i}^{2})(\sum w_{i}y_{i}^{2})}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msqrt> <mo stretchy="false">(</mo> <mo>∑</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>∑</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rr_{xy,w}={\frac {\sum w_{i}x_{i}y_{i}}{\sqrt {(\sum w_{i}x_{i}^{2})(\sum w_{i}y_{i}^{2})}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3473f0dd4cd5dd5adcfd0399d4dadb08a9caec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:31.453ex; height:8.509ex;" alt="{\displaystyle rr_{xy,w}={\frac {\sum w_{i}x_{i}y_{i}}{\sqrt {(\sum w_{i}x_{i}^{2})(\sum w_{i}y_{i}^{2})}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Scaled_correlation_coefficient">Scaled correlation coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=29" title="Edit section: Scaled correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Scaled_correlation" title="Scaled correlation">Scaled correlation</a></div> <p>Scaled correlation is a variant of Pearson's correlation in which the range of the data is restricted intentionally and in a controlled manner to reveal correlations between fast components in <a href="/wiki/Time_series" title="Time series">time series</a>.<sup id="cite_ref-Nikolicetal_44-0" class="reference"><a href="#cite_note-Nikolicetal-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> Scaled correlation is defined as average correlation across short segments of data. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> be the number of segments that can fit into the total length of the signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> for a given scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\operatorname {round} \left({\frac {T}{s}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>round</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mi>s</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\operatorname {round} \left({\frac {T}{s}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a18a5b1b26d158dc37bd2b34fe0921ee1c73dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.656ex; height:6.176ex;" alt="{\displaystyle K=\operatorname {round} \left({\frac {T}{s}}\right).}"></span></dd></dl> <p>The scaled correlation across the entire signals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {r}}_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {r}}_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fb173b0ab9244d34f1fc5ceb57f7cb091bc5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.295ex; height:2.343ex;" alt="{\displaystyle {\bar {r}}_{s}}"></span> is then computed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {r}}_{s}={\frac {1}{K}}\sum \limits _{k=1}^{K}r_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>K</mi> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {r}}_{s}={\frac {1}{K}}\sum \limits _{k=1}^{K}r_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c9d6d707342d0e2f0ce3b221d114124ac6ef20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.209ex; height:7.343ex;" alt="{\displaystyle {\bar {r}}_{s}={\frac {1}{K}}\sum \limits _{k=1}^{K}r_{k},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b28e0e640d099f3676330bd4f604ae15c37bb4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.137ex; height:2.009ex;" alt="{\displaystyle r_{k}}"></span> is Pearson's coefficient of correlation for segment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. </p><p>By choosing the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, the range of values is reduced and the correlations on long time scale are filtered out, only the correlations on short time scales being revealed. Thus, the contributions of slow components are removed and those of fast components are retained. </p> <div class="mw-heading mw-heading3"><h3 id="Pearson's_distance"><span id="Pearson.27s_distance"></span>Pearson's distance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=30" title="Edit section: Pearson's distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A distance metric for two variables <i>X</i> and <i>Y</i> known as <i>Pearson's distance</i> can be defined from their correlation coefficient as<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{X,Y}=1-\rho _{X,Y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{X,Y}=1-\rho _{X,Y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68aeaeed324bc9a50c6e3215f77ef7986038a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.846ex; height:2.843ex;" alt="{\displaystyle d_{X,Y}=1-\rho _{X,Y}.}"></span></dd></dl> <p>Considering that the Pearson correlation coefficient falls between [−1, +1], the Pearson distance lies in [0, 2]. The Pearson distance has been used in <a href="/wiki/Cluster_analysis" title="Cluster analysis">cluster analysis</a> and data detection for communications and storage with unknown gain and offset.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p>The Pearson "distance" defined this way assigns distance greater than 1 to negative correlations. In reality, both strong positive correlation and negative correlations are meaningful, so care must be taken when Pearson "distance" is used for nearest neighbor algorithm as such algorithm will only include neighbors with positive correlation and exclude neighbors with negative correlation. Alternatively, an absolute valued distance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{X,Y}=1-|\rho _{X,Y}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{X,Y}=1-|\rho _{X,Y}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b598720b176798a135be16f4beb058a133454da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.493ex; height:3.009ex;" alt="{\displaystyle d_{X,Y}=1-|\rho _{X,Y}|}"></span>, can be applied, which will take both positive and negative correlations into consideration. The information on positive and negative association can be extracted separately, later. </p> <div class="mw-heading mw-heading3"><h3 id="Circular_correlation_coefficient">Circular correlation coefficient<span class="anchor" id="Circular"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=31" title="Edit section: Circular correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Circular_statistics" class="mw-redirect" title="Circular statistics">Circular statistics</a></div> <p>For variables <i>X</i> = {<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>} and <i>Y</i> = {<i>y</i><sub>1</sub>,...,<i>y</i><sub><i>n</i></sub>} that are defined on the unit circle <span class="texhtml">[0, 2π)</span>, it is possible to define a circular analog of Pearson's coefficient.<sup id="cite_ref-SRJ_47-0" class="reference"><a href="#cite_note-SRJ-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> This is done by transforming data points in <i>X</i> and <i>Y</i> with a <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> function such that the correlation coefficient is given as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{circular}}={\frac {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})\sin(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}\sin(y_{i}-{\bar {y}})^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>circular</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{circular}}={\frac {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})\sin(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}\sin(y_{i}-{\bar {y}})^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9ca950ffae319129c11e0da879728b4143e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:50.564ex; height:8.676ex;" alt="{\displaystyle r_{\text{circular}}={\frac {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})\sin(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}\sin(y_{i}-{\bar {y}})^{2}}}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span> are the <a href="/wiki/Mean_of_circular_quantities" class="mw-redirect" title="Mean of circular quantities">circular means</a> of <i>X</i> and <i>Y</i>. This measure can be useful in fields like meteorology where the angular direction of data is important. </p> <div class="mw-heading mw-heading3"><h3 id="Partial_correlation">Partial correlation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=32" title="Edit section: Partial correlation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></div> <p>If a population or data-set is characterized by more than two variables, a <a href="/wiki/Partial_correlation" title="Partial correlation">partial correlation</a> coefficient measures the strength of dependence between a pair of variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables. </p> <div class="mw-heading mw-heading3"><h3 id="Pearson_correlation_coefficient_in_quantum_systems">Pearson correlation coefficient in quantum systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=33" title="Edit section: Pearson correlation coefficient in quantum systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For two observables, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, in a bipartite quantum system Pearson correlation coefficient is defined as <sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Cor} (X,Y)={\frac {\mathbb {E} [X\otimes Y]-\mathbb {E} [X]\cdot \mathbb {E} [Y]}{\sqrt {\mathbb {V} [X]\cdot \mathbb {V} [Y]}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> <mi mathvariant="double-struck">o</mi> <mi mathvariant="double-struck">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>⊗</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Cor} (X,Y)={\frac {\mathbb {E} [X\otimes Y]-\mathbb {E} [X]\cdot \mathbb {E} [Y]}{\sqrt {\mathbb {V} [X]\cdot \mathbb {V} [Y]}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e11a93834596dc5412bb175d74f57a217d5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.716ex; height:7.009ex;" alt="{\displaystyle \mathbb {Cor} (X,Y)={\frac {\mathbb {E} [X\otimes Y]-\mathbb {E} [X]\cdot \mathbb {E} [Y]}{\sqrt {\mathbb {V} [X]\cdot \mathbb {V} [Y]}}}\,,}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09de7acbba84104ff260708b6e9b8bae32c3fafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.824ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X]}"></span> is the expectation value of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74f6124a0e4a70bc6fda6c2b0f7c43d13ee0e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.617ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [Y]}"></span> is the expectation value of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [X\otimes Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>⊗</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X\otimes Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0153ecd83015d566f496dbbeb8e25d9641193ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.438ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X\otimes Y]}"></span> is the expectation value of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\otimes Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊗</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\otimes Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73cb0079947d63aa9b62873f3b215171615e435c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="{\displaystyle X\otimes Y}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/025a7353276eda435ee8a3d04758a41b363061c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {V} [X]}"></span> is the variance of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} [Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} [Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8414159ddf57b8d840ce8288bddb634ecf563f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle \mathbb {V} [Y]}"></span> is the variance of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>.</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Cor} (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> <mi mathvariant="double-struck">o</mi> <mi mathvariant="double-struck">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Cor} (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/002ba30c258619c4f214f841f97fa647e40ac8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.349ex; height:2.843ex;" alt="{\displaystyle \mathbb {Cor} (X,Y)}"></span> is symmetric, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Cor} (X,Y)=\mathbb {Cor} (Y,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> <mi mathvariant="double-struck">o</mi> <mi mathvariant="double-struck">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> <mi mathvariant="double-struck">o</mi> <mi mathvariant="double-struck">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Cor} (X,Y)=\mathbb {Cor} (Y,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b7ade648b27848de5b440e02b2994f2cdf3c0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.796ex; height:2.843ex;" alt="{\displaystyle \mathbb {Cor} (X,Y)=\mathbb {Cor} (Y,X)}"></span>, and its absolute value is invariant under affine transformations. </p> <div class="mw-heading mw-heading2"><h2 id="Decorrelation_of_n_random_variables">Decorrelation of <i>n</i> random variables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=34" title="Edit section: Decorrelation of n random variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Decorrelation" title="Decorrelation">Decorrelation</a></div> <p>It is always possible to remove the correlations between all pairs of an arbitrary number of random variables by using a data transformation, even if the relationship between the variables is nonlinear. A presentation of this result for population distributions is given by Cox & Hinkley.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>A corresponding result exists for reducing the sample correlations to zero. Suppose a vector of <i>n</i> random variables is observed <i>m</i> times. Let <i>X</i> be a matrix where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b9e6670d691d42489f7be6e034cbcca7a92f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.859ex; height:2.843ex;" alt="{\displaystyle X_{i,j}}"></span> is the <i>j</i>th variable of observation <i>i</i>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{m,m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{m,m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40ff160c7b41d1bcc4a806d848128347d3c38f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.163ex; height:2.843ex;" alt="{\displaystyle Z_{m,m}}"></span> be an <i>m</i> by <i>m</i> square matrix with every element 1. Then <i>D</i> is the data transformed so every random variable has zero mean, and <i>T</i> is the data transformed so all variables have zero mean and zero correlation with all other variables – the sample <a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">correlation matrix</a> of <i>T</i> will be the identity matrix. This has to be further divided by the standard deviation to get unit variance. The transformed variables will be uncorrelated, even though they may not be <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=X-{\frac {1}{m}}Z_{m,m}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=X-{\frac {1}{m}}Z_{m,m}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03456ef75e5fee295db26dc7ed15ec1d6b102866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.862ex; height:5.176ex;" alt="{\displaystyle D=X-{\frac {1}{m}}Z_{m,m}X}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=D(D^{\mathsf {T}}D)^{-{\frac {1}{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=D(D^{\mathsf {T}}D)^{-{\frac {1}{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872f960553d8d1f9a2019224522c25f232f6f986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.329ex; height:4.009ex;" alt="{\displaystyle T=D(D^{\mathsf {T}}D)^{-{\frac {1}{2}}},}"></span></dd></dl> <p>where an exponent of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac">−<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span> represents the <a href="/wiki/Matrix_square_root" class="mw-redirect" title="Matrix square root">matrix square root</a> of the <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of a matrix. The correlation matrix of <i>T</i> will be the identity matrix. If a new data observation <i>x</i> is a row vector of <i>n</i> elements, then the same transform can be applied to <i>x</i> to get the transformed vectors <i>d</i> and <i>t</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=x-{\frac {1}{m}}Z_{1,m}X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=x-{\frac {1}{m}}Z_{1,m}X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec18c03e3baee8fe5087f73be402f05b3b8db048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.53ex; height:5.176ex;" alt="{\displaystyle d=x-{\frac {1}{m}}Z_{1,m}X,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=d(D^{\mathsf {T}}D)^{-{\frac {1}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=d(D^{\mathsf {T}}D)^{-{\frac {1}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba2f18311264d9b6e7c8dcdac92ff6e0097b43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.824ex; height:4.009ex;" alt="{\displaystyle t=d(D^{\mathsf {T}}D)^{-{\frac {1}{2}}}.}"></span></dd></dl> <p>This decorrelation is related to <a href="/wiki/Principal_components_analysis" class="mw-redirect" title="Principal components analysis">principal components analysis</a> for multivariate data. </p> <div class="mw-heading mw-heading2"><h2 id="Software_implementations">Software implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=35" title="Edit section: Software implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>'s statistics base-package implements the correlation coefficient with <code>cor(x, y)</code>, or (with the P value also) with <a rel="nofollow" class="external text" href="http://stat.ethz.ch/R-manual/R-patched/library/stats/html/cor.test.html"><code>cor.test(x, y)</code></a>.</li> <li>The <a href="/wiki/SciPy" title="SciPy">SciPy</a> <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> library via <a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html"><code>pearsonr(x, y)</code></a>.</li> <li>The <a href="/wiki/Pandas_(software)" title="Pandas (software)">Pandas</a> Python library implements Pearson correlation coefficient calculation as the default option for the method <a rel="nofollow" class="external text" href="https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corr.html"><code>pandas.DataFrame.corr</code></a></li> <li><a href="/wiki/Wolfram_Mathematica" title="Wolfram Mathematica">Wolfram Mathematica</a> via the <a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/Correlation.html"><code>Correlation</code></a> function, or (with the P value) with <a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/CorrelationTest.html"><code>CorrelationTest</code></a>.</li> <li>The <a href="/wiki/Boost_(C%2B%2B_libraries)" title="Boost (C++ libraries)">Boost</a> <a href="/wiki/C%2B%2B" title="C++">C++</a> library via the <a rel="nofollow" class="external text" href="https://www.boost.org/doc/libs/1_76_0/libs/math/doc/html/math_toolkit/bivariate_statistics.html"><code>correlation_coefficient</code></a> function.</li> <li><a href="/wiki/Excel" class="mw-redirect" title="Excel">Excel</a> has an in-built <a rel="nofollow" class="external text" href="https://support.microsoft.com/en-us/office/correl-function-995dcef7-0c0a-4bed-a3fb-239d7b68ca92"><code>correl(array1, array2)</code></a> function for calculating the pearson's correlation coefficient.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=36" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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class="mw-redirect" title="Yule's Y">Yule's Y</a></li></ul></li> <li><a href="/wiki/Coefficient_of_multiple_correlation" title="Coefficient of multiple correlation">Coefficient of multiple correlation</a></li> <li><a href="/wiki/Concordance_correlation_coefficient" title="Concordance correlation coefficient">Concordance correlation coefficient</a></li> <li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation and dependence</a></li> <li><a href="/wiki/Correlation_ratio" title="Correlation ratio">Correlation ratio</a></li> <li><a href="/wiki/Disattenuation" class="mw-redirect" title="Disattenuation">Disattenuation</a></li> <li><a href="/wiki/Distance_correlation" title="Distance correlation">Distance correlation</a></li> <li><a href="/wiki/Maximal_information_coefficient" title="Maximal information coefficient">Maximal information coefficient</a></li> <li><a href="/wiki/Multiple_correlation" class="mw-redirect" title="Multiple correlation">Multiple correlation</a></li> <li><a href="/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent" class="mw-redirect" title="Normally distributed and uncorrelated does not imply independent">Normally distributed and uncorrelated does not imply independent</a></li> <li><a href="/wiki/Odds_ratio" title="Odds ratio">Odds ratio</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Polychoric_correlation" title="Polychoric correlation">Polychoric correlation</a></li> <li><a href="/wiki/Quadrant_count_ratio" title="Quadrant count ratio">Quadrant count ratio</a></li> <li><a href="/wiki/RV_coefficient" title="RV coefficient">RV coefficient</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's rank correlation coefficient</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=37" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-lower-alpha"> <ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Also known as <b>Pearson's <i>r</i></b>, the <b>Pearson product-moment correlation coefficient</b> (<b>PPMCC</b>), the <b>bivariate correlation</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or simply the unqualified <b>correlation coefficient</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">As early as 1877, Galton was using the term "reversion" and the symbol "<i>r</i>" for what would become "regression".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=38" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://libguides.library.kent.edu/SPSS/PearsonCorr">"SPSS Tutorials: Pearson Correlation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=SPSS+Tutorials%3A+Pearson+Correlation&rft_id=http%3A%2F%2Flibguides.library.kent.edu%2FSPSS%2FPearsonCorr&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.statisticshowto.com/probability-and-statistics/correlation-coefficient-formula/">"Correlation Coefficient: Simple Definition, Formula, Easy Steps"</a>. <i>Statistics How To</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Statistics+How+To&rft.atitle=Correlation+Coefficient%3A+Simple+Definition%2C+Formula%2C+Easy+Steps&rft_id=https%3A%2F%2Fwww.statisticshowto.com%2Fprobability-and-statistics%2Fcorrelation-coefficient-formula%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalton1877" class="citation journal cs1">Galton, F. (5–19 April 1877). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eskKAAAAYAAJ&pg=PA512">"Typical laws of heredity"</a>. <i>Nature</i>. <b>15</b> (388, 389, 390): 492–495, 512–514, 532–533. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1877Natur..15..492.">1877Natur..15..492.</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F015492a0">10.1038/015492a0</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4136393">4136393</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Typical+laws+of+heredity&rft.volume=15&rft.issue=388%2C+389%2C+390&rft.pages=492-495%2C+512-514%2C+532-533&rft.date=1877-04-05%2F1877-04-19&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4136393%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F015492a0&rft_id=info%3Abibcode%2F1877Natur..15..492.&rft.aulast=Galton&rft.aufirst=F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeskKAAAAYAAJ%26pg%3DPA512&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span> In the "Appendix" on page 532, Galton uses the term "reversion" and the symbol <i>r</i>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalton1885" class="citation journal cs1">Galton, F. (24 September 1885). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lN3RjXLUuWsC&pg=PA499">"The British Association: Section II, Anthropology: Opening address by Francis Galton, F.R.S., etc., President of the Anthropological Institute, President of the Section"</a>. <i>Nature</i>. <b>32</b> (830): 507–510.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=The+British+Association%3A+Section+II%2C+Anthropology%3A+Opening+address+by+Francis+Galton%2C+F.R.S.%2C+etc.%2C+President+of+the+Anthropological+Institute%2C+President+of+the+Section&rft.volume=32&rft.issue=830&rft.pages=507-510&rft.date=1885-09-24&rft.aulast=Galton&rft.aufirst=F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlN3RjXLUuWsC%26pg%3DPA499&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalton1886" class="citation journal cs1">Galton, F. (1886). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JPcRAAAAYAAJ&pg=PA246">"Regression towards mediocrity in hereditary stature"</a>. <i>Journal of the Anthropological Institute of Great Britain and Ireland</i>. <b>15</b>: 246–263. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2841583">10.2307/2841583</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2841583">2841583</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Anthropological+Institute+of+Great+Britain+and+Ireland&rft.atitle=Regression+towards+mediocrity+in+hereditary+stature&rft.volume=15&rft.pages=246-263&rft.date=1886&rft_id=info%3Adoi%2F10.2307%2F2841583&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2841583%23id-name%3DJSTOR&rft.aulast=Galton&rft.aufirst=F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJPcRAAAAYAAJ%26pg%3DPA246&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPearson1895" class="citation journal cs1">Pearson, Karl (20 June 1895). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=60aL0zlT-90C&pg=PA240">"Notes on regression and inheritance in the case of two parents"</a>. <i>Proceedings of the Royal Society of London</i>. <b>58</b>: 240–242. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1895RSPS...58..240P">1895RSPS...58..240P</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=Notes+on+regression+and+inheritance+in+the+case+of+two+parents&rft.volume=58&rft.pages=240-242&rft.date=1895-06-20&rft_id=info%3Abibcode%2F1895RSPS...58..240P&rft.aulast=Pearson&rft.aufirst=Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D60aL0zlT-90C%26pg%3DPA240&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler1989" class="citation journal cs1">Stigler, Stephen M. 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(1 July 1989). <a rel="nofollow" class="external text" href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.913">"Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification"</a>. <i>Physical Review A</i>. <b>40</b> (2): 913–923. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.40.913">10.1103/PhysRevA.40.913</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Demonstration+of+the+Einstein-Podolsky-Rosen+paradox+using+nondegenerate+parametric+amplification&rft.volume=40&rft.issue=2&rft.pages=913-923&rft.date=1989-07-01&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.40.913&rft.aulast=Reid&rft.aufirst=M.+D.&rft_id=https%3A%2F%2Fjournals.aps.org%2Fpra%2Fabstract%2F10.1103%2FPhysRevA.40.913&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaccone,_L.Dagmar,_B.Macchiavello,_C.2015" class="citation journal cs1">Maccone, L.; Dagmar, B.; Macchiavello, C. 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(1974). <i>Theoretical Statistics</i>. Chapman & Hall. Appendix 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-12420-3" title="Special:BookSources/0-412-12420-3"><bdi>0-412-12420-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Statistics&rft.pages=Appendix+3&rft.pub=Chapman+%26+Hall&rft.date=1974&rft.isbn=0-412-12420-3&rft.au=Cox%2C+D.R.&rft.au=Hinkley%2C+D.V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pearson_correlation_coefficient&action=edit&section=39" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Linear_correlation" class="extiw" title="v:Linear correlation">Linear correlation</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://comparingcorrelations.org">"cocor"</a>. <i>comparingcorrelations.org</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=comparingcorrelations.org&rft.atitle=cocor&rft_id=http%3A%2F%2Fcomparingcorrelations.org&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span> – A free web interface and R package for the statistical comparison of two dependent or independent correlations with overlapping or non-overlapping variables.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nagysandor.eu/AsimovTeka/correlation_en/index.html">"Correlation"</a>. <i>nagysandor.eu</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nagysandor.eu&rft.atitle=Correlation&rft_id=http%3A%2F%2Fnagysandor.eu%2FAsimovTeka%2Fcorrelation_en%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span> – an interactive Flash simulation on the correlation of two normally distributed variables.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.hackmath.net/en/calculator/linear-regression">"Correlation coefficient calculator"</a>. <i>hackmath.net</i>. Linear regression.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=hackmath.net&rft.atitle=Correlation+coefficient+calculator&rft_id=http%3A%2F%2Fwww.hackmath.net%2Fen%2Fcalculator%2Flinear-regression&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://frank.mtsu.edu/~dkfuller/tables/correlationtable.pdf">"Critical values for Pearson's correlation coefficient"</a> <span class="cs1-format">(PDF)</span>. <i>frank.mtsu.edu/~dkfuller</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=frank.mtsu.edu%2F~dkfuller&rft.atitle=Critical+values+for+Pearson%27s+correlation+coefficient&rft_id=http%3A%2F%2Ffrank.mtsu.edu%2F~dkfuller%2Ftables%2Fcorrelationtable.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span> – large table.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://guessthecorrelation.com">"Guess the Correlation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Guess+the+Correlation&rft_id=http%3A%2F%2Fguessthecorrelation.com&rfr_id=info%3Asid%2Fen.wikipedia.org%3APearson+correlation+coefficient" class="Z3988"></span> – A game where players guess how correlated two variables in a scatter plot are, in order to gain a better understanding of the concept of correlation.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a class="mw-selflink selflink">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier 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title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a 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navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning_evaluation_metrics" title="Template:Machine learning evaluation metrics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning_evaluation_metrics" title="Template talk:Machine learning evaluation metrics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning_evaluation_metrics" title="Special:EditPage/Template:Machine learning evaluation metrics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Machine_learning_evaluation_metrics" style="font-size:114%;margin:0 4em"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> evaluation metrics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_squared_error" title="Mean squared error">MSE</a></li> <li><a href="/wiki/Mean_absolute_error" title="Mean absolute error">MAE</a></li> <li><a href="/wiki/Symmetric_mean_absolute_percentage_error" title="Symmetric mean absolute percentage error">sMAPE</a></li> <li><a href="/wiki/Mean_absolute_percentage_error" title="Mean absolute percentage error">MAPE</a></li> <li><a href="/wiki/Mean_absolute_scaled_error" title="Mean absolute scaled error">MASE</a></li> <li><a href="/wiki/Mean_squared_prediction_error" title="Mean squared prediction error">MSPE</a></li> <li><a href="/wiki/Root_mean_square" title="Root mean square">RMS</a></li> <li><a href="/wiki/Root-mean-square_deviation" class="mw-redirect" title="Root-mean-square deviation">RMSE/RMSD</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">R<sup>2</sup></a></li> <li><a href="/wiki/Mean_directional_accuracy" title="Mean directional accuracy">MDA</a></li> <li><a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">MAD</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/F-score" title="F-score">F-score</a></li> <li><a href="/wiki/P4-metric" title="P4-metric">P4</a></li> <li><a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">Accuracy</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Precision</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Recall</a></li> <li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Kappa</a></li> <li><a href="/wiki/Phi_coefficient" title="Phi coefficient">MCC</a></li> <li><a href="/wiki/Receiver_operating_characteristic#Area_under_the_curve" title="Receiver operating characteristic">AUC</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC</a></li> <li><a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">Sensitivity and specificity</a></li> <li><a href="/wiki/Cross-entropy#Cross-entropy_loss_function_and_logistic_regression" title="Cross-entropy">Logarithmic Loss</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Silhouette_(clustering)" title="Silhouette (clustering)">Silhouette</a></li> <li><a href="/wiki/Calinski-Harabasz_index" class="mw-redirect" title="Calinski-Harabasz index">Calinski-Harabasz index</a></li> <li><a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies-Bouldin</a></li> <li><a href="/wiki/Dunn_index" title="Dunn index">Dunn index</a></li> <li><a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a></li> <li><a href="/wiki/Jaccard_index" title="Jaccard index">Jaccard index</a></li> <li><a href="/wiki/Rand_index" title="Rand index">Rand index</a></li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a></li> <li><a href="/wiki/Simple_matching_coefficient" title="Simple matching coefficient">SMC</a></li> <li><a href="/wiki/SimHash" title="SimHash">SimHash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ranking_(information_retrieval)" title="Ranking (information retrieval)">Ranking</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_reciprocal_rank" title="Mean reciprocal rank">MRR</a></li> <li><a href="/wiki/NDCG" class="mw-redirect" title="NDCG">NDCG</a></li> <li><a href="/wiki/Average_precision" class="mw-redirect" title="Average precision">AP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_Vision" class="mw-redirect" title="Computer Vision">Computer Vision</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/PSNR" class="mw-redirect" title="PSNR">PSNR</a></li> <li><a href="/wiki/SSIM" class="mw-redirect" title="SSIM">SSIM</a></li> <li><a href="/wiki/Intersection_over_union" class="mw-redirect" title="Intersection over union">IoU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Natural_language_processing" title="Natural language processing">NLP</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perplexity" title="Perplexity">Perplexity</a></li> <li><a href="/wiki/BLEU" title="BLEU">BLEU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Deep Learning Related Metrics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Inception_score" title="Inception score">Inception score</a></li> <li><a href="/wiki/Fr%C3%A9chet_inception_distance" title="Fréchet inception distance">FID</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Recommender_system" title="Recommender system">Recommender system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coverage_probability" title="Coverage probability">Coverage</a></li> <li><a href="/w/index.php?title=Intra-list_Similarity&action=edit&redlink=1" class="new" title="Intra-list Similarity (page does not exist)">Intra-list Similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Similarity</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cosine_similarity" title="Cosine similarity">Cosine similarity</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a class="mw-selflink selflink">Pearson correlation coefficient</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" 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Pearson correlation coefficient - Wikipedia