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class="vector-toc-numb">2</span> <span>Definition and notation</span> </div> </a> <ul id="toc-Definition_and_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Uses"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Uses</span> </div> </a> <ul id="toc-Uses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_distributions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Probability_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Probability distributions</span> </div> </a> <button aria-controls="toc-Probability_distributions-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 Probability distributions subsection</span> </button> <ul id="toc-Probability_distributions-sublist" class="vector-toc-list"> <li id="toc-Medians_of_particular_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medians_of_particular_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Medians of particular distributions</span> </div> </a> <ul id="toc-Medians_of_particular_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Optimality_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimality_property"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Optimality property</span> </div> </a> <ul id="toc-Optimality_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inequality_relating_means_and_medians" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inequality_relating_means_and_medians"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Inequality relating means and medians</span> </div> </a> <ul id="toc-Inequality_relating_means_and_medians-sublist" class="vector-toc-list"> <li id="toc-Unimodal_distributions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Unimodal_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.1</span> <span>Unimodal distributions</span> </div> </a> <ul id="toc-Unimodal_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mean,_median,_and_skew" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mean,_median,_and_skew"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Mean, median, and skew</span> </div> </a> <ul id="toc-Mean,_median,_and_skew-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Jensen's_inequality_for_medians" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Jensen's_inequality_for_medians"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Jensen's inequality for medians</span> </div> </a> <ul id="toc-Jensen's_inequality_for_medians-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medians_for_samples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Medians_for_samples"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Medians for samples</span> </div> </a> <button aria-controls="toc-Medians_for_samples-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 Medians for samples subsection</span> </button> <ul id="toc-Medians_for_samples-sublist" class="vector-toc-list"> <li id="toc-Efficient_computation_of_the_sample_median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Efficient_computation_of_the_sample_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Efficient computation of the sample median</span> </div> </a> <ul id="toc-Efficient_computation_of_the_sample_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sampling_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sampling_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Sampling distribution</span> </div> </a> <ul id="toc-Sampling_distribution-sublist" class="vector-toc-list"> <li id="toc-Derivation_of_the_asymptotic_distribution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derivation_of_the_asymptotic_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.1</span> <span>Derivation of the asymptotic distribution</span> </div> </a> <ul id="toc-Derivation_of_the_asymptotic_distribution-sublist" class="vector-toc-list"> <li id="toc-Empirical_local_density" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Empirical_local_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.1.1</span> <span>Empirical local density</span> </div> </a> <ul id="toc-Empirical_local_density-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Estimation_of_variance_from_sample_data" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation_of_variance_from_sample_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Estimation of variance from sample data</span> </div> </a> <ul id="toc-Estimation_of_variance_from_sample_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Efficiency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Efficiency</span> </div> </a> <ul id="toc-Efficiency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_estimators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Other estimators</span> </div> </a> <ul id="toc-Other_estimators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Multivariate_median" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Multivariate_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Multivariate median</span> </div> </a> <button aria-controls="toc-Multivariate_median-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 Multivariate median subsection</span> </button> <ul id="toc-Multivariate_median-sublist" class="vector-toc-list"> <li id="toc-Marginal_median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Marginal_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Marginal median</span> </div> </a> <ul id="toc-Marginal_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Geometric median</span> </div> </a> <ul id="toc-Geometric_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Median_in_all_directions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Median_in_all_directions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Median in all directions</span> </div> </a> <ul id="toc-Median_in_all_directions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Centerpoint" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Centerpoint"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Centerpoint</span> </div> </a> <ul id="toc-Centerpoint-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conditional_median" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conditional_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Conditional median</span> </div> </a> <ul id="toc-Conditional_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_median-related_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_median-related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Other median-related concepts</span> </div> </a> <button aria-controls="toc-Other_median-related_concepts-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 Other median-related concepts subsection</span> </button> <ul id="toc-Other_median-related_concepts-sublist" class="vector-toc-list"> <li id="toc-Interpolated_median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpolated_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Interpolated median</span> </div> </a> <ul id="toc-Interpolated_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudo-median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudo-median"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Pseudo-median</span> </div> </a> <ul id="toc-Pseudo-median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variants_of_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variants_of_regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Variants of regression</span> </div> </a> <ul id="toc-Variants_of_regression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Median_filter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Median_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.4</span> <span>Median filter</span> </div> </a> <ul id="toc-Median_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cluster_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cluster_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.5</span> <span>Cluster analysis</span> </div> </a> <ul id="toc-Cluster_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Median–median_line" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Median–median_line"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.6</span> <span>Median–median line</span> </div> </a> <ul id="toc-Median–median_line-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Median-unbiased_estimators" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Median-unbiased_estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Median-unbiased estimators</span> </div> </a> <ul id="toc-Median-unbiased_estimators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>History</span> </div> </a> <ul id="toc-History-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">13</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">15</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">16</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" title="Table of Contents" > <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">Median</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 to an article in another language. Available in 67 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-67" 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">67 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Mediaan" title="Mediaan – Afrikaans" lang="af" hreflang="af" data-title="Mediaan" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%88%D8%B3%D9%8A%D8%B7_(%D8%A5%D8%AD%D8%B5%D8%A7%D8%A1)" title="وسيط (إحصاء) – Arabic" lang="ar" hreflang="ar" data-title="وسيط (إحصاء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Mediana_(statistika)" title="Mediana (statistika) – Azerbaijani" lang="az" hreflang="az" data-title="Mediana (statistika)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A6%A7%E0%A7%8D%E0%A6%AF%E0%A6%95" title="মধ্যক – Bangla" lang="bn" hreflang="bn" data-title="মধ্যক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D1%8B%D1%8F%D0%BD%D0%B0_(%D1%81%D1%82%D0%B0%D1%82%D1%8B%D1%81%D1%82%D1%8B%D0%BA%D0%B0)" title="Медыяна (статыстыка) – Belarusian" lang="be" hreflang="be" data-title="Медыяна (статыстыка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D0%B0%D0%BD%D0%B0_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Медиана (статистика) – Bulgarian" lang="bg" hreflang="bg" data-title="Медиана (статистика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Medijana_(statistika)" title="Medijana (statistika) – Bosnian" lang="bs" hreflang="bs" data-title="Medijana (statistika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mediana" title="Mediana – Catalan" lang="ca" hreflang="ca" data-title="Mediana" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Medi%C3%A1n" title="Medián – Czech" lang="cs" hreflang="cs" data-title="Medián" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Chipakati" title="Chipakati – Shona" lang="sn" hreflang="sn" data-title="Chipakati" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Canolrif" title="Canolrif – Welsh" lang="cy" hreflang="cy" data-title="Canolrif" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Median" title="Median – Danish" lang="da" hreflang="da" data-title="Median" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Median" title="Median – German" lang="de" hreflang="de" data-title="Median" 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/Mediaan" title="Mediaan – Estonian" lang="et" hreflang="et" data-title="Mediaan" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CE%BC%CE%B5%CF%83%CE%BF%CF%82" title="Διάμεσος – Greek" lang="el" hreflang="el" data-title="Διάμεσος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mediana_(estad%C3%ADstica)" title="Mediana (estadística) – Spanish" lang="es" hreflang="es" data-title="Mediana (estadística)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Mediano_(statistiko)" title="Mediano (statistiko) – Esperanto" lang="eo" hreflang="eo" data-title="Mediano (statistiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Mediana" title="Mediana – Basque" lang="eu" hreflang="eu" data-title="Mediana" 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/%D9%85%DB%8C%D8%A7%D9%86%D9%87_(%D8%A2%D9%85%D8%A7%D8%B1)" title="میانه (آمار) – Persian" lang="fa" hreflang="fa" data-title="میانه (آمار)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9diane_(statistiques)" title="Médiane (statistiques) – French" lang="fr" hreflang="fr" data-title="Médiane (statistiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mediana_(c%C3%A1lculo)" title="Mediana (cálculo) – Galician" lang="gl" hreflang="gl" data-title="Mediana (cálculo)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A4%91%EC%95%99%EA%B0%92" title="중앙값 – Korean" lang="ko" hreflang="ko" data-title="중앙값" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%BB%D5%B6%D5%A1%D6%80%D5%AA%D5%A5%D6%84" title="Միջնարժեք – Armenian" lang="hy" hreflang="hy" data-title="Միջնարժեք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A4%BE" title="माध्यिका – Hindi" lang="hi" hreflang="hi" data-title="माध्यिका" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Medijan" title="Medijan – Croatian" lang="hr" hreflang="hr" data-title="Medijan" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Mediano" title="Mediano – Ido" lang="io" hreflang="io" data-title="Mediano" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Median" title="Median – Indonesian" lang="id" hreflang="id" data-title="Median" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Mi%C3%B0gildi" title="Miðgildi – Icelandic" lang="is" hreflang="is" data-title="Miðgildi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Mediana_(statistica)" title="Mediana (statistica) – Italian" lang="it" hreflang="it" data-title="Mediana (statistica)" 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%97%D7%A6%D7%99%D7%95%D7%9F" title="חציון – Hebrew" lang="he" hreflang="he" data-title="חציון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Kati_(Takwimu)" title="Kati (Takwimu) – Swahili" lang="sw" hreflang="sw" data-title="Kati (Takwimu)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Medi%C4%81na_(statistika)" title="Mediāna (statistika) – Latvian" lang="lv" hreflang="lv" data-title="Mediāna (statistika)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mediana" title="Mediana – Lithuanian" lang="lt" hreflang="lt" data-title="Mediana" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Medi%C3%A1n" title="Medián – Hungarian" lang="hu" hreflang="hu" data-title="Medián" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D1%98%D0%B0%D0%BD%D0%B0" title="Медијана – Macedonian" lang="mk" hreflang="mk" data-title="Медијана" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Median" title="Median – Malay" lang="ms" hreflang="ms" data-title="Median" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Mediaan_(statistiek)" title="Mediaan (statistiek) – Dutch" lang="nl" hreflang="nl" data-title="Mediaan (statistiek)" 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/%E4%B8%AD%E5%A4%AE%E5%80%A4" title="中央値 – Japanese" lang="ja" hreflang="ja" data-title="中央値" 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/Median" title="Median – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Median" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Median" title="Median – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Median" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mediana_(statistika)" title="Mediana (statistika) – Uzbek" lang="uz" hreflang="uz" data-title="Mediana (statistika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mediana" title="Mediana – Polish" lang="pl" hreflang="pl" data-title="Mediana" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://pt.wikipedia.org/wiki/Mediana_(estat%C3%ADstica)" title="Mediana (estatística) – Portuguese" lang="pt" hreflang="pt" data-title="Mediana (estatística)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Median%C4%83_(statistic%C4%83)" title="Mediană (statistică) – Romanian" lang="ro" hreflang="ro" data-title="Mediană (statistică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D0%B0%D0%BD%D0%B0_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Медиана (статистика) – Russian" lang="ru" hreflang="ru" data-title="Медиана (статистика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mediana_(statistik%C3%AB)" title="Mediana (statistikë) – Albanian" lang="sq" hreflang="sq" data-title="Mediana (statistikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Mediana" title="Mediana – Sicilian" lang="scn" hreflang="scn" data-title="Mediana" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B8%E0%B6%B0%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%83%E0%B7%8A%E0%B6%AE%E0%B6%BA" title="මධ්‍යස්ථය – Sinhala" lang="si" hreflang="si" data-title="මධ්‍යස්ථය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Median" title="Median – Simple English" lang="en-simple" hreflang="en-simple" data-title="Median" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%88%DA%86%D9%88%D9%86_%D8%B9%D8%AF%D8%AF" title="وچون عدد – Sindhi" lang="sd" hreflang="sd" data-title="وچون عدد" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Medi%C3%A1n" title="Medián – Slovak" lang="sk" hreflang="sk" data-title="Medián" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mediana" title="Mediana – Slovenian" lang="sl" hreflang="sl" data-title="Mediana" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%D8%A7%D9%88%DB%95%DA%95%D8%A7%D8%B3%D8%AA%DB%95_(%D8%A6%D8%A7%D9%85%D8%A7%D8%B1)" title="ناوەڕاستە (ئامار) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ناوەڕاستە (ئامار)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D1%98%D0%B0%D0%BD%D0%B0_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Медијана (статистика) – Serbian" lang="sr" hreflang="sr" data-title="Медијана (статистика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Medijan" title="Medijan – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Medijan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Median" title="Median – Sundanese" lang="su" hreflang="su" data-title="Median" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Mediaani" title="Mediaani – Finnish" lang="fi" hreflang="fi" data-title="Mediaani" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Median" title="Median – Swedish" lang="sv" hreflang="sv" data-title="Median" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%9F%E0%AF%88%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88%E0%AE%AF%E0%AE%B3%E0%AE%B5%E0%AF%81" title="இடைநிலையளவு – Tamil" lang="ta" hreflang="ta" data-title="இடைநிலையளவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A1%E0%B8%B1%E0%B8%98%E0%B8%A2%E0%B8%90%E0%B8%B2%E0%B8%99" title="มัธยฐาน – Thai" lang="th" hreflang="th" data-title="มัธยฐาน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D0%B0%D0%BD%D0%B0" title="Медиана – Tajik" lang="tg" hreflang="tg" data-title="Медиана" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Medyan" title="Medyan – Turkish" lang="tr" hreflang="tr" data-title="Medyan" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D1%96%D0%B0%D0%BD%D0%B0_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Медіана (статистика) – Ukrainian" lang="uk" hreflang="uk" data-title="Медіана (статистика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_trung_v%E1%BB%8B" title="Số trung vị – Vietnamese" lang="vi" hreflang="vi" data-title="Số trung vị" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%B8%AD%E4%BD%8D%E6%95%B0" title="中位数 – Wu" lang="wuu" hreflang="wuu" data-title="中位数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%AD%E4%BD%8D%E6%95%B8" title="中位數 – Cantonese" lang="yue" hreflang="yue" data-title="中位數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E4%BD%8D%E6%95%B8" title="中位數 – Chinese" lang="zh" hreflang="zh" data-title="中位數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q226995#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Median" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Median" rel="discussion" title="Discuss 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searchaux" style="display:none">Middle quantile of a data set or probability distribution</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">This article is about the statistical concept. For other uses, see <a href="/wiki/Median_(disambiguation)" class="mw-disambig" title="Median (disambiguation)">Median (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Finding_the_median.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Finding_the_median.png/220px-Finding_the_median.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Finding_the_median.png/330px-Finding_the_median.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Finding_the_median.png/440px-Finding_the_median.png 2x" data-file-width="11154" data-file-height="8194" /></a><figcaption>Calculating the median in data sets of odd (above) and even (below) observations</figcaption></figure> <p>The <b>median</b> of a set of numbers is the value separating the higher half from the lower half of a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">data sample</a>, a <a href="/wiki/Statistical_population" title="Statistical population">population</a>, or a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>. For a <a href="/wiki/Data_set" title="Data set">data set</a>, it may be thought of as the “middle" value. The basic feature of the median in describing data compared to the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean</a> (often simply described as the "average") is that it is not <a href="/wiki/Skewness" title="Skewness">skewed</a> by a small proportion of extremely large or small values, and therefore provides a better representation of the center. <a href="/wiki/Median_income" title="Median income">Median income</a>, for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Finite_set_of_numbers">Finite set of numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=1" title="Edit section: Finite set of numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. </p><p>If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order). For example, the following list of seven numbers, </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;">1, 3, 3, <b>6</b>, 7, 8, 9</div> <p>has the median of <i>6</i>, which is the fourth value. </p><p>If the data set has an even number of observations, there is no distinct middle value and the median is usually defined to be the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of the two middle values.<sup id="cite_ref-StatisticalMedian_1-0" class="reference"><a href="#cite_note-StatisticalMedian-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> For example, this data set of 8 numbers </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">1, 2, 3, <b>4, 5</b>, 6, 8, 9</div> <p>has a median value of <i>4.5</i>, that 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 (4+5)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4+5)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9b4524b7175527adfb4bcafff8213f101ccdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle (4+5)/2}" /></span>. (In more technical terms, this interprets the median as the fully <a href="/wiki/Trimmed_estimator" title="Trimmed estimator">trimmed</a> <a href="/wiki/Mid-range" title="Mid-range">mid-range</a>). </p><p>In general, with this convention, the median can be defined as follows: For a data set <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> 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> elements, ordered from smallest to greatest, </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">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 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 odd, <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 {med} (x)=x_{(n+1)/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>med</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {med} (x)=x_{(n+1)/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2527a9add1db94280f529b5abbab205cde103f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.07ex; height:3.176ex;" alt="{\displaystyle \operatorname {med} (x)=x_{(n+1)/2}}" /></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">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 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 even, <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 {med} (x)={\frac {x_{(n/2)}+x_{((n/2)+1)}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>med</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {med} (x)={\frac {x_{(n/2)}+x_{((n/2)+1)}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a068af66136ddeaaf19fe63c5a2684b3b9cb6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.498ex; height:5.509ex;" alt="{\displaystyle \operatorname {med} (x)={\frac {x_{(n/2)}+x_{((n/2)+1)}}{2}}}" /></span></div> <table class="wikitable"> <caption>Comparison of common <a href="/wiki/Average" title="Average">averages</a> of values [ 1, 2, 2, 3, 4, 7, 9 ] </caption> <tbody><tr> <th>Type </th> <th>Description </th> <th>Example </th> <th>Result </th></tr> <tr> <td align="center"><a href="/wiki/Mid-range" title="Mid-range">Midrange</a> </td> <td>Midway point between the minimum and the maximum of a data set </td> <td align="center"><b>1</b>, 2, 2, 3, 4, 7, <b>9</b> </td> <td align="center"><b>5</b> </td></tr> <tr> <td align="center"><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic mean</a> </td> <td>Sum of values of a data set divided by number of values: <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> </td> <td align="center"><span class="nowrap">(1 + 2 + 2 + 3 + 4 + 7 + 9) / 7</span> </td> <td align="center"><b>4</b> </td></tr> <tr> <td align="center">Median </td> <td>Middle value separating the greater and lesser halves of a data set </td> <td align="center">1, 2, 2, <b>3</b>, 4, 7, 9 </td> <td align="center"><b>3</b> </td></tr> <tr> <td align="center"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a> </td> <td>Most frequent value in a data set </td> <td align="center">1, <b>2</b>, <b>2</b>, 3, 4, 7, 9 </td> <td align="center"><b>2</b> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Definition_and_notation">Definition and notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=2" title="Edit section: Definition and notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, a median of a <a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">population</a> is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains more than half the population, then some of the population is exactly equal to the unique median. </p><p>The median is well-defined for any <a href="/wiki/Weak_ordering" title="Weak ordering">ordered</a> (one-dimensional) data and is independent of any <a href="/wiki/Distance_metric" class="mw-redirect" title="Distance metric">distance metric</a>. The median can thus be applied to school classes which are ranked but not numerical (e.g. working out a median grade when student test scores are graded from F to A), although the result might be halfway between classes if there is an even number of classes. (For odd number classes, one specific class is determined as the median.) </p><p>A <a href="/wiki/Geometric_median" title="Geometric median">geometric median</a>, on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the <a href="/wiki/Medoid" title="Medoid">medoid</a>. </p><p>There is no widely accepted standard notation for the median, but some authors represent the median of a variable <i>x</i> as med(<i>x</i>), <i>x͂</i>,<sup id="cite_ref-Bissell1994_3-0" class="reference"><a href="#cite_note-Bissell1994-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> as <i>μ</i><sub>1/2</sub>,<sup id="cite_ref-StatisticalMedian_1-1" class="reference"><a href="#cite_note-StatisticalMedian-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or as <i>M</i>.<sup id="cite_ref-Bissell1994_3-1" class="reference"><a href="#cite_note-Bissell1994-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sheskin2003_4-0" class="reference"><a href="#cite_note-Sheskin2003-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced. </p><p>The median is a special case of other <a href="/wiki/Location_parameter" title="Location parameter">ways of summarizing the typical values associated with a statistical distribution</a>: it is the 2nd <a href="/wiki/Quartile" title="Quartile">quartile</a>, 5th <a href="/wiki/Decile" title="Decile">decile</a>, and 50th <a href="/wiki/Percentile" title="Percentile">percentile</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Uses">Uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=3" title="Edit section: Uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The median can be used as a measure of <a href="/wiki/Location_parameter" title="Location parameter">location</a> when one attaches reduced importance to extreme values, typically because a distribution is <a href="/wiki/Skewness" title="Skewness">skewed</a>, extreme values are not known, or <a href="/wiki/Outlier" title="Outlier">outliers</a> are untrustworthy, i.e., may be measurement or transcription errors. </p><p>For example, consider the <a href="/wiki/Multiset" title="Multiset">multiset</a> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">1, 2, 2, 2, 3, 14.</div> <p>The median is 2 in this case, as is the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>, and it might be seen as a better indication of the <a href="/wiki/Central_tendency" title="Central tendency">center</a> than the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of 4, which is larger than all but one of the values. However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. At most, one can say that the two statistics cannot be "too far" apart; see <a href="#Inequality_relating_means_and_medians">§ Inequality relating means and medians</a> below.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>As a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.<sup id="cite_ref-Robson_6-0" class="reference"><a href="#cite_note-Robson-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Because the median is simple to understand and easy to calculate, while also a robust approximation to the <a href="/wiki/Mean" title="Mean">mean</a>, the median is a popular <a href="/wiki/Summary_statistic" class="mw-redirect" title="Summary statistic">summary statistic</a> in <a href="/wiki/Descriptive_statistics" title="Descriptive statistics">descriptive statistics</a>. In this context, there are several choices for a measure of <a href="/wiki/Variability_(statistics)" class="mw-redirect" title="Variability (statistics)">variability</a>: the <a href="/wiki/Range_(statistics)" title="Range (statistics)">range</a>, the <a href="/wiki/Interquartile_range" title="Interquartile range">interquartile range</a>, the <a href="/wiki/Mean_absolute_deviation" class="mw-redirect" title="Mean absolute deviation">mean absolute deviation</a>, and the <a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">median absolute deviation</a>. </p><p>For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficiency</a> of candidate estimators shows that the sample mean is more statistically efficient <a href="/wiki/If_and_only_if" title="If and only if">when—and only when—</a> data is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions.<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. (February 2020)">citation needed</span></a></i>]</sup> Even then, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean.<sup id="cite_ref-Williams_2001_165_7-0" class="reference"><a href="#cite_note-Williams_2001_165-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Probability_distributions">Probability distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=4" title="Edit section: Probability distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <a href="/wiki/Real_number" title="Real number">real</a>-valued <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> with <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> <i>F</i>, a median is defined as any real number <i>m</i> that satisfies the inequalities <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 \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>m</mi> <mo>−</mo> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90cd3c449b8b6a2fbd3b27d76f1da9cb509bc11b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.358ex; height:5.343ex;" alt="{\displaystyle \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)}" /></span> (cf. the <a href="/wiki/Expected_value#Uhl2023Bild1" title="Expected value">drawing</a> in the <a href="/wiki/Expected_value#Arbitrary_real-valued_random_variables" title="Expected value">definition of expected value for arbitrary real-valued random variables</a>). An equivalent phrasing uses a random variable <i>X</i> distributed according to <i>F</i>: <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 {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4c3d93b2afcd7c9632c607f41a22253798ecdf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.546ex; height:5.176ex;" alt="{\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.}" /></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visualisation_mode_median_mean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Visualisation_mode_median_mean.svg/250px-Visualisation_mode_median_mean.svg.png" decoding="async" width="170" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Visualisation_mode_median_mean.svg/330px-Visualisation_mode_median_mean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Visualisation_mode_median_mean.svg/500px-Visualisation_mode_median_mean.svg.png 2x" data-file-width="512" data-file-height="878" /></a><figcaption><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a>, median and mean (<a href="/wiki/Expected_value" title="Expected value">expected value</a>) of a probability density function<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Note that this definition does not require <i>X</i> to have an <a href="/wiki/Absolute_continuity" title="Absolute continuity">absolutely continuous distribution</a> (which has a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> <i>f</i>), nor does it require a <a href="/wiki/Discrete_distribution" class="mw-redirect" title="Discrete distribution">discrete one</a>. In the former case, the inequalities can be upgraded to equality: a median satisfies <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 {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e241edcc66ca0107b53d05d1c522ba98cb135cdb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.89ex; height:6.009ex;" alt="{\displaystyle \operatorname {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}}" /></span> and <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 {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46969770287ff3dfe3f71446a1d4e31377150f09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.817ex; height:5.843ex;" alt="{\displaystyle \operatorname {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.}" /></span> </p><p>Any <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> on the real number set <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 {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 \mathbb {R} }" /></span> has at least one median, but in pathological cases there may be more than one median: if <i>F</i> is constant 1/2 on an interval (so that <i>f</i> = 0 there), then any value of that interval is a median. </p> <div class="mw-heading mw-heading3"><h3 id="Medians_of_particular_distributions">Medians of particular distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=5" title="Edit section: Medians of particular distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking a well-defined mean, such as the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>: </p> <ul><li>The median of a symmetric <a href="/wiki/Unimodal_distribution" class="mw-redirect" title="Unimodal distribution">unimodal distribution</a> coincides with the mode.</li> <li>The median of a <a href="/wiki/Symmetric_distribution" class="mw-redirect" title="Symmetric distribution">symmetric distribution</a> which possesses a mean <i>μ</i> also takes the value <i>μ</i>. <ul><li>The median of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean <i>μ</i> and variance <i>σ</i><sup>2</sup> is μ. In fact, for a normal distribution, mean = median = mode.</li> <li>The median of a <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> in the interval [<i>a</i>, <i>b</i>] is (<i>a</i> + <i>b</i>) / 2, which is also the mean.</li></ul></li> <li>The median of a <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> with location parameter <i>x</i><sub>0</sub> and scale parameter <i>y</i> is <i>x</i><sub>0</sub>, the location parameter.</li> <li>The median of a <a href="/wiki/Power_law" title="Power law">power law distribution</a> <i>x</i><sup>−<i>a</i></sup>, with exponent <i>a</i> > 1 is 2<sup>1/(<i>a</i> − 1)</sup><i>x</i><sub>min</sub>, where <i>x</i><sub>min</sub> is the minimum value for which the power law holds<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>The median of an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate parameter</a> <i>λ</i> is the natural logarithm of 2 divided by the rate parameter: <i>λ</i><sup>−1</sup>ln 2.</li> <li>The median of a <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a> with shape parameter <i>k</i> and scale parameter <i>λ</i> is <i>λ</i>(ln 2)<sup>1/<i>k</i></sup>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Optimality_property">Optimality property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=7" title="Edit section: Optimality property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i><a href="/wiki/Mean_absolute_error" title="Mean absolute error">mean absolute error</a></i> of a real variable <i>c</i> with respect to the <a href="/wiki/Random_variable" title="Random variable">random variable</a> <i>X</i> is <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 {E} \left[\left|X-c\right|\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>|</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>c</mi> </mrow> <mo>|</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[\left|X-c\right|\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4072449930e8e16114f696667e9d513d9c5318d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.997ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} \left[\left|X-c\right|\right]}" /></span> Provided that the probability distribution of <i>X</i> is such that the above expectation exists, then <i>m</i> is a median of <i>X</i> if and only if <i>m</i> is a minimizer of the mean absolute error with respect to <i>X</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In particular, if <i>m</i> is a sample median, then it minimizes the arithmetic mean of the absolute deviations.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Note, however, that in cases where the sample contains an even number of elements, this minimizer is not unique. </p><p>More generally, a median is defined as a minimum of <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 {E} \left[\left|X-c\right|-\left|X\right|\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>|</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>c</mi> </mrow> <mo>|</mo> </mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>X</mi> <mo>|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[\left|X-c\right|-\left|X\right|\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef531900354aef96355c40ec0f33f1251215e086" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.758ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} \left[\left|X-c\right|-\left|X\right|\right],}" /></span> as discussed below in the section on <a href="/wiki/Multivariate_median" class="mw-redirect" title="Multivariate median">multivariate medians</a> (specifically, the <a href="/wiki/Spatial_median" class="mw-redirect" title="Spatial median">spatial median</a>). </p><p>This optimization-based definition of the median is useful in statistical data-analysis, for example, in <a href="/wiki/K-medians_clustering" title="K-medians clustering"><i>k</i>-medians clustering</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Inequality_relating_means_and_medians">Inequality relating means and medians</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=8" title="Edit section: Inequality relating means and medians"><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:Comparison_mean_median_mode.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/300px-Comparison_mean_median_mode.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/450px-Comparison_mean_median_mode.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/600px-Comparison_mean_median_mode.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Comparison of <a href="/wiki/Mean" title="Mean">mean</a>, median and <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> of two <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distributions</a> with different <a href="/wiki/Skewness" title="Skewness">skewness</a></figcaption></figure> <p>If the distribution has finite variance, then the distance between the median <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 {\tilde {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 {\tilde {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b6c2d2aa76b9cf010d897dc2ce988acf539624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.676ex;" alt="{\displaystyle {\tilde {X}}}" /></span> and the mean <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/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}" /></span> is bounded by one <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. </p><p>This bound was proved by Book and Sher in 1979 for discrete samples,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and more generally by Page and Murty in 1982.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In a comment on a subsequent proof by O'Cinneide,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Mallows in 1991 presented a compact proof that uses <a href="/wiki/Jensen%27s_inequality" title="Jensen's inequality">Jensen's inequality</a> twice,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> as follows. Using |·| for the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>, we have </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 {\begin{aligned}\left|\mu -m\right|=\left|\operatorname {E} (X-m)\right|&\leq \operatorname {E} \left(\left|X-m\right|\right)\\[2ex]&\leq \operatorname {E} \left(\left|X-\mu \right|\right)\\[1ex]&\leq {\sqrt {\operatorname {E} \left({\left(X-\mu \right)}^{2}\right)}}=\sigma .\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="1.16em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mi>μ</mi> <mo>−</mo> <mi>m</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>|</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>m</mi> </mrow> <mo>|</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≤</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>|</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mo>|</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mi>σ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|\mu -m\right|=\left|\operatorname {E} (X-m)\right|&\leq \operatorname {E} \left(\left|X-m\right|\right)\\[2ex]&\leq \operatorname {E} \left(\left|X-\mu \right|\right)\\[1ex]&\leq {\sqrt {\operatorname {E} \left({\left(X-\mu \right)}^{2}\right)}}=\sigma .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79b646a0aa8d3a45969aaf2b24996d23c47ff99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:46.914ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\left|\mu -m\right|=\left|\operatorname {E} (X-m)\right|&\leq \operatorname {E} \left(\left|X-m\right|\right)\\[2ex]&\leq \operatorname {E} \left(\left|X-\mu \right|\right)\\[1ex]&\leq {\sqrt {\operatorname {E} \left({\left(X-\mu \right)}^{2}\right)}}=\sigma .\end{aligned}}}" /></span> </p><p>The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes the <a href="/wiki/Absolute_deviation" class="mw-redirect" title="Absolute deviation">absolute deviation</a> function <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 a\mapsto \operatorname {E} [|X-a|]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto \operatorname {E} [|X-a|]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8fea8d50c9d819a35eee4405904f7af77746cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.064ex; height:2.843ex;" alt="{\displaystyle a\mapsto \operatorname {E} [|X-a|]}" /></span>. </p><p>Mallows's proof can be generalized to obtain a multivariate version of the inequality<sup id="cite_ref-PicheRandomVectorsSequences_17-0" class="reference"><a href="#cite_note-PicheRandomVectorsSequences-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> simply by replacing the absolute value with a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>: <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 \left\|\mu -m\right\|\leq {\sqrt {\operatorname {E} \left({\left\|X-\mu \right\|}^{2}\right)}}={\sqrt {\operatorname {trace} \left(\operatorname {var} (X)\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>μ</mi> <mo>−</mo> <mi>m</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>trace</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mu -m\right\|\leq {\sqrt {\operatorname {E} \left({\left\|X-\mu \right\|}^{2}\right)}}={\sqrt {\operatorname {trace} \left(\operatorname {var} (X)\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a325055cb19313abb34c2eb3c826482306ef58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.355ex; height:6.176ex;" alt="{\displaystyle \left\|\mu -m\right\|\leq {\sqrt {\operatorname {E} \left({\left\|X-\mu \right\|}^{2}\right)}}={\sqrt {\operatorname {trace} \left(\operatorname {var} (X)\right)}}}" /></span> </p><p>where <i>m</i> is a <a href="/wiki/Spatial_median" class="mw-redirect" title="Spatial median">spatial median</a>, that is, a minimizer of the function <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 a\mapsto \operatorname {E} (\|X-a\|).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo>−</mo> <mi>a</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50930f6c028807a007dcc0fb7130e5550eed970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.645ex; height:2.843ex;" alt="{\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,}" /></span> The spatial median is unique when the data-set's dimension is two or more.<sup id="cite_ref-Kemperman_18-0" class="reference"><a href="#cite_note-Kemperman-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-MilasevicDucharme_19-0" class="reference"><a href="#cite_note-MilasevicDucharme-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>An alternative proof uses the one-sided Chebyshev inequality; it appears in <a href="/wiki/An_inequality_on_location_and_scale_parameters#An_application_-_distance_between_the_mean_and_the_median" class="mw-redirect" title="An inequality on location and scale parameters">an inequality on location and scale parameters</a>. This formula also follows directly from <a href="/wiki/Cantelli%27s_inequality" title="Cantelli's inequality">Cantelli's inequality</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Unimodal_distributions">Unimodal distributions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=9" title="Edit section: Unimodal distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the case of <a href="/wiki/Unimodality" title="Unimodality">unimodal</a> distributions, one can achieve a sharper bound on the distance between the median and the mean:<sup id="cite_ref-unimodal_21-0" class="reference"><a href="#cite_note-unimodal-21"><span class="cite-bracket">[</span>21<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 \left|{\tilde {X}}-{\bar {X}}\right|\leq \left({\frac {3}{5}}\right)^{1/2}\sigma \approx 0.7746\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <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>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>σ</mi> <mo>≈</mo> <mn>0.7746</mn> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\tilde {X}}-{\bar {X}}\right|\leq \left({\frac {3}{5}}\right)^{1/2}\sigma \approx 0.7746\sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/417d7db861ad7a32d5edff9553531fa61c06828c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.174ex; height:6.676ex;" alt="{\displaystyle \left|{\tilde {X}}-{\bar {X}}\right|\leq \left({\frac {3}{5}}\right)^{1/2}\sigma \approx 0.7746\sigma .}" /></span> </p><p>A similar relation holds between the median and the mode: </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 \left|{\tilde {X}}-\mathrm {mode} \right|\leq 3^{1/2}\sigma \approx 1.732\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <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"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>σ</mi> <mo>≈</mo> <mn>1.732</mn> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\tilde {X}}-\mathrm {mode} \right|\leq 3^{1/2}\sigma \approx 1.732\sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d60161fad1e715e68e1d63a02134c28bc4d9a68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.198ex; height:3.676ex;" alt="{\displaystyle \left|{\tilde {X}}-\mathrm {mode} \right|\leq 3^{1/2}\sigma \approx 1.732\sigma .}" /></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg/220px-Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg/330px-Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg/440px-Proof_without_words-_The_mean_is_greater_than_the_median_for_monotonic_distributions.svg.png 2x" data-file-width="768" data-file-height="768" /></a><figcaption>The mean is greater than the median for monotonic distributions.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Mean,_median,_and_skew"><span id="Mean.2C_median.2C_and_skew"></span>Mean, median, and skew</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=10" title="Edit section: Mean, median, and skew"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A typical heuristic is that positively skewed distributions have mean > median. This is true for all members of the <a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson distribution family</a>. However this is not always true. For example, the <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution family</a> has members with positive mean, but mean < median. Violations of the rule are particularly common for discrete distributions. For example, any Poisson distribution has positive skew, but its mean < median whenever <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 {\bmod {1}}>\ln 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> <mo>></mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu {\bmod {1}}>\ln 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce7e417f6a3a65090608f59375fb6b0b8d8b562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.833ex; height:2.676ex;" alt="{\displaystyle \mu {\bmod {1}}>\ln 2}" /></span>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> See <sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> for a proof sketch. </p><p>When the distribution has a monotonically decreasing probability density, then the median is less than the mean, as shown in the figure. </p> <div class="mw-heading mw-heading2"><h2 id="Jensen's_inequality_for_medians"><span id="Jensen.27s_inequality_for_medians"></span>Jensen's inequality for medians</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=11" title="Edit section: Jensen's inequality for medians"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jensen's inequality states that for any random variable <i>X</i> with a finite expectation <i>E</i>[<i>X</i>] and for any convex function <i>f</i> </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 f(\operatorname {E} (x))\leq \operatorname {E} (f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\operatorname {E} (x))\leq \operatorname {E} (f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e56335dbe49f5804b40bfe46c437f00aee0444f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.718ex; height:2.843ex;" alt="{\displaystyle f(\operatorname {E} (x))\leq \operatorname {E} (f(x))}" /></span> </p><p>This inequality generalizes to the median as well. We say a function <span class="texhtml"><i>f</i>: <b>R</b> → <b>R</b></span> is a <b>C function</b> if, for any <i>t</i>, </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 f^{-1}\left(\,(-\infty ,t]\,\right)=\{x\in \mathbb {R} \mid f(x)\leq t\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace"></mspace> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\left(\,(-\infty ,t]\,\right)=\{x\in \mathbb {R} \mid f(x)\leq t\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb6a4a02d8480c441a0f73bea93cc4fffb9b08d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.745ex; height:3.176ex;" alt="{\displaystyle f^{-1}\left(\,(-\infty ,t]\,\right)=\{x\in \mathbb {R} \mid f(x)\leq t\}}" /></span> is a <a href="/wiki/Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> (allowing the degenerate cases of a <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">single point</a> or an <a href="/wiki/Empty_set" title="Empty set">empty set</a>). Every convex function is a C function, but the reverse does not hold. If <i>f</i> is a C function, then </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 f(\operatorname {med} [X])\leq \operatorname {med} [f(X)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>med</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>med</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\operatorname {med} [X])\leq \operatorname {med} [f(X)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1522fc2d118c152e6d9313b577ac71b210223d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.343ex; height:2.843ex;" alt="{\displaystyle f(\operatorname {med} [X])\leq \operatorname {med} [f(X)]}" /></span> </p><p>If the medians are not unique, the statement holds for the corresponding suprema.<sup id="cite_ref-Merkle2005_24-0" class="reference"><a href="#cite_note-Merkle2005-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Medians_for_samples">Medians for samples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=12" title="Edit section: Medians for samples"><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">This section discusses the theory of estimating a population median from a sample. To calculate the median of a sample "by hand," see <a href="#Finite_data_set_of_numbers">§ Finite data set of numbers</a> above.</div> <div class="mw-heading mw-heading3"><h3 id="Efficient_computation_of_the_sample_median"><span class="anchor" id="Ninther"></span> <span class="anchor" id="Remedian"></span> Efficient computation of the sample median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=13" title="Edit section: Efficient computation of the sample median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Even though <a href="/wiki/Sorting_algorithm" title="Sorting algorithm">comparison-sorting</a> <i>n</i> items requires <span class="texhtml"><a href="/wiki/Big_O_notation" title="Big O notation">Ω</a>(<i>n</i> log <i>n</i>)</span> operations, <a href="/wiki/Selection_algorithm" title="Selection algorithm">selection algorithms</a> can compute the <a href="/wiki/Order_statistic" title="Order statistic"><span class="texhtml mvar" style="font-style:italic;">k</span>th-smallest of <span class="texhtml mvar" style="font-style:italic;">n</span> items</a> with only <span class="texhtml">Θ(<i>n</i>)</span> operations. This includes the median, which is the <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.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="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>th order statistic (or for an even number of samples, the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of the two middle order statistics).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Selection algorithms still have the downside of requiring <span class="texhtml">Ω(<i>n</i>)</span> memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the <a href="/wiki/Quicksort" title="Quicksort">quicksort</a> sorting algorithm, which uses an estimate of its input's median. A more <a href="/wiki/Robust_estimator" class="mw-redirect" title="Robust estimator">robust estimator</a> is <a href="/wiki/John_Tukey" title="John Tukey">Tukey</a>'s <i>ninther</i>, which is the median of three rule applied with limited recursion:<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> if <span class="texhtml mvar" style="font-style:italic;">A</span> is the sample laid out as an <a href="/wiki/Array_(data_structure)" title="Array (data structure)">array</a>, and </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;"><span class="texhtml">med3(<i>A</i>) = med(<i>A</i>[1], <i>A</i>[<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>], <i>A</i>[<i>n</i>])</span>,</div> <p>then </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;"><span class="texhtml">ninther(<i>A</i>) = med3(med3(<i>A</i>[1 ... <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span><i>n</i>]), med3(<i>A</i>[<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span><i>n</i> ... <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span><i>n</i>]), med3(<i>A</i>[<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span><i>n</i> ... <i>n</i>]))</span></div> <p>The <i>remedian</i> is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Sampling_distribution">Sampling distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=14" title="Edit section: Sampling distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The distributions of both the sample mean and the sample median were determined by <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a>.<sup id="cite_ref-Stigler1973_28-0" class="reference"><a href="#cite_note-Stigler1973-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The distribution of the sample median from a population with a density function <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" /></span> is asymptotically normal with mean <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> and variance<sup id="cite_ref-Rider1960_29-0" class="reference"><a href="#cite_note-Rider1960-29"><span class="cite-bracket">[</span>29<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 {\frac {1}{4nf(m)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <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 {\frac {1}{4nf(m)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d71542bb340c6f7e0695f940d990affc6e3ea8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.576ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{4nf(m)^{2}}}}" /></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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /></span> is the median 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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(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 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 the sample size: </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 {\text{Sample median}}\sim {\mathcal {N}}{\left(\mu {=}m,\,\sigma ^{2}{=}{\frac {1}{4nf(m)^{2}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sample median</mtext> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mi>m</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Sample median}}\sim {\mathcal {N}}{\left(\mu {=}m,\,\sigma ^{2}{=}{\frac {1}{4nf(m)^{2}}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36d9eea42f72eb17be447715a78682bebc774d8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.54ex; height:6.343ex;" alt="{\displaystyle {\text{Sample median}}\sim {\mathcal {N}}{\left(\mu {=}m,\,\sigma ^{2}{=}{\frac {1}{4nf(m)^{2}}}\right)}}" /></span> </p><p>A modern proof follows below. Laplace's result is now understood as a special case of <a href="/wiki/Quantile#Estimating_quantiles_from_a_sample" title="Quantile">the asymptotic distribution of arbitrary quantiles</a>. </p><p>For normal samples, the density 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 f(m)=1/{\sqrt {2\pi \sigma ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(m)=1/{\sqrt {2\pi \sigma ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21288bb4ad37ef3719a11fb7bd31a4490d88382f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.755ex; height:3.509ex;" alt="{\displaystyle f(m)=1/{\sqrt {2\pi \sigma ^{2}}}}" /></span>, thus for large samples the variance of the median equals <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 ({\pi }/{2})\cdot (\sigma ^{2}/n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\pi }/{2})\cdot (\sigma ^{2}/n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44f9962437fbde2063f0cb1220ee80de3ec7461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.543ex; height:3.176ex;" alt="{\displaystyle ({\pi }/{2})\cdot (\sigma ^{2}/n).}" /></span><sup id="cite_ref-Williams_2001_165_7-1" class="reference"><a href="#cite_note-Williams_2001_165-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> (See also section <a href="#Efficiency">#Efficiency</a> below.) </p> <div class="mw-heading mw-heading4"><h4 id="Derivation_of_the_asymptotic_distribution">Derivation of the asymptotic distribution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=15" title="Edit section: Derivation of the asymptotic distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/60px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/120px-Question_book-new.svg.png 1.5x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Median" title="Special:EditPage/Median">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">November 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>We take the sample size to be an odd number <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=2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a536c21ca360c4a045989f18def069cef5497b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.722ex; height:2.343ex;" alt="{\displaystyle N=2n+1}" /></span> and assume our variable continuous; the formula for the case of discrete variables is given below in <a href="#Empirical_local_density">§ Empirical local density</a>. The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities <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(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf80e05072fd210ede15b7bb42cdc261abc2929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.678ex; height:2.843ex;" alt="{\displaystyle F(v)}" /></span>, <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(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b2b785b7b6d9a22484d466da88d6328ed0b197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.215ex; height:2.843ex;" alt="{\displaystyle f(v)}" /></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 1-F(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-F(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8581a0eff8215b9be9a2003dc4957ff16bf2156a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.681ex; height:2.843ex;" alt="{\displaystyle 1-F(v)}" /></span>. For a continuous variable, the probability of multiple sample values being exactly equal to the median is 0, so one can calculate the density of at the point <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span> directly from the trinomial distribution: </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 \Pr[\operatorname {med} =v]\,dv={\frac {(2n+1)!}{n!n!}}F(v)^{n}(1-F(v))^{n}f(v)\,dv.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>med</mi> <mo>=</mo> <mi>v</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[\operatorname {med} =v]\,dv={\frac {(2n+1)!}{n!n!}}F(v)^{n}(1-F(v))^{n}f(v)\,dv.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc50e31f1f6f59cc7db71c125dc3034b7034beb8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.154ex; height:5.843ex;" alt="{\displaystyle \Pr[\operatorname {med} =v]\,dv={\frac {(2n+1)!}{n!n!}}F(v)^{n}(1-F(v))^{n}f(v)\,dv.}" /></span> </p><p>Now we introduce the beta function. For integer arguments <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span>, this can be expressed as <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} (\alpha ,\beta )={\frac {(\alpha -1)!(\beta -1)!}{(\alpha +\beta -1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (\alpha ,\beta )={\frac {(\alpha -1)!(\beta -1)!}{(\alpha +\beta -1)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91af9c56b551af314bf3c8149f105052bf78ec80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.981ex; height:6.509ex;" alt="{\displaystyle \mathrm {B} (\alpha ,\beta )={\frac {(\alpha -1)!(\beta -1)!}{(\alpha +\beta -1)!}}}" /></span>. Also, recall 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 f(v)\,dv=dF(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(v)\,dv=dF(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8ad417bae845ebc17674bd041e03e44715ecf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.938ex; height:2.843ex;" alt="{\displaystyle f(v)\,dv=dF(v)}" /></span>. Using these relationships and setting both <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> equal to <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+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" /></span> allows the last expression to 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 {\frac {F(v)^{n}(1-F(v))^{n}}{\mathrm {B} (n+1,n+1)}}\,dF(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F(v)^{n}(1-F(v))^{n}}{\mathrm {B} (n+1,n+1)}}\,dF(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee12ac4b9a1ec7157af5cd0e0eed0cf2651eded" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.721ex; height:6.509ex;" alt="{\displaystyle {\frac {F(v)^{n}(1-F(v))^{n}}{\mathrm {B} (n+1,n+1)}}\,dF(v)}" /></span> </p><p>Hence the density function of the median is a symmetric beta distribution <a href="/wiki/Pushforward_measure" title="Pushforward measure">pushed forward</a> 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 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>. Its mean, as we would expect, is 0.5 and its variance 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 1/(4(N+2))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/(4(N+2))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079faf019c9dd618fa97095ce508bc406917b926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.172ex; height:2.843ex;" alt="{\displaystyle 1/(4(N+2))}" /></span>. By the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>, the corresponding variance of the sample median is </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 {\frac {1}{4(N+2)f(m)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <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 {\frac {1}{4(N+2)f(m)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16e9d04925b2ceaf6e3eca2b9dedfd10016f725" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.704ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{4(N+2)f(m)^{2}}}.}" /></span> </p><p>The additional 2 is negligible <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">in the limit</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Empirical_local_density">Empirical local density</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=16" title="Edit section: Empirical local density"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In practice, the functions <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/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></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 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> above are often not known or assumed. However, they can be estimated from an observed frequency distribution. In this section, we give an example. Consider the following table, representing a sample of 3,800 (discrete-valued) observations: </p> <table class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;"> <tbody><tr> <th><span class="texhtml mvar" style="font-style:italic;">v</span></th> <th>0</th> <th>0.5</th> <th>1</th> <th>1.5</th> <th>2</th> <th>2.5</th> <th>3</th> <th>3.5</th> <th>4</th> <th>4.5</th> <th>5 </th></tr> <tr> <th><span class="texhtml"><i>f</i>(<i>v</i>)</span> </th> <td>0.000</td> <td>0.008</td> <td>0.010</td> <td>0.013</td> <td>0.083</td> <td>0.108</td> <td>0.328</td> <td>0.220</td> <td>0.202</td> <td>0.023</td> <td>0.005 </td></tr> <tr> <th><span class="texhtml"><i>F</i>(<i>v</i>)</span> </th> <td>0.000</td> <td>0.008</td> <td>0.018</td> <td>0.031</td> <td>0.114</td> <td>0.222</td> <td>0.550</td> <td>0.770</td> <td>0.972</td> <td>0.995</td> <td>1.000 </td></tr></tbody></table> <p>Because the observations are discrete-valued, constructing the exact distribution of the median is not an immediate translation of the above expression 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 \Pr(\operatorname {med} =v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>med</mi> <mo>=</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(\operatorname {med} =v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b4ee20b139d9b3abaa645bdccd1176b4c3b794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.791ex; height:2.843ex;" alt="{\displaystyle \Pr(\operatorname {med} =v)}" /></span>; one may (and typically does) have multiple instances of the median in one's sample. So we must sum over all these possibilities: </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 \Pr(\operatorname {med} =v)=\sum _{i=0}^{n}\sum _{k=0}^{n}{\frac {N!}{i!(N-i-k)!k!}}F(v-1)^{i}(1-F(v))^{k}f(v)^{N-i-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>med</mi> <mo>=</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>!</mo> </mrow> <mrow> <mi>i</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>i</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>i</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(\operatorname {med} =v)=\sum _{i=0}^{n}\sum _{k=0}^{n}{\frac {N!}{i!(N-i-k)!k!}}F(v-1)^{i}(1-F(v))^{k}f(v)^{N-i-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2aeff6a883cb2f5e097d60b1c550ae45211a01f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:70.678ex; height:7.009ex;" alt="{\displaystyle \Pr(\operatorname {med} =v)=\sum _{i=0}^{n}\sum _{k=0}^{n}{\frac {N!}{i!(N-i-k)!k!}}F(v-1)^{i}(1-F(v))^{k}f(v)^{N-i-k}}" /></span> </p><p>Here, <i>i</i> is the number of points strictly less than the median and <i>k</i> the number strictly greater. </p><p>Using these preliminaries, it is possible to investigate the effect of sample size on the standard errors of the mean and median. The observed mean is 3.16, the observed raw median is 3 and the observed interpolated median is 3.174. The following table gives some comparison statistics. </p> <table class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;"> <tbody><tr> <th style="background:var(--background-color-neutral,#eaecf0);color:inherit;background:linear-gradient(to top right,var(--background-color-neutral,#eaecf0) 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,var(--background-color-neutral,#eaecf0) 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Sample size</div><div style="margin-right:2em;text-align:left">Statistic</div></th> <th>3</th> <th>9</th> <th>15</th> <th>21 </th></tr> <tr> <th>Expected value of median </th> <td>3.198</td> <td>3.191</td> <td>3.174</td> <td>3.161 </td></tr> <tr> <th>Standard error of median (above formula) </th> <td>0.482</td> <td>0.305</td> <td>0.257</td> <td>0.239 </td></tr> <tr> <th>Standard error of median (asymptotic approximation) </th> <td>0.879</td> <td>0.508</td> <td>0.393</td> <td>0.332 </td></tr> <tr> <th>Standard error of mean </th> <td>0.421</td> <td>0.243</td> <td>0.188</td> <td>0.159 </td></tr></tbody></table> <p>The expected value of the median falls slightly as sample size increases while, as would be expected, the standard errors of both the median and the mean are proportionate to the inverse square root of the sample size. The asymptotic approximation errs on the side of caution by overestimating the standard error. </p> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_variance_from_sample_data">Estimation of variance from sample data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=17" title="Edit section: Estimation of variance from sample data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value 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 (2f(x))^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2f(x))^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73924b0d56a2a150229b1a77cec20e923da3405e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.722ex; height:3.176ex;" alt="{\displaystyle (2f(x))^{-2}}" /></span>—the asymptotic value 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^{-1/2}(\nu -m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{-1/2}(\nu -m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2781bccc9d6ce761991a55b7271162304fd7a45b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.294ex; height:3.343ex;" alt="{\displaystyle n^{-1/2}(\nu -m)}" /></span> 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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" /></span> is the population median—has been studied by several authors. The standard "delete one" <a href="/wiki/Resampling_(statistics)#Jackknife" title="Resampling (statistics)">jackknife</a> method produces <a href="/wiki/Consistent_estimator" title="Consistent estimator">inconsistent</a> results.<sup id="cite_ref-Efron1982_30-0" class="reference"><a href="#cite_note-Efron1982-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> An alternative—the "delete k" method—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 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> grows with the sample size has been shown to be asymptotically consistent.<sup id="cite_ref-Shao1989_31-0" class="reference"><a href="#cite_note-Shao1989-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,<sup id="cite_ref-Efron1979_32-0" class="reference"><a href="#cite_note-Efron1979-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> but converges very slowly (<a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">order</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 n^{-{\frac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{-{\frac {1}{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6da6ef58413ac87c17b18707b69ac2aec0ad3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.409ex; height:3.509ex;" alt="{\displaystyle n^{-{\frac {1}{4}}}}" /></span>).<sup id="cite_ref-Hall1988_33-0" class="reference"><a href="#cite_note-Hall1988-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Other methods have been proposed but their behavior may differ between large and small samples.<sup id="cite_ref-Jimenez-Gamero2004_34-0" class="reference"><a href="#cite_note-Jimenez-Gamero2004-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Efficiency">Efficiency<span class="anchor" id="Efficiency"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=18" title="Edit section: Efficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficiency</a> of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of 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=2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a536c21ca360c4a045989f18def069cef5497b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.722ex; height:2.343ex;" alt="{\displaystyle N=2n+1}" /></span> from the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the efficiency for large N is </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 {\frac {2}{\pi }}{\frac {N+2}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\pi }}{\frac {N+2}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef4b9467991ec400031c0879a0a276e41e50d76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.071ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{\pi }}{\frac {N+2}{N}}}" /></span> </p><p>The efficiency tends to <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 {\frac {2}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d9a566a0f4e1892e61f7140dacb7bed7f7b44e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{\pi }}}" /></span> as <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> tends to infinity. </p><p>In other words, the relative variance of the median will be <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 \pi /2\approx 1.57}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>≈</mo> <mn>1.57</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2\approx 1.57}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a36ff9e1888aa40c1aded554c531ed6c74ac50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.89ex; height:2.843ex;" alt="{\displaystyle \pi /2\approx 1.57}" /></span>, or 57% greater than the variance of the mean – the relative <a href="/wiki/Standard_error" title="Standard error">standard error</a> of the median will be <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 (\pi /2)^{\frac {1}{2}}\approx 1.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>≈</mo> <mn>1.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi /2)^{\frac {1}{2}}\approx 1.25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13072a95e688805a98e810845f4bae6be9e48ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.435ex; height:4.009ex;" alt="{\displaystyle (\pi /2)^{\frac {1}{2}}\approx 1.25}" /></span>, or 25% greater than the <a href="/wiki/Standard_error_of_the_mean" class="mw-redirect" title="Standard error of the mean">standard error of the mean</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 /{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma /{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592dd858beccc6d81b0d6156194e6b40bddc2d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.823ex; height:3.009ex;" alt="{\displaystyle \sigma /{\sqrt {n}}}" /></span> (see also section <a href="#Sampling_distribution">#Sampling distribution</a> above.).<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_estimators">Other estimators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=19" title="Edit section: Other estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For univariate distributions that are <i>symmetric</i> about one median, the <a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a> is a <a href="/wiki/Robust_statistics" title="Robust statistics">robust</a> and highly <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficient estimator</a> of the population median.<sup id="cite_ref-HM_36-0" class="reference"><a href="#cite_note-HM-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>If data is represented by a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> specifying a particular family of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution. <a href="/wiki/Pareto_interpolation" title="Pareto interpolation">Pareto interpolation</a> is an application of this when the population is assumed to have a <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Multivariate_median">Multivariate median</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=20" title="Edit section: Multivariate median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one.<sup id="cite_ref-HM_36-1" class="reference"><a href="#cite_note-HM-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Marginal_median">Marginal median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=21" title="Edit section: Marginal median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.<sup id="cite_ref-HM_36-2" class="reference"><a href="#cite_note-HM-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_median">Geometric median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=22" title="Edit section: Geometric median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Geometric_median" title="Geometric median">geometric median</a> of a discrete set of sample points <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_{1},\ldots x_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots x_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0ba83662fefde0a54f0df3e3b0ec9b6793270f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.549ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots x_{N}}" /></span> in a Euclidean space is the<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> point minimizing the sum of distances to the sample points. </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 {\hat {\mu }}={\underset {\mu \in \mathbb {R} ^{m}}{\operatorname {arg\,min} }}\sum _{n=1}^{N}\left\|\mu -x_{n}\right\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mi>μ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </munder> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}={\underset {\mu \in \mathbb {R} ^{m}}{\operatorname {arg\,min} }}\sum _{n=1}^{N}\left\|\mu -x_{n}\right\|_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff00c345101de1e41ad6dc740144899eca2a353" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.298ex; height:7.343ex;" alt="{\displaystyle {\hat {\mu }}={\underset {\mu \in \mathbb {R} ^{m}}{\operatorname {arg\,min} }}\sum _{n=1}^{N}\left\|\mu -x_{n}\right\|_{2}}" /></span> </p><p>In contrast to the marginal median, the geometric median is <a href="/wiki/Equivariant" class="mw-redirect" title="Equivariant">equivariant</a> with respect to Euclidean <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity transformations</a> such as <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a> and <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Median_in_all_directions">Median in all directions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=23" title="Edit section: Median in all directions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the marginal medians for all coordinate systems coincide, then their common location may be termed the "median in all directions".<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> This concept is relevant to voting theory on account of the <a href="/wiki/Median_voter_theorem" title="Median voter theorem">median voter theorem</a>. When it exists, the median in all directions coincides with the geometric median (at least for discrete distributions). </p> <div class="mw-heading mw-heading3"><h3 id="Centerpoint">Centerpoint</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=24" title="Edit section: Centerpoint"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Centerpoint_(geometry)" title="Centerpoint (geometry)">Centerpoint (geometry)</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Centerpoint_(geometry)&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>, the notion of <a href="/wiki/Centerpoint_(geometry)" title="Centerpoint (geometry)">centerpoint</a> is a generalization of the median to data in higher-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. Given a set of points in <i>d</i>-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(<i>d</i> + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.</div></div> <div class="mw-heading mw-heading2"><h2 id="Conditional_median">Conditional median</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=25" title="Edit section: Conditional median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conditional median occurs in the setting where we seek to estimate a random variable <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> from a random variable <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>, which is a noisy version 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>. The conditional median in this setting is given by </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 m(X|Y=y)=F_{X|Y=y}^{-1}\left({\frac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(X|Y=y)=F_{X|Y=y}^{-1}\left({\frac {1}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6ddd31c7c97a7672b02658327855b3950cdf82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.343ex; height:6.176ex;" alt="{\displaystyle m(X|Y=y)=F_{X|Y=y}^{-1}\left({\frac {1}{2}}\right)}" /></span> 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 t\mapsto F_{X|Y=y}^{-1}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto F_{X|Y=y}^{-1}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7472030d396c1910a5710bfb6a611abe425b198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:14.036ex; height:3.843ex;" alt="{\displaystyle t\mapsto F_{X|Y=y}^{-1}(t)}" /></span> is the inverse of the conditional cdf (i.e., conditional quantile function) 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\mapsto F_{X|Y}(x|y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto F_{X|Y}(x|y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1be47a221a7e0d99130cc2db64d03adae47270c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.723ex; height:3.176ex;" alt="{\displaystyle x\mapsto F_{X|Y}(x|y)}" /></span>. For example, a popular model 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 Y=X+Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>X</mi> <mo>+</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X+Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422d143341d1096308ad0f7f010d9150efa0d1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.373ex; height:2.343ex;" alt="{\displaystyle Y=X+Z}" /></span> 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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" /></span> is standard normal independent 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>. The conditional median is the optimal Bayesian <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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" /></span> estimator: </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 m(X|Y=y)=\arg \min _{f}\operatorname {E} \left[|X-f(Y)|\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(X|Y=y)=\arg \min _{f}\operatorname {E} \left[|X-f(Y)|\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dfa4187ace82496af4041705d82a396a021c69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.34ex; height:4.343ex;" alt="{\displaystyle m(X|Y=y)=\arg \min _{f}\operatorname {E} \left[|X-f(Y)|\right]}" /></span> </p><p>It is known that for the model <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=X+Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>X</mi> <mo>+</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X+Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422d143341d1096308ad0f7f010d9150efa0d1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.373ex; height:2.343ex;" alt="{\displaystyle Y=X+Z}" /></span> 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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" /></span> is standard normal independent 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>, the estimator is linear if and only 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/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> is Gaussian.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_median-related_concepts">Other median-related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=26" title="Edit section: Other median-related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Interpolated_median">Interpolated median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=27" title="Edit section: Interpolated median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a <a href="/wiki/Likert_scale" title="Likert scale">Likert scale</a>, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /></span> is 3 since the median is the smallest value 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/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> for which <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" /></span> is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50. First we add half of the interval width <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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" /></span> to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark. In other words, we split up the interval width pro rata to the numbers of observations. In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" /></span> are known, the interpolated median can be calculated from </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 m_{\text{int}}=m+w\left[{\frac {1}{2}}-{\frac {F(m)-{\frac {1}{2}}}{f(m)}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>int</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mi>w</mi> <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> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{int}}=m+w\left[{\frac {1}{2}}-{\frac {F(m)-{\frac {1}{2}}}{f(m)}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e823608d9eba650d4796825d3043ef41d06370e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.823ex; height:7.676ex;" alt="{\displaystyle m_{\text{int}}=m+w\left[{\frac {1}{2}}-{\frac {F(m)-{\frac {1}{2}}}{f(m)}}\right].}" /></span> </p><p>Alternatively, if in an observed sample there are <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> scores above the median category, <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /></span> scores in it 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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> scores below it then the interpolated median is given by </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 m_{\text{int}}=m+{\frac {w}{2}}\left[{\frac {k-i}{j}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>int</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> <mi>j</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{int}}=m+{\frac {w}{2}}\left[{\frac {k-i}{j}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8519008345d5bd2863ff18203d1b6144f851ae95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.329ex; height:6.176ex;" alt="{\displaystyle m_{\text{int}}=m+{\frac {w}{2}}\left[{\frac {k-i}{j}}\right].}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Pseudo-median">Pseudo-median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=28" title="Edit section: Pseudo-median"><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/Pseudomedian" title="Pseudomedian">Pseudomedian</a></div> <p>For univariate distributions that are <i>symmetric</i> about one median, the <a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a> is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population <i>pseudo-median</i>, which is the median of a symmetrized distribution and which is close to the population median.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> The Hodges–Lehmann estimator has been generalized to multivariate distributions.<sup id="cite_ref-Oja_2010_xiv+232_46-0" class="reference"><a href="#cite_note-Oja_2010_xiv+232-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Variants_of_regression">Variants of regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=29" title="Edit section: Variants of regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Theil%E2%80%93Sen_estimator" title="Theil–Sen estimator">Theil–Sen estimator</a> is a method for <a href="/wiki/Robust_statistics" title="Robust statistics">robust</a> <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a> based on finding medians of <a href="/wiki/Slope" title="Slope">slopes</a>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Median_filter">Median filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=30" title="Edit section: Median filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Median_filter" title="Median filter">median filter</a> is an important tool of <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, that can effectively remove any <a href="/wiki/Salt_and_pepper_noise" class="mw-redirect" title="Salt and pepper noise">salt and pepper noise</a> from <a href="/wiki/Grayscale" title="Grayscale">grayscale</a> images. </p> <div class="mw-heading mw-heading3"><h3 id="Cluster_analysis">Cluster analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=31" title="Edit section: Cluster 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">Main article: <a href="/wiki/K-medians_clustering" title="K-medians clustering">k-medians clustering</a></div> <p>In <a href="/wiki/Cluster_analysis" title="Cluster analysis">cluster analysis</a>, the <a href="/wiki/K-medians_clustering" title="K-medians clustering">k-medians clustering</a> algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in <a href="/wiki/K-means_clustering" title="K-means clustering">k-means clustering</a>, is replaced by maximising the distance between cluster-medians. </p> <div class="mw-heading mw-heading3"><h3 id="Median–median_line"><span id="Median.E2.80.93median_line"></span>Median–median line</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=32" title="Edit section: Median–median line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is a method of robust regression. The idea dates back to <a href="/wiki/Abraham_Wald" title="Abraham Wald">Wald</a> in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent 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 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>: a left half with values less than the median and a right half with values greater than the median.<sup id="cite_ref-Wald1940_48-0" class="reference"><a href="#cite_note-Wald1940-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> He suggested taking the means of the dependent <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> and independent <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> variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set. </p><p>Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples.<sup id="cite_ref-Nair1942_49-0" class="reference"><a href="#cite_note-Nair1942-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.<sup id="cite_ref-Brown1951_50-0" class="reference"><a href="#cite_note-Brown1951-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.<sup id="cite_ref-Tukey1971_51-0" class="reference"><a href="#cite_note-Tukey1971-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Median-unbiased_estimators">Median-unbiased estimators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=33" title="Edit section: Median-unbiased estimators"><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/Bias_of_an_estimator#Median-unbiased_estimators" title="Bias of an estimator">Bias of an estimator § Median-unbiased estimators</a></div> <p>Any <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator"><i>mean</i>-unbiased estimator</a> minimizes the <a href="/wiki/Risk" title="Risk">risk</a> (<a href="/wiki/Expected_loss" title="Expected loss">expected loss</a>) with respect to the squared-error <a href="/wiki/Loss_function" title="Loss function">loss function</a>, as observed by <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>. A <a href="/wiki/Bias_of_an_estimator#Median_unbiased_estimators,_and_bias_with_respect_to_other_loss_functions" title="Bias of an estimator"><i>median</i>-unbiased estimator</a> minimizes the risk with respect to the <a href="/wiki/Absolute_deviation" class="mw-redirect" title="Absolute deviation">absolute-deviation</a> loss function, as observed by <a href="/wiki/Laplace" class="mw-redirect" title="Laplace">Laplace</a>. Other <a href="/wiki/Loss_functions" class="mw-redirect" title="Loss functions">loss functions</a> are used in <a href="/wiki/Statistical_theory" title="Statistical theory">statistical theory</a>, particularly in <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistics</a>. </p><p>The theory of median-unbiased estimators was revived by George W. Brown in 1947:<sup id="cite_ref-Brown_52-0" class="reference"><a href="#cite_note-Brown-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>An estimate of a one-dimensional parameter θ will be said to be median-unbiased if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.</p><div class="templatequotecite">— <cite>page 584</cite></div></blockquote> <p>Further properties of median-unbiased estimators have been reported.<sup id="cite_ref-Lehmann_53-0" class="reference"><a href="#cite_note-Lehmann-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Birnbaum_54-0" class="reference"><a href="#cite_note-Birnbaum-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-vdW_55-0" class="reference"><a href="#cite_note-vdW-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pf_56-0" class="reference"><a href="#cite_note-Pf-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>There are methods of constructing median-unbiased estimators that are optimal (in a sense analogous to the minimum-variance property for mean-unbiased estimators). Such constructions exist for probability distributions having <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio">monotone likelihood-functions</a>.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> One such procedure is an analogue of the <a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwell procedure</a> for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of <a href="/wiki/Loss_function" title="Loss function">loss functions</a>.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=34" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Scientific researchers in the ancient near east appear not to have used summary statistics altogether, instead choosing values that offered maximal consistency with a broader theory that integrated a wide variety of phenomena.<sup id="cite_ref-:0_60-0" class="reference"><a href="#cite_note-:0-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> Within the Mediterranean (and, later, European) scholarly community, statistics like the mean are fundamentally a medieval and early modern development. (The history of the median outside Europe and its predecessors remains relatively unstudied.) </p><p>The idea of the median appeared in the 6th century in the <a href="/wiki/Talmud" title="Talmud">Talmud</a>, in order to fairly analyze divergent <a href="/wiki/Economic_appraisal" title="Economic appraisal">appraisals</a>.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> However, the concept did not spread to the broader scientific community. </p><p>Instead, the closest ancestor of the modern median is the <a href="/wiki/Mid-range" title="Mid-range">mid-range</a>, invented by <a href="/wiki/Al-Biruni" title="Al-Biruni">Al-Biruni</a><sup id="cite_ref-Eisenhart_63-0" class="reference"><a href="#cite_note-Eisenhart-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 31">: 31 </span></sup><sup id="cite_ref-:2_64-0" class="reference"><a href="#cite_note-:2-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> Transmission of his work to later scholars is unclear. He applied his technique to <a href="/wiki/Assay" title="Assay">assaying</a> currency metals, but, after he published his work, most assayers still adopted the most unfavorable value from their results, lest they appear to <a href="/wiki/Debasement" title="Debasement">cheat</a>.<sup id="cite_ref-Eisenhart_63-1" class="reference"><a href="#cite_note-Eisenhart-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 35–8">: 35–8 </span></sup> <sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> However, increased navigation at sea during the <a href="/wiki/Age_of_Discovery" title="Age of Discovery">Age of Discovery</a> meant that ship's navigators increasingly had to attempt to determine latitude in unfavorable weather against hostile shores, leading to renewed interest in summary statistics. Whether rediscovered or independently invented, the mid-range is recommended to nautical navigators in Harriot's "Instructions for Raleigh's Voyage to Guiana, 1595".<sup id="cite_ref-Eisenhart_63-2" class="reference"><a href="#cite_note-Eisenhart-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 45–8">: 45–8 </span></sup> </p><p>The idea of the median may have first appeared in <a href="/wiki/Edward_Wright_(mathematician)" title="Edward Wright (mathematician)">Edward Wright</a>'s 1599 book <i>Certaine Errors in Navigation</i> on a section about <a href="/wiki/Compass" title="Compass">compass</a> navigation.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> Wright was reluctant to discard measured values, and may have felt that the median — incorporating a greater proportion of the dataset than the <a href="/wiki/Mid-range" title="Mid-range">mid-range</a> — was more likely to be correct. However, Wright did not give examples of his technique's use, making it hard to verify that he described the modern notion of median.<sup id="cite_ref-:0_60-1" class="reference"><a href="#cite_note-:0-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_64-1" class="reference"><a href="#cite_note-:2-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> The median (in the context of probability) certainly appeared in the correspondence of <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>, but as an example of a statistic that was inappropriate for <a href="/wiki/Actuarial_science" title="Actuarial science">actuarial practice</a>.<sup id="cite_ref-:0_60-2" class="reference"><a href="#cite_note-:0-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>The earliest recommendation of the median dates to 1757, when <a href="/wiki/Roger_Joseph_Boscovich" title="Roger Joseph Boscovich">Roger Joseph Boscovich</a> developed a regression method based on the <a href="/wiki/L1_norm" class="mw-redirect" title="L1 norm"><i>L</i><sup>1</sup> norm</a> and therefore implicitly on the median.<sup id="cite_ref-:0_60-3" class="reference"><a href="#cite_note-:0-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stigler1986_68-0" class="reference"><a href="#cite_note-Stigler1986-68"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> In 1774, <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> made this desire explicit: he suggested the median be used as the standard estimator of the value of a posterior <a href="/wiki/Probability_density_function" title="Probability density function">PDF</a>. The specific criterion was to minimize the expected magnitude of the error; <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 |\alpha -\alpha ^{*}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha -\alpha ^{*}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10283056ed7c50813b1afd7e6b2fd0e5f795cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.164ex; height:2.843ex;" alt="{\displaystyle |\alpha -\alpha ^{*}|}" /></span> 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 \alpha ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326a86ec82d76637f77c4c79178745517c4ef77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.343ex;" alt="{\displaystyle \alpha ^{*}}" /></span> is the estimate 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is the true value. To this end, Laplace determined the distributions of both the sample mean and the sample median in the early 1800s.<sup id="cite_ref-Stigler1973_28-1" class="reference"><a href="#cite_note-Stigler1973-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Laplace1818_69-0" class="reference"><a href="#cite_note-Laplace1818-69"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> However, a decade later, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> and <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> developed the <a href="/wiki/Least_squares" title="Least squares">least squares</a> method, which minimizes <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 (\alpha -\alpha ^{*})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha -\alpha ^{*})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ff4a21d663a9bd925b55459f85e507ba7da211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.733ex; height:3.176ex;" alt="{\displaystyle (\alpha -\alpha ^{*})^{2}}" /></span> to obtain the mean; the strong justification of this estimator by reference to <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a> based on a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> means it has mostly replaced Laplace's original suggestion.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a> in 1843 was the first<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> to use the term <i>median</i> (<i>valeur médiane</i>) for the value that divides a probability distribution into two equal halves. <a href="/wiki/Gustav_Theodor_Fechner" class="mw-redirect" title="Gustav Theodor Fechner">Gustav Theodor Fechner</a> used the median (<i>Centralwerth</i>) in sociological and psychological phenomena.<sup id="cite_ref-Keynes1921_72-0" class="reference"><a href="#cite_note-Keynes1921-72"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> It had earlier been used only in astronomy and related fields. <a href="/wiki/Gustav_Theodor_Fechner" class="mw-redirect" title="Gustav Theodor Fechner">Gustav Fechner</a> popularized the median into the formal analysis of data, although it had been used previously by Laplace,<sup id="cite_ref-Keynes1921_72-1" class="reference"><a href="#cite_note-Keynes1921-72"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> and the median appeared in a textbook by <a href="/wiki/Francis_Ysidro_Edgeworth" title="Francis Ysidro Edgeworth">F. Y. Edgeworth</a>.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a> used the term <i>median</i> in 1881,<sup id="cite_ref-Galton1881_74-0" class="reference"><a href="#cite_note-Galton1881-74"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> having earlier used the terms <i>middle-most value</i> in 1869, and the <i>medium</i> in 1880.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p><p><br /> </p> <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=Median&action=edit&section=35" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.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 .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" 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class="mw-redirect" title="Lipschitz functions">Lipschitz functions</a> – Strong form of uniform continuity<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Median_graph" title="Median graph">Median graph</a> – Graph with a median for each three vertices</li> <li><a href="/wiki/Median_of_medians" title="Median of medians">Median of medians</a> – Fast approximate median algorithm – Algorithm to calculate the approximate median in linear time</li> <li><a href="/wiki/Median_search" class="mw-redirect" title="Median search">Median search</a> – Method for finding kth smallest value<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Median_slope" class="mw-redirect" title="Median slope">Median slope</a> – Statistical method for fitting a line<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Median_voter_theory" class="mw-redirect" title="Median voter theory">Median voter theory</a> – Theorem in political science<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Medoid" title="Medoid">Medoid</a> – representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span>s – Generalization of the median in higher dimensions</li> <li><a href="/wiki/Moving_average#Moving_median" title="Moving average">Moving average#Moving median</a> – Type of statistical measure over subsets of a dataset</li> <li><a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">Median absolute deviation</a> – Statistical measure of variability</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median&action=edit&section=36" title="Edit section: Notes"><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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">The geometric median is unique unless the sample is collinear.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text">Subsequent scholars appear to concur with Eisenhart that Boroughs' 1580 figures, while suggestive of the median, in fact describe an arithmetic mean.;<sup id="cite_ref-Eisenhart_63-3" class="reference"><a href="#cite_note-Eisenhart-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 62–3">: 62–3 </span></sup> Boroughs is mentioned in no other work.</span> </li> </ol></div></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=Median&action=edit&section=37" 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-StatisticalMedian-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-StatisticalMedian_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-StatisticalMedian_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Statistical_Median"><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 id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/StatisticalMedian.html">"Statistical Median"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Statistical+Median&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FStatisticalMedian.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></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">Simon, Laura J.; <a rel="nofollow" class="external text" href="http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm">"Descriptive statistics"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100730032416/http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm">Archived</a> 2010-07-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i>Statistical Education Resource Kit</i>, Pennsylvania State Department of Statistics</span> </li> <li id="cite_note-Bissell1994-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bissell1994_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bissell1994_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDerek_Bissell1994" class="citation book cs1">Derek Bissell (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cTwwtyBX7PAC&pg=PA26"><i>Statistical Methods for Spc and Tqm</i></a>. CRC Press. pp. 26–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-39440-9" title="Special:BookSources/978-0-412-39440-9"><bdi>978-0-412-39440-9</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 February</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Methods+for+Spc+and+Tqm&rft.pages=26-&rft.pub=CRC+Press&rft.date=1994&rft.isbn=978-0-412-39440-9&rft.au=Derek+Bissell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcTwwtyBX7PAC%26pg%3DPA26&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Sheskin2003-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sheskin2003_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavid_J._Sheskin2003" class="citation book cs1">David J. Sheskin (27 August 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bmwhcJqq01cC&pg=PA7"><i>Handbook of Parametric and Nonparametric Statistical Procedures</i></a> (Third ed.). CRC Press. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-3626-8" title="Special:BookSources/978-1-4200-3626-8"><bdi>978-1-4200-3626-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 February</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Parametric+and+Nonparametric+Statistical+Procedures&rft.pages=7&rft.edition=Third&rft.pub=CRC+Press&rft.date=2003-08-27&rft.isbn=978-1-4200-3626-8&rft.au=David+J.+Sheskin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbmwhcJqq01cC%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></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="CITEREFPaul_T._von_Hippel2005" class="citation journal cs1">Paul T. von Hippel (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081014045349/http://www.amstat.org/publications/jse/v13n2/vonhippel.html">"Mean, Median, and Skew: Correcting a Textbook Rule"</a>. <i>Journal of Statistics Education</i>. <b>13</b> (2). Archived from <a rel="nofollow" class="external text" href="http://www.amstat.org/publications/jse/v13n2/vonhippel.html">the original</a> on 2008-10-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-06-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistics+Education&rft.atitle=Mean%2C+Median%2C+and+Skew%3A+Correcting+a+Textbook+Rule&rft.volume=13&rft.issue=2&rft.date=2005&rft.au=Paul+T.+von+Hippel&rft_id=http%3A%2F%2Fwww.amstat.org%2Fpublications%2Fjse%2Fv13n2%2Fvonhippel.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Robson-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Robson_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRobson1994" class="citation book cs1">Robson, Colin (1994). <i>Experiment, Design and Statistics in Psychology</i>. Penguin. pp. <span class="nowrap">42–</span>45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-017648-9" title="Special:BookSources/0-14-017648-9"><bdi>0-14-017648-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experiment%2C+Design+and+Statistics+in+Psychology&rft.pages=%3Cspan+class%3D%22nowrap%22%3E42-%3C%2Fspan%3E45&rft.pub=Penguin&rft.date=1994&rft.isbn=0-14-017648-9&rft.aulast=Robson&rft.aufirst=Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Williams_2001_165-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Williams_2001_165_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Williams_2001_165_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilliams2001" class="citation book cs1">Williams, D. (2001). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/weighingoddscour00will_530"><i>Weighing the Odds</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/weighingoddscour00will_530/page/n184">165</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/052100618X" title="Special:BookSources/052100618X"><bdi>052100618X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Weighing+the+Odds&rft.pages=165&rft.pub=Cambridge+University+Press&rft.date=2001&rft.isbn=052100618X&rft.aulast=Williams&rft.aufirst=D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fweighingoddscour00will_530&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" 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="CITEREFMaindonaldBraun2010" class="citation book cs1">Maindonald, John; Braun, W. John (2010-05-06). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8bMj8m-4RDQC&pg=PA104"><i>Data Analysis and Graphics Using R: An Example-Based Approach</i></a>. Cambridge University Press. p. 104. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-48667-5" title="Special:BookSources/978-1-139-48667-5"><bdi>978-1-139-48667-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+Analysis+and+Graphics+Using+R%3A+An+Example-Based+Approach&rft.pages=104&rft.pub=Cambridge+University+Press&rft.date=2010-05-06&rft.isbn=978-1-139-48667-5&rft.aulast=Maindonald&rft.aufirst=John&rft.au=Braun%2C+W.+John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8bMj8m-4RDQC%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150408230922/https://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions">"AP Statistics Review - Density Curves and the Normal Distributions"</a>. Archived from <a rel="nofollow" class="external text" href="https://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions">the original</a> on 8 April 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">16 March</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=AP+Statistics+Review+-+Density+Curves+and+the+Normal+Distributions&rft_id=http%3A%2F%2Fapstatsreview.tumblr.com%2Fpost%2F50058615236%2Fdensity-curves-and-the-normal-distributions&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNewman2005" class="citation journal cs1">Newman, M. E. J. (2005). "Power laws, Pareto distributions and Zipf's law". <i>Contemporary Physics</i>. <b>46</b> (5): <span class="nowrap">323–</span>351. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0412004">cond-mat/0412004</a></span>. <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/2005ConPh..46..323N">2005ConPh..46..323N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00107510500052444">10.1080/00107510500052444</a>. <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:2871747">2871747</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Contemporary+Physics&rft.atitle=Power+laws%2C+Pareto+distributions+and+Zipf%27s+law&rft.volume=46&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E323-%3C%2Fspan%3E351&rft.date=2005&rft_id=info%3Aarxiv%2Fcond-mat%2F0412004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2871747%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F00107510500052444&rft_id=info%3Abibcode%2F2005ConPh..46..323N&rft.aulast=Newman&rft.aufirst=M.+E.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStroock2011" class="citation book cs1">Stroock, Daniel (2011). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00stro"><i>Probability Theory</i></a></span>. Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00stro/page/n66">43</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-13250-3" title="Special:BookSources/978-0-521-13250-3"><bdi>978-0-521-13250-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+Theory&rft.pages=43&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-0-521-13250-3&rft.aulast=Stroock&rft.aufirst=Daniel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprobabilitytheor00stro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDeGroot1970" class="citation book cs1">DeGroot, Morris H. (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7rDY2_r4bmEC&pg=PA232"><i>Optimal Statistical Decisions</i></a>. McGraw-Hill Book Co., New York-London-Sydney. p. 232. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471680291" title="Special:BookSources/9780471680291"><bdi>9780471680291</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0356303">0356303</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+Statistical+Decisions&rft.pages=232&rft.pub=McGraw-Hill+Book+Co.%2C+New+York-London-Sydney&rft.date=1970&rft.isbn=9780471680291&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0356303%23id-name%3DMR&rft.aulast=DeGroot&rft.aufirst=Morris+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7rDY2_r4bmEC%26pg%3DPA232&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStephen_A._BookLawrence_Sher1979" class="citation journal cs1">Stephen A. Book; Lawrence Sher (1979). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3026748">"How close are the mean and the median?"</a>. <i>The Two-Year College Mathematics Journal</i>. <b>10</b> (3): <span class="nowrap">202–</span>204. <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%2F3026748">10.2307/3026748</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/3026748">3026748</a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=How+close+are+the+mean+and+the+median%3F&rft.volume=10&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E202-%3C%2Fspan%3E204&rft.date=1979&rft_id=info%3Adoi%2F10.2307%2F3026748&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3026748%23id-name%3DJSTOR&rft.au=Stephen+A.+Book&rft.au=Lawrence+Sher&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3026748&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarren_PageVedula_N._Murty1982" class="citation journal cs1">Warren Page; Vedula N. Murty (1982). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00494925.1982.11972639?journalCode=ucmj19">"Nearness Relations Among Measures of Central Tendency and Dispersion: Part 1"</a>. <i>The Two-Year College Mathematics Journal</i>. <b>13</b> (5): <span class="nowrap">315–</span>327. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00494925.1982.11972639">10.1080/00494925.1982.11972639</a> (inactive 1 November 2024)<span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=Nearness+Relations+Among+Measures+of+Central+Tendency+and+Dispersion%3A+Part+1&rft.volume=13&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E315-%3C%2Fspan%3E327&rft.date=1982&rft_id=info%3Adoi%2F10.1080%2F00494925.1982.11972639&rft.au=Warren+Page&rft.au=Vedula+N.+Murty&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00494925.1982.11972639%3FjournalCode%3Ducmj19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'Cinneide1990" class="citation journal cs1">O'Cinneide, Colm Art (1990). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00031305.1990.10475743?journalCode=utas20">"The mean is within one standard deviation of any median"</a>. <i>The American Statistician</i>. <b>44</b> (4): <span class="nowrap">292–</span>293. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1990.10475743">10.1080/00031305.1990.10475743</a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=The+mean+is+within+one+standard+deviation+of+any+median&rft.volume=44&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E292-%3C%2Fspan%3E293&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F00031305.1990.10475743&rft.aulast=O%27Cinneide&rft.aufirst=Colm+Art&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.1990.10475743%3FjournalCode%3Dutas20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMallows1991" class="citation journal cs1">Mallows, Colin (August 1991). "Another comment on O'Cinneide". <i>The American Statistician</i>. <b>45</b> (3): 257. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1991.10475815">10.1080/00031305.1991.10475815</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Another+comment+on+O%27Cinneide&rft.volume=45&rft.issue=3&rft.pages=257&rft.date=1991-08&rft_id=info%3Adoi%2F10.1080%2F00031305.1991.10475815&rft.aulast=Mallows&rft.aufirst=Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-PicheRandomVectorsSequences-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-PicheRandomVectorsSequences_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPiché2012" class="citation book cs1">Piché, Robert (2012). <i>Random Vectors and Random Sequences</i>. Lambert Academic Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3659211966" title="Special:BookSources/978-3659211966"><bdi>978-3659211966</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Vectors+and+Random+Sequences&rft.pub=Lambert+Academic+Publishing&rft.date=2012&rft.isbn=978-3659211966&rft.aulast=Pich%C3%A9&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Kemperman-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kemperman_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKemperman1987" class="citation journal cs1">Kemperman, Johannes H. B. (1987). Dodge, Yadolah (ed.). "The median of a finite measure on a Banach space: Statistical data analysis based on the L1-norm and related methods". <i>Papers from the First International Conference Held at Neuchâtel, August 31–September 4, 1987</i>. Amsterdam: North-Holland Publishing Co.: <span class="nowrap">217–</span>230. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0949228">0949228</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Papers+from+the+First+International+Conference+Held+at+Neuch%C3%A2tel%2C+August+31%E2%80%93September+4%2C+1987&rft.atitle=The+median+of+a+finite+measure+on+a+Banach+space%3A+Statistical+data+analysis+based+on+the+L1-norm+and+related+methods&rft.pages=%3Cspan+class%3D%22nowrap%22%3E217-%3C%2Fspan%3E230&rft.date=1987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D949228%23id-name%3DMR&rft.aulast=Kemperman&rft.aufirst=Johannes+H.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-MilasevicDucharme-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-MilasevicDucharme_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMilasevicDucharme1987" class="citation journal cs1">Milasevic, Philip; Ducharme, Gilles R. (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350511">"Uniqueness of the spatial median"</a>. <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a></i>. <b>15</b> (3): <span class="nowrap">1332–</span>1333. <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/10.1214%2Faos%2F1176350511">10.1214/aos/1176350511</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0902264">0902264</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=Uniqueness+of+the+spatial+median&rft.volume=15&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1332-%3C%2Fspan%3E1333&rft.date=1987&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176350511&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D902264%23id-name%3DMR&rft.aulast=Milasevic&rft.aufirst=Philip&rft.au=Ducharme%2C+Gilles+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176350511&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.montefiore.ulg.ac.be/~kvansteen/MATH0008-2/ac20112012/Class3/Chapter2_ac1112_vfinalPartII.pdf">K.Van Steen <i>Notes on probability and statistics</i> </a></span> </li> <li id="cite_note-unimodal-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-unimodal_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBasuDasgupta1997" class="citation journal cs1">Basu, S.; Dasgupta, A. (1997). "The Mean, Median, and Mode of Unimodal Distributions:A Characterization". <i>Theory of Probability and Its Applications</i>. <b>41</b> (2): <span class="nowrap">210–</span>223. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0040585X97975447">10.1137/S0040585X97975447</a>. <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:54593178">54593178</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+of+Probability+and+Its+Applications&rft.atitle=The+Mean%2C+Median%2C+and+Mode+of+Unimodal+Distributions%3AA+Characterization&rft.volume=41&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E210-%3C%2Fspan%3E223&rft.date=1997&rft_id=info%3Adoi%2F10.1137%2FS0040585X97975447&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54593178%23id-name%3DS2CID&rft.aulast=Basu&rft.aufirst=S.&rft.au=Dasgupta%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvon_Hippel2005" class="citation journal cs1">von Hippel, Paul T. (January 2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10691898.2005.11910556">"Mean, Median, and Skew: Correcting a Textbook Rule"</a>. <i>Journal of Statistics Education</i>. <b>13</b> (2). <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/10.1080%2F10691898.2005.11910556">10.1080/10691898.2005.11910556</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1069-1898">1069-1898</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+Statistics+Education&rft.atitle=Mean%2C+Median%2C+and+Skew%3A+Correcting+a+Textbook+Rule&rft.volume=13&rft.issue=2&rft.date=2005-01&rft_id=info%3Adoi%2F10.1080%2F10691898.2005.11910556&rft.issn=1069-1898&rft.aulast=von+Hippel&rft.aufirst=Paul+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%252F10691898.2005.11910556&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGroeneveldMeeden1977" class="citation journal cs1">Groeneveld, Richard A.; Meeden, Glen (August 1977). <a rel="nofollow" class="external text" href="http://www.tandfonline.com/doi/abs/10.1080/00031305.1977.10479215">"The Mode, Median, and Mean Inequality"</a>. <i>The American Statistician</i>. <b>31</b> (3): <span class="nowrap">120–</span>121. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1977.10479215">10.1080/00031305.1977.10479215</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-1305">0003-1305</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=The+Mode%2C+Median%2C+and+Mean+Inequality&rft.volume=31&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E120-%3C%2Fspan%3E121&rft.date=1977-08&rft_id=info%3Adoi%2F10.1080%2F00031305.1977.10479215&rft.issn=0003-1305&rft.aulast=Groeneveld&rft.aufirst=Richard+A.&rft.au=Meeden%2C+Glen&rft_id=http%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.1977.10479215&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Merkle2005-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Merkle2005_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerkle2005" class="citation journal cs1">Merkle, M. (2005). "Jensen's inequality for medians". <i>Statistics & Probability Letters</i>. <b>71</b> (3): <span class="nowrap">277–</span>281. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spl.2004.11.010">10.1016/j.spl.2004.11.010</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+%26+Probability+Letters&rft.atitle=Jensen%27s+inequality+for+medians&rft.volume=71&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E277-%3C%2Fspan%3E281&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2Fj.spl.2004.11.010&rft.aulast=Merkle&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlfred_V._Aho_and_John_E._Hopcroft_and_Jeffrey_D._Ullman1974" class="citation book cs1">Alfred V. Aho and John E. Hopcroft and Jeffrey D. Ullman (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/designanalysisof00ahoarich"><i>The Design and Analysis of Computer Algorithms</i></a></span>. Reading/MA: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-00029-6" title="Special:BookSources/0-201-00029-6"><bdi>0-201-00029-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Design+and+Analysis+of+Computer+Algorithms&rft.place=Reading%2FMA&rft.pub=Addison-Wesley&rft.date=1974&rft.isbn=0-201-00029-6&rft.au=Alfred+V.+Aho+and+John+E.+Hopcroft+and+Jeffrey+D.+Ullman&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdesignanalysisof00ahoarich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span> Here: Section 3.6 "Order Statistics", p.97-99, in particular Algorithm 3.6 and Theorem 3.9.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBentleyMcIlroy1993" class="citation journal cs1">Bentley, Jon L.; McIlroy, M. Douglas (1993). <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.8162">"Engineering a sort function"</a>. <i>Software: Practice and Experience</i>. <b>23</b> (11): <span class="nowrap">1249–</span>1265. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fspe.4380231105">10.1002/spe.4380231105</a>. <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:8822797">8822797</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Software%3A+Practice+and+Experience&rft.atitle=Engineering+a+sort+function&rft.volume=23&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1249-%3C%2Fspan%3E1265&rft.date=1993&rft_id=info%3Adoi%2F10.1002%2Fspe.4380231105&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8822797%23id-name%3DS2CID&rft.aulast=Bentley&rft.aufirst=Jon+L.&rft.au=McIlroy%2C+M.+Douglas&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.14.8162&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRousseeuwBassett1990" class="citation journal cs1">Rousseeuw, Peter J.; Bassett, Gilbert W. Jr. (1990). <a rel="nofollow" class="external text" href="http://wis.kuleuven.be/stat/robust/papers/publications-1990/rousseeuwbassett-remedian-jasa-1990.pdf">"The remedian: a robust averaging method for large data sets"</a> <span class="cs1-format">(PDF)</span>. <i>J. Amer. Statist. Assoc</i>. <b>85</b> (409): <span class="nowrap">97–</span>104. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1990.10475311">10.1080/01621459.1990.10475311</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Amer.+Statist.+Assoc.&rft.atitle=The+remedian%3A+a+robust+averaging+method+for+large+data+sets&rft.volume=85&rft.issue=409&rft.pages=%3Cspan+class%3D%22nowrap%22%3E97-%3C%2Fspan%3E104&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F01621459.1990.10475311&rft.aulast=Rousseeuw&rft.aufirst=Peter+J.&rft.au=Bassett%2C+Gilbert+W.+Jr.&rft_id=http%3A%2F%2Fwis.kuleuven.be%2Fstat%2Frobust%2Fpapers%2Fpublications-1990%2Frousseeuwbassett-remedian-jasa-1990.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Stigler1973-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stigler1973_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stigler1973_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStigler1973" class="citation journal cs1"><a href="/wiki/Stephen_Stigler" title="Stephen Stigler">Stigler, Stephen</a> (December 1973). "Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency". <i>Biometrika</i>. <b>60</b> (3): <span class="nowrap">439–</span>445. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F60.3.439">10.1093/biomet/60.3.439</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/2334992">2334992</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0326872">0326872</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Studies+in+the+History+of+Probability+and+Statistics.+XXXII%3A+Laplace%2C+Fisher+and+the+Discovery+of+the+Concept+of+Sufficiency&rft.volume=60&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E439-%3C%2Fspan%3E445&rft.date=1973-12&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0326872%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2334992%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F60.3.439&rft.aulast=Stigler&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Rider1960-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rider1960_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRider1960" class="citation journal cs1">Rider, Paul R. (1960). "Variance of the median of small samples from several special populations". <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">J. Amer. Statist. Assoc.</a></i> <b>55</b> (289): <span class="nowrap">148–</span>150. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1960.10482056">10.1080/01621459.1960.10482056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Amer.+Statist.+Assoc.&rft.atitle=Variance+of+the+median+of+small+samples+from+several+special+populations&rft.volume=55&rft.issue=289&rft.pages=%3Cspan+class%3D%22nowrap%22%3E148-%3C%2Fspan%3E150&rft.date=1960&rft_id=info%3Adoi%2F10.1080%2F01621459.1960.10482056&rft.aulast=Rider&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Efron1982-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Efron1982_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEfron1982" class="citation book cs1">Efron, B. (1982). <i>The Jackknife, the Bootstrap and other Resampling Plans</i>. Philadelphia: SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0898711797" title="Special:BookSources/0898711797"><bdi>0898711797</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Jackknife%2C+the+Bootstrap+and+other+Resampling+Plans&rft.place=Philadelphia&rft.pub=SIAM&rft.date=1982&rft.isbn=0898711797&rft.aulast=Efron&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Shao1989-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shao1989_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShaoWu1989" class="citation journal cs1">Shao, J.; Wu, C. F. (1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176347263">"A General Theory for Jackknife Variance Estimation"</a>. <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Ann. Stat.</a></i> <b>17</b> (3): <span class="nowrap">1176–</span>1197. <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/10.1214%2Faos%2F1176347263">10.1214/aos/1176347263</a></span>. <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/2241717">2241717</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Stat.&rft.atitle=A+General+Theory+for+Jackknife+Variance+Estimation&rft.volume=17&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1176-%3C%2Fspan%3E1197&rft.date=1989&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176347263&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2241717%23id-name%3DJSTOR&rft.aulast=Shao&rft.aufirst=J.&rft.au=Wu%2C+C.+F.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176347263&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Efron1979-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Efron1979_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEfron1979" class="citation journal cs1">Efron, B. (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">"Bootstrap Methods: Another Look at the Jackknife"</a>. <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Ann. Stat.</a></i> <b>7</b> (1): <span class="nowrap">1–</span>26. <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/10.1214%2Faos%2F1176344552">10.1214/aos/1176344552</a></span>. <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/2958830">2958830</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Stat.&rft.atitle=Bootstrap+Methods%3A+Another+Look+at+the+Jackknife&rft.volume=7&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E26&rft.date=1979&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176344552&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2958830%23id-name%3DJSTOR&rft.aulast=Efron&rft.aufirst=B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176344552&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Hall1988-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hall1988_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHallMartin1988" class="citation journal cs1">Hall, P.; Martin, M. A. (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00356105">"Exact Convergence Rate of Bootstrap Quantile Variance Estimator"</a>. <i>Probab Theory Related Fields</i>. <b>80</b> (2): <span class="nowrap">261–</span>268. <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/10.1007%2FBF00356105">10.1007/BF00356105</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:119701556">119701556</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Probab+Theory+Related+Fields&rft.atitle=Exact+Convergence+Rate+of+Bootstrap+Quantile+Variance+Estimator&rft.volume=80&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E261-%3C%2Fspan%3E268&rft.date=1988&rft_id=info%3Adoi%2F10.1007%2FBF00356105&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119701556%23id-name%3DS2CID&rft.aulast=Hall&rft.aufirst=P.&rft.au=Martin%2C+M.+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF00356105&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Jimenez-Gamero2004-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jimenez-Gamero2004_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJiménez-GameroMunoz-GarcíaPino-Mejías2004" class="citation journal cs1">Jiménez-Gamero, M. D.; Munoz-García, J.; Pino-Mejías, R. (2004). <a rel="nofollow" class="external text" href="http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=14&num=4&art=11">"Reduced bootstrap for the median"</a>. <i>Statistica Sinica</i>. <b>14</b> (4): <span class="nowrap">1179–</span>1198.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistica+Sinica&rft.atitle=Reduced+bootstrap+for+the+median&rft.volume=14&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1179-%3C%2Fspan%3E1198&rft.date=2004&rft.aulast=Jim%C3%A9nez-Gamero&rft.aufirst=M.+D.&rft.au=Munoz-Garc%C3%ADa%2C+J.&rft.au=Pino-Mej%C3%ADas%2C+R.&rft_id=http%3A%2F%2Fwww3.stat.sinica.edu.tw%2Fstatistica%2Fpassword.asp%3Fvol%3D14%26num%3D4%26art%3D11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaindonaldJohn_Braun2010" class="citation book cs1">Maindonald, John; John Braun, W. (2010-05-06). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8bMj8m-4RDQC&q=standard%20error%20of%20the%20median&pg=PA104"><i>Data Analysis and Graphics Using R: An Example-Based Approach</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781139486675" title="Special:BookSources/9781139486675"><bdi>9781139486675</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+Analysis+and+Graphics+Using+R%3A+An+Example-Based+Approach&rft.pub=Cambridge+University+Press&rft.date=2010-05-06&rft.isbn=9781139486675&rft.aulast=Maindonald&rft.aufirst=John&rft.au=John+Braun%2C+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8bMj8m-4RDQC%26q%3Dstandard%2520error%2520of%2520the%2520median%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-HM-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-HM_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HM_36-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HM_36-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHettmanspergerMcKean1998" class="citation book cs1">Hettmansperger, Thomas P.; McKean, Joseph W. (1998). <i>Robust nonparametric statistical methods</i>. Kendall's Library of Statistics. Vol. 5. London: Edward Arnold. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-340-54937-8" title="Special:BookSources/0-340-54937-8"><bdi>0-340-54937-8</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1604954">1604954</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+nonparametric+statistical+methods&rft.place=London&rft.series=Kendall%27s+Library+of+Statistics&rft.pub=Edward+Arnold&rft.date=1998&rft.isbn=0-340-54937-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1604954%23id-name%3DMR&rft.aulast=Hettmansperger&rft.aufirst=Thomas+P.&rft.au=McKean%2C+Joseph+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Small, Christopher G. "A survey of multidimensional medians." International Statistical Review/Revue Internationale de Statistique (1990): 263–277. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><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%2F1403809">10.2307/1403809</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/1403809">1403809</a></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Niinimaa, A., and H. Oja. "Multivariate median." Encyclopedia of statistical sciences (1999).</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">Mosler, Karl. Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach. Vol. 165. Springer Science & Business Media, 2012.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Puri, Madan L.; Sen, Pranab K.; <i>Nonparametric Methods in Multivariate Analysis</i>, John Wiley & Sons, New York, NY, 1971. (Reprinted by Krieger Publishing)</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVardiZhang2000" class="citation journal cs1">Vardi, Yehuda; Zhang, Cun-Hui (2000). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC26449">"The multivariate <i>L</i><sub>1</sub>-median and associated data depth"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. <b>97</b> (4): 1423–1426 (electronic). <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/2000PNAS...97.1423V">2000PNAS...97.1423V</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/10.1073%2Fpnas.97.4.1423">10.1073/pnas.97.4.1423</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1740461">1740461</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC26449">26449</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10677477">10677477</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+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=The+multivariate+L%3Csub%3E1%3C%2Fsub%3E-median+and+associated+data+depth&rft.volume=97&rft.issue=4&rft.pages=1423-1426+%28electronic%29&rft.date=2000&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC26449%23id-name%3DPMC&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1740461%23id-name%3DMR&rft_id=info%3Abibcode%2F2000PNAS...97.1423V&rft_id=info%3Apmid%2F10677477&rft_id=info%3Adoi%2F10.1073%2Fpnas.97.4.1423&rft.aulast=Vardi&rft.aufirst=Yehuda&rft.au=Zhang%2C+Cun-Hui&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC26449&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavisDeGrootHinich1972" class="citation journal cs1">Davis, Otto A.; DeGroot, Morris H.; Hinich, Melvin J. (January 1972). <a rel="nofollow" class="external text" href="https://www.cmu.edu/dietrich/sds/docs/davis/Social%20Preference%20Orderings%20and%20Majority%20Rule.pdf">"Social Preference Orderings and Majority Rule"</a> <span class="cs1-format">(PDF)</span>. <i>Econometrica</i>. <b>40</b> (1): <span class="nowrap">147–</span>157. <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%2F1909727">10.2307/1909727</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/1909727">1909727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Social+Preference+Orderings+and+Majority+Rule&rft.volume=40&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E147-%3C%2Fspan%3E157&rft.date=1972-01&rft_id=info%3Adoi%2F10.2307%2F1909727&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1909727%23id-name%3DJSTOR&rft.aulast=Davis&rft.aufirst=Otto+A.&rft.au=DeGroot%2C+Morris+H.&rft.au=Hinich%2C+Melvin+J.&rft_id=https%3A%2F%2Fwww.cmu.edu%2Fdietrich%2Fsds%2Fdocs%2Fdavis%2FSocial%2520Preference%2520Orderings%2520and%2520Majority%2520Rule.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span> The authors, working in a topic in which uniqueness is assumed, actually use the expression "<i>unique</i> median in all directions".</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarnesDytsoJingboPoor2024" class="citation journal cs1">Barnes, Leighton; Dytso, Alex J.; Jingbo, Liu; Poor, H.Vincent (2024-08-22). "L1 Estimation: On the Optimality of Linear Estimators". <i>IEEE Transactions on Information Theory</i>. <b>70</b> (11): <span class="nowrap">8026–</span>8039. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2024.3440929">10.1109/TIT.2024.3440929</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=L1+Estimation%3A+On+the+Optimality+of+Linear+Estimators&rft.volume=70&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E8026-%3C%2Fspan%3E8039&rft.date=2024-08-22&rft_id=info%3Adoi%2F10.1109%2FTIT.2024.3440929&rft.aulast=Barnes&rft.aufirst=Leighton&rft.au=Dytso%2C+Alex+J.&rft.au=Jingbo%2C+Liu&rft.au=Poor%2C+H.Vincent&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPrattCooperKabir1985" class="citation journal cs1">Pratt, William K.; Cooper, Ted J.; Kabir, Ihtisham (1985-07-11). Corbett, Francis J (ed.). "Pseudomedian Filter". <i>Architectures and Algorithms for Digital Image Processing II</i>. <b>0534</b>: 34. <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/1985SPIE..534...34P">1985SPIE..534...34P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1117%2F12.946562">10.1117/12.946562</a>. <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:173183609">173183609</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Architectures+and+Algorithms+for+Digital+Image+Processing+II&rft.atitle=Pseudomedian+Filter&rft.volume=0534&rft.pages=34&rft.date=1985-07-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A173183609%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1117%2F12.946562&rft_id=info%3Abibcode%2F1985SPIE..534...34P&rft.aulast=Pratt&rft.aufirst=William+K.&rft.au=Cooper%2C+Ted+J.&rft.au=Kabir%2C+Ihtisham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Oja_2010_xiv+232-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-Oja_2010_xiv+232_46-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOja2010" class="citation book cs1">Oja, Hannu (2010). <i>Multivariate nonparametric methods with </i>R<i>: An approach based on spatial signs and ranks</i>. Lecture Notes in Statistics. Vol. 199. New York, NY: Springer. pp. xiv+232. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4419-0468-3">10.1007/978-1-4419-0468-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-0467-6" title="Special:BookSources/978-1-4419-0467-6"><bdi>978-1-4419-0467-6</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2598854">2598854</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multivariate+nonparametric+methods+with+R%3A+An+approach+based+on+spatial+signs+and+ranks&rft.place=New+York%2C+NY&rft.series=Lecture+Notes+in+Statistics&rft.pages=xiv%2B232&rft.pub=Springer&rft.date=2010&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2598854%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-0468-3&rft.isbn=978-1-4419-0467-6&rft.aulast=Oja&rft.aufirst=Hannu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilcox2001" class="citation cs2">Wilcox, Rand R. (2001), "Theil–Sen estimator", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YSFb4QX2UIoC&pg=PA207"><i>Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy</i></a>, Springer-Verlag, pp. <span class="nowrap">207–</span>210, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95157-7" title="Special:BookSources/978-0-387-95157-7"><bdi>978-0-387-95157-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theil%E2%80%93Sen+estimator&rft.btitle=Fundamentals+of+Modern+Statistical+Methods%3A+Substantially+Improving+Power+and+Accuracy&rft.pages=%3Cspan+class%3D%22nowrap%22%3E207-%3C%2Fspan%3E210&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=978-0-387-95157-7&rft.aulast=Wilcox&rft.aufirst=Rand+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYSFb4QX2UIoC%26pg%3DPA207&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span>.</span> </li> <li id="cite_note-Wald1940-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wald1940_48-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWald1940" class="citation journal cs1">Wald, A. (1940). <a rel="nofollow" class="external text" href="http://dml.cz/bitstream/handle/10338.dmlcz/103573/AplMat_20-1975-2_3.pdf">"The Fitting of Straight Lines if Both Variables are Subject to Error"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>11</b> (3): <span class="nowrap">282–</span>300. <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/10.1214%2Faoms%2F1177731868">10.1214/aoms/1177731868</a></span>. <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/2235677">2235677</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=The+Fitting+of+Straight+Lines+if+Both+Variables+are+Subject+to+Error&rft.volume=11&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E282-%3C%2Fspan%3E300&rft.date=1940&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177731868&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2235677%23id-name%3DJSTOR&rft.aulast=Wald&rft.aufirst=A.&rft_id=http%3A%2F%2Fdml.cz%2Fbitstream%2Fhandle%2F10338.dmlcz%2F103573%2FAplMat_20-1975-2_3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Nair1942-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nair1942_49-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNairShrivastava1942" class="citation journal cs1">Nair, K. R.; Shrivastava, M. P. (1942). "On a Simple Method of Curve Fitting". <i>Sankhyā: The Indian Journal of Statistics</i>. <b>6</b> (2): <span class="nowrap">121–</span>132. <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/25047749">25047749</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sankhy%C4%81%3A+The+Indian+Journal+of+Statistics&rft.atitle=On+a+Simple+Method+of+Curve+Fitting&rft.volume=6&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E121-%3C%2Fspan%3E132&rft.date=1942&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25047749%23id-name%3DJSTOR&rft.aulast=Nair&rft.aufirst=K.+R.&rft.au=Shrivastava%2C+M.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Brown1951-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brown1951_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrownMood1951" class="citation book cs1">Brown, G. W.; Mood, A. M. (1951). "On Median Tests for Linear Hypotheses". <i>Proc Second Berkeley Symposium on Mathematical Statistics and Probability</i>. Berkeley, CA: University of California Press. pp. <span class="nowrap">159–</span>166. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0045.08606">0045.08606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+Median+Tests+for+Linear+Hypotheses&rft.btitle=Proc+Second+Berkeley+Symposium+on+Mathematical+Statistics+and+Probability&rft.place=Berkeley%2C+CA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E159-%3C%2Fspan%3E166&rft.pub=University+of+California+Press&rft.date=1951&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0045.08606%23id-name%3DZbl&rft.aulast=Brown&rft.aufirst=G.+W.&rft.au=Mood%2C+A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Tukey1971-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tukey1971_51-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTukey1977" class="citation book cs1">Tukey, J. W. (1977). <a rel="nofollow" class="external text" href="https://archive.org/details/exploratorydataa00tuke_0"><i>Exploratory Data Analysis</i></a>. Reading, MA: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0201076160" title="Special:BookSources/0201076160"><bdi>0201076160</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exploratory+Data+Analysis&rft.place=Reading%2C+MA&rft.pub=Addison-Wesley&rft.date=1977&rft.isbn=0201076160&rft.aulast=Tukey&rft.aufirst=J.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fexploratorydataa00tuke_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Brown-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brown_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrown1947" class="citation journal cs1">Brown, George W. (1947). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177730349">"On Small-Sample Estimation"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>18</b> (4): <span class="nowrap">582–</span>585. <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/10.1214%2Faoms%2F1177730349">10.1214/aoms/1177730349</a></span>. <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/2236236">2236236</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=On+Small-Sample+Estimation&rft.volume=18&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E582-%3C%2Fspan%3E585&rft.date=1947&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177730349&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236236%23id-name%3DJSTOR&rft.aulast=Brown&rft.aufirst=George+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177730349&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Lehmann-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lehmann_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLehmann1951" class="citation journal cs1"><a href="/wiki/Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, Erich L.</a> (1951). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177729549">"A General Concept of Unbiasedness"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>22</b> (4): <span class="nowrap">587–</span>592. <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/10.1214%2Faoms%2F1177729549">10.1214/aoms/1177729549</a></span>. <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/2236928">2236928</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=A+General+Concept+of+Unbiasedness&rft.volume=22&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E587-%3C%2Fspan%3E592&rft.date=1951&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177729549&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236928%23id-name%3DJSTOR&rft.aulast=Lehmann&rft.aufirst=Erich+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177729549&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Birnbaum-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-Birnbaum_54-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBirnbaum1961" class="citation journal cs1"><a href="/wiki/Allan_Birnbaum" title="Allan Birnbaum">Birnbaum, Allan</a> (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177705145">"A Unified Theory of Estimation, I"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>32</b> (1): <span class="nowrap">112–</span>135. <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/10.1214%2Faoms%2F1177705145">10.1214/aoms/1177705145</a></span>. <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/2237612">2237612</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=A+Unified+Theory+of+Estimation%2C+I&rft.volume=32&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E112-%3C%2Fspan%3E135&rft.date=1961&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177705145&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2237612%23id-name%3DJSTOR&rft.aulast=Birnbaum&rft.aufirst=Allan&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177705145&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-vdW-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-vdW_55-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_der_Vaart1961" class="citation journal cs1">van der Vaart, H. Robert (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177705051">"Some Extensions of the Idea of Bias"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>32</b> (2): <span class="nowrap">436–</span>447. <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/10.1214%2Faoms%2F1177705051">10.1214/aoms/1177705051</a></span>. <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/2237754">2237754</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0125674">0125674</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Some+Extensions+of+the+Idea+of+Bias&rft.volume=32&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E436-%3C%2Fspan%3E447&rft.date=1961&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D125674%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2237754%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177705051&rft.aulast=van+der+Vaart&rft.aufirst=H.+Robert&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177705051&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Pf-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pf_56-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPfanzaglwith_the_assistance_of_R._Hamböker1994" class="citation book cs1">Pfanzagl, Johann; with the assistance of R. Hamböker (1994). <i>Parametric Statistical Theory</i>. Walter de Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-11-013863-8" title="Special:BookSources/3-11-013863-8"><bdi>3-11-013863-8</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1291393">1291393</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parametric+Statistical+Theory&rft.pub=Walter+de+Gruyter&rft.date=1994&rft.isbn=3-11-013863-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1291393%23id-name%3DMR&rft.aulast=Pfanzagl&rft.aufirst=Johann&rft.au=with+the+assistance+of+R.+Hamb%C3%B6ker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text">Pfanzagl, Johann. "On optimal median unbiased estimators in the presence of nuisance parameters." The Annals of Statistics (1979): 187–193.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrownCohenStrawderman1976" class="citation journal cs1">Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.aos/1176343543">"A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications"</a>. <i>Ann. Statist</i>. <b>4</b> (4): <span class="nowrap">712–</span>722. <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/10.1214%2Faos%2F1176343543">10.1214/aos/1176343543</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Statist.&rft.atitle=A+Complete+Class+Theorem+for+Strict+Monotone+Likelihood+Ratio+With+Applications&rft.volume=4&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E712-%3C%2Fspan%3E722&rft.date=1976&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176343543&rft.aulast=Brown&rft.aufirst=L.+D.&rft.au=Cohen%2C+Arthur&rft.au=Strawderman%2C+W.+E.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.aos%2F1176343543&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPageBrownCohenStrawderman1976" class="citation journal cs1">Page; Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.aos/1176343543">"A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications"</a>. <i>Ann. Statist</i>. <b>4</b> (4): <span class="nowrap">712–</span>722. <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/10.1214%2Faos%2F1176343543">10.1214/aos/1176343543</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Statist.&rft.atitle=A+Complete+Class+Theorem+for+Strict+Monotone+Likelihood+Ratio+With+Applications&rft.volume=4&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E712-%3C%2Fspan%3E722&rft.date=1976&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176343543&rft.au=Page&rft.au=Brown%2C+L.+D.&rft.au=Cohen%2C+Arthur&rft.au=Strawderman%2C+W.+E.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.aos%2F1176343543&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-:0-60"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_60-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_60-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_60-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_60-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBakkerGravemeijer2006" class="citation journal cs1">Bakker, Arthur; Gravemeijer, Koeno P. E. (2006-06-01). "An Historical Phenomenology of Mean and Median". <i>Educational Studies in Mathematics</i>. <b>62</b> (2): <span class="nowrap">149–</span>168. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10649-006-7099-8">10.1007/s10649-006-7099-8</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1573-0816">1573-0816</a>. <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:143708116">143708116</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Educational+Studies+in+Mathematics&rft.atitle=An+Historical+Phenomenology+of+Mean+and+Median&rft.volume=62&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E149-%3C%2Fspan%3E168&rft.date=2006-06-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143708116%23id-name%3DS2CID&rft.issn=1573-0816&rft_id=info%3Adoi%2F10.1007%2Fs10649-006-7099-8&rft.aulast=Bakker&rft.aufirst=Arthur&rft.au=Gravemeijer%2C+Koeno+P.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdler2014" class="citation web cs1">Adler, Dan (31 December 2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151206134315/http://danadler.com/blog/2014/12/31/talmud-and-modern-economics/">"Talmud and Modern Economics"</a>. <i>Jewish American and Israeli Issues</i>. Archived from <a rel="nofollow" class="external text" href="http://danadler.com/blog/2014/12/31/talmud-and-modern-economics/">the original</a> on 6 December 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">22 February</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Jewish+American+and+Israeli+Issues&rft.atitle=Talmud+and+Modern+Economics&rft.date=2014-12-31&rft.aulast=Adler&rft.aufirst=Dan&rft_id=http%3A%2F%2Fdanadler.com%2Fblog%2F2014%2F12%2F31%2Ftalmud-and-modern-economics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.wisdom.weizmann.ac.il/math/AABeyond12/presentations/Aumann.pdf">Modern Economic Theory in the Talmud</a> by <a href="/wiki/Yisrael_Aumann" class="mw-redirect" title="Yisrael Aumann">Yisrael Aumann</a></span> </li> <li id="cite_note-Eisenhart-63"><span class="mw-cite-backlink">^ <a href="#cite_ref-Eisenhart_63-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Eisenhart_63-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Eisenhart_63-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Eisenhart_63-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEisenhart1971" class="citation speech cs1"><a href="/wiki/Churchill_Eisenhart" title="Churchill Eisenhart">Eisenhart, Churchill</a> (24 August 1971). <a rel="nofollow" class="external text" href="https://www.stat.uchicago.edu/~stigler/eisenhart.pdf"><i>The Development of the Concept of the Best Mean of a Set of Measurements from Antiquity to the Present Day</i></a> <span class="cs1-format">(PDF)</span> (Speech). 131st Annual Meeting of the American Statistical Association. Colorado State University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Development+of+the+Concept+of+the+Best+Mean+of+a+Set+of+Measurements+from+Antiquity+to+the+Present+Day&rft.place=Colorado+State+University&rft.date=1971-08-24&rft.aulast=Eisenhart&rft.aufirst=Churchill&rft_id=https%3A%2F%2Fwww.stat.uchicago.edu%2F~stigler%2Feisenhart.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-:2-64"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_64-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_64-1"><sup><i><b>b</b></i></sup></a></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="http://priceonomics.com/how-the-average-triumphed-over-the-median/">"How the Average Triumphed Over the Median"</a>. <i>Priceonomics</i>. 5 April 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-02-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Priceonomics&rft.atitle=How+the+Average+Triumphed+Over+the+Median&rft.date=2016-04-05&rft_id=http%3A%2F%2Fpriceonomics.com%2Fhow-the-average-triumphed-over-the-median%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSangster2021" class="citation journal cs1">Sangster, Alan (March 2021). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1111/abac.12218">"The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator"</a>. <i>Abacus</i>. <b>57</b> (1): <span class="nowrap">126–</span>152. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fabac.12218">10.1111/abac.12218</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2164%2F16100">2164/16100</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-3072">0001-3072</a>. <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:233917744">233917744</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Abacus&rft.atitle=The+Life+and+Works+of+Luca+Pacioli+%281446%2F7%E2%80%931517%29%2C+Humanist+Educator&rft.volume=57&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E126-%3C%2Fspan%3E152&rft.date=2021-03&rft_id=info%3Ahdl%2F2164%2F16100&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A233917744%23id-name%3DS2CID&rft.issn=0001-3072&rft_id=info%3Adoi%2F10.1111%2Fabac.12218&rft.aulast=Sangster&rft.aufirst=Alan&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fabac.12218&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWrightParsonsMorris1939" class="citation journal cs1">Wright, Edward; Parsons, E. J. S.; Morris, W. F. (1939). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1149920">"Edward Wright and His Work"</a>. <i>Imago Mundi</i>. <b>3</b>: <span class="nowrap">61–</span>71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03085693908591862">10.1080/03085693908591862</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0308-5694">0308-5694</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/1149920">1149920</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Imago+Mundi&rft.atitle=Edward+Wright+and+His+Work&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E61-%3C%2Fspan%3E71&rft.date=1939&rft.issn=0308-5694&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1149920%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F03085693908591862&rft.aulast=Wright&rft.aufirst=Edward&rft.au=Parsons%2C+E.+J.+S.&rft.au=Morris%2C+W.+F.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1149920&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Stigler1986-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stigler1986_68-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStigler1986" class="citation book cs1">Stigler, S. M. (1986). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofstatist00stig"><i>The History of Statistics: The Measurement of Uncertainty Before 1900</i></a>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0674403401" title="Special:BookSources/0674403401"><bdi>0674403401</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Statistics%3A+The+Measurement+of+Uncertainty+Before+1900&rft.pub=Harvard+University+Press&rft.date=1986&rft.isbn=0674403401&rft.aulast=Stigler&rft.aufirst=S.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofstatist00stig&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Laplace1818-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-Laplace1818_69-0">^</a></b></span> <span class="reference-text">Laplace PS de (1818) <i>Deuxième supplément à la Théorie Analytique des Probabilités</i>, Paris, Courcier</span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJaynes2007" class="citation book cs1">Jaynes, E.T. (2007). <i>Probability theory : the logic of science</i> (5. print. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 172. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-59271-0" title="Special:BookSources/978-0-521-59271-0"><bdi>978-0-521-59271-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+theory+%3A+the+logic+of+science&rft.place=Cambridge+%5Bu.a.%5D&rft.pages=172&rft.edition=5.+print.&rft.pub=Cambridge+Univ.+Press&rft.date=2007&rft.isbn=978-0-521-59271-0&rft.aulast=Jaynes&rft.aufirst=E.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHowarth2017" class="citation book cs1">Howarth, Richard (2017). <i>Dictionary of Mathematical Geosciences: With Historical Notes</i>. Springer. p. 374.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Mathematical+Geosciences%3A+With+Historical+Notes&rft.pages=374&rft.pub=Springer&rft.date=2017&rft.aulast=Howarth&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Keynes1921-72"><span class="mw-cite-backlink">^ <a href="#cite_ref-Keynes1921_72-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Keynes1921_72-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Keynes, J.M. (1921) <i><a href="/wiki/A_Treatise_on_Probability" title="A Treatise on Probability">A Treatise on Probability</a></i>. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781596055308" title="Special:BookSources/9781596055308">9781596055308</a> : multiple other reprints)</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStigler2002" class="citation book cs1">Stigler, Stephen M. (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qQusWukdPa4C&q=stigler+%22statistics+on+the+table%22"><i>Statistics on the Table: The History of Statistical Concepts and Methods</i></a>. Harvard University Press. pp. <span class="nowrap">105–</span>7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-00979-0" title="Special:BookSources/978-0-674-00979-0"><bdi>978-0-674-00979-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+on+the+Table%3A+The+History+of+Statistical+Concepts+and+Methods&rft.pages=%3Cspan+class%3D%22nowrap%22%3E105-%3C%2Fspan%3E7&rft.pub=Harvard+University+Press&rft.date=2002&rft.isbn=978-0-674-00979-0&rft.aulast=Stigler&rft.aufirst=Stephen+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqQusWukdPa4C%26q%3Dstigler%2B%2522statistics%2Bon%2Bthe%2Btable%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-Galton1881-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-Galton1881_74-0">^</a></b></span> <span class="reference-text">Galton F (1881) "Report of the Anthropometric Committee" pp 245–260. <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/94448"><i>Report of the 51st Meeting of the British Association for the Advancement of Science</i></a></span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavid1995" class="citation journal cs1">David, H. A. (1995). "First (?) Occurrence of Common Terms in Mathematical Statistics". <i>The American Statistician</i>. <b>49</b> (2): <span class="nowrap">121–</span>133. <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%2F2684625">10.2307/2684625</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-1305">0003-1305</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/2684625">2684625</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=First+%28%3F%29+Occurrence+of+Common+Terms+in+Mathematical+Statistics&rft.volume=49&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E121-%3C%2Fspan%3E133&rft.date=1995&rft.issn=0003-1305&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2684625%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2684625&rft.aulast=David&rft.aufirst=H.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Galton,_Francis"><i>encyclopediaofmath.org</i></a></span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.personal.psu.edu/users/e/c/ecb5/Courses/M475W/WeeklyReadings/Week%2013/DevelopmentOfModernStatistics.pdf"><i>personal.psu.edu</i></a></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=Median&action=edit&section=38" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Median_(in_statistics)">"Median (in statistics)"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Median+%28in+statistics%29&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMedian_%28in_statistics%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.accessecon.com/pubs/EB/2004/Volume3/EB-04C10011A.pdf">Median as a weighted arithmetic mean of all Sample Observations</a></li> <li><a rel="nofollow" class="external text" href="http://www.poorcity.richcity.org/cgi-bin/inequality.cgi">On-line calculator</a></li> <li><a rel="nofollow" class="external text" href="http://www.statcan.gc.ca/edu/power-pouvoir/ch11/median-mediane/5214872-eng.htm">Calculating the median</a></li> <li><a rel="nofollow" class="external text" href="http://mathschallenge.net/index.php?section=problems&show=true&titleid=average_problem">A problem involving the mean, the median, and the mode.</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Statistical_Median"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/StatisticalMedian.html">"Statistical Median"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Statistical+Median&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FStatisticalMedian.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.poorcity.richcity.org/oei/#GiniHooverTheil">Python script</a> for Median computations and <a href="/wiki/Income_inequality_metrics" title="Income inequality metrics">income inequality metrics</a></li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0806.3301">Fast Computation of the Median by Successive Binning</a></li> <li><a rel="nofollow" class="external text" href="http://www.celiagreen.com/charlesmccreery/statistics/meanmedianmode.pdf">'Mean, median, mode and skewness'</a>, A tutorial devised for first-year psychology students at Oxford University, based on a worked example.</li> <li><a rel="nofollow" class="external text" href="https://www.popularmechanics.com/science/math/a28614640/complex-sat-math-problem/">The Complex SAT Math Problem Even the College Board Got Wrong</a>: Andrew Daniels in <i><a href="/wiki/Popular_Mechanics" title="Popular Mechanics">Popular Mechanics</a></i></li></ul> <p><i>This article incorporates material from Median of a distribution on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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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="Statistics636" 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_statistics636" 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 class="mw-selflink selflink">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 href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">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_collection636" 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_inference636" 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_analysis636" 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/Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</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_analysis636" 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 estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</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/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</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/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></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="Applications636" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</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/Biostatistics" title="Biostatistics">Biostatistics</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/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</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/Chemometrics" 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" 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