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<span>Regressions</span> </div> </a> <ul id="toc-Regressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_uses_of_the_word_"error"_in_statistics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_uses_of_the_word_"error"_in_statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other uses of the word "error" in statistics</span> </div> </a> <ul id="toc-Other_uses_of_the_word_"error"_in_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Errors and residuals</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 20 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-20" 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">20 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Cheng-chha" title="Cheng-chha – Minnan" lang="nan" hreflang="nan" data-title="Cheng-chha" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Errors_i_residus" title="Errors i residus – Catalan" lang="ca" hreflang="ca" data-title="Errors i residus" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/St%C3%B6rgr%C3%B6%C3%9Fe_und_Residuum" title="Störgröße und Residuum – German" lang="de" hreflang="de" data-title="Störgröße und Residuum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%B7%D8%A7%D9%87%D8%A7_%D9%88_%D8%A8%D8%A7%D9%82%DB%8C%D9%85%D8%A7%D9%86%D8%AF%D9%87%E2%80%8C%D9%87%D8%A7" title="خطاها و باقیمانده‌ها – Persian" lang="fa" hreflang="fa" data-title="خطاها و باقیمانده‌ها" 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/R%C3%A9sidu_(statistiques)" title="Résidu (statistiques) – French" lang="fr" hreflang="fr" data-title="Résidu (statistiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%B0%A8%EC%99%80_%EC%9E%94%EC%B0%A8" title="오차와 잔차 – Korean" lang="ko" hreflang="ko" data-title="오차와 잔차" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Galat" title="Galat – Indonesian" lang="id" hreflang="id" data-title="Galat" 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-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atsitiktin%C4%97_paklaida" title="Atsitiktinė paklaida – Lithuanian" lang="lt" hreflang="lt" data-title="Atsitiktinė paklaida" 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/Hib%C3%A1k_%C3%A9s_rezidu%C3%A1lisok" title="Hibák és reziduálisok – Hungarian" lang="hu" hreflang="hu" data-title="Hibák és reziduálisok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ralat" title="Ralat – Malay" lang="ms" hreflang="ms" data-title="Ralat" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AA%A4%E5%B7%AE" title="誤差 – Japanese" lang="ja" hreflang="ja" data-title="誤差" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_b%C5%82%C4%99d%C3%B3w" title="Rachunek błędów – Polish" lang="pl" hreflang="pl" data-title="Rachunek błędów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_dos_erros" title="Teoria dos erros – Portuguese" lang="pt" hreflang="pt" data-title="Teoria dos erros" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Errors_and_residuals_in_statistics" title="Errors and residuals in statistics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Errors and residuals in statistics" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Errors_and_residuals_in_statistics" title="Errors and residuals in statistics – Sundanese" lang="su" hreflang="su" data-title="Errors and residuals in statistics" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Slumpfel" title="Slumpfel – Swedish" lang="sv" hreflang="sv" data-title="Slumpfel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hata_terimi" title="Hata terimi – Turkish" lang="tr" hreflang="tr" data-title="Hata terimi" 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%9F%D0%BE%D1%85%D0%B8%D0%B1%D0%BA%D0%B8_%D1%82%D0%B0_%D0%B7%D0%B0%D0%BB%D0%B8%D1%88%D0%BA%D0%B8" title="Похибки та залишки – Ukrainian" lang="uk" hreflang="uk" data-title="Похибки та залишки" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AE%98%E5%B7%AE%E5%90%8C%E8%AA%A4%E5%B7%AE" title="殘差同誤差 – Cantonese" lang="yue" hreflang="yue" data-title="殘差同誤差" 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/%E8%AF%AF%E5%B7%AE" title="误差 – Chinese" lang="zh" hreflang="zh" data-title="误差" 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/Q1502698#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading"> Models</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple regression</a></li> <li><a href="/wiki/Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li> <li><a href="/wiki/Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li> <li><a href="/wiki/Discrete_choice" title="Discrete choice">Discrete choice</a></li> <li><a href="/wiki/Binomial_regression" title="Binomial regression">Binomial regression</a></li> <li><a href="/wiki/Binary_regression" title="Binary regression">Binary regression</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li> <li><a href="/wiki/Mixed_logit" title="Mixed logit">Mixed logit</a></li> <li><a href="/wiki/Probit_model" title="Probit model">Probit</a></li> <li><a href="/wiki/Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li> <li><a href="/wiki/Ordered_logit" title="Ordered logit">Ordered logit</a></li> <li><a href="/wiki/Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Multilevel_model" title="Multilevel model">Multilevel model</a></li> <li><a href="/wiki/Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li> <li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Linear mixed-effects model</a></li> <li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <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/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Principal_component_regression" title="Principal component regression">Principal components</a></li> <li><a href="/wiki/Least-angle_regression" title="Least-angle regression">Least angle</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Errors-in-variables_models" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Linear_least_squares" title="Linear least squares">Linear</a></li> <li><a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted</a></li> <li><a href="/wiki/Generalized_least_squares" title="Generalized least squares">Generalized</a></li> <li><a href="/wiki/Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li> <li><a href="/wiki/Total_least_squares" title="Total least squares">Total</a></li> <li><a href="/wiki/Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li> <li><a href="/wiki/Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li> <li><a href="/wiki/Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li> <li><a href="/wiki/Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li> <li><a href="/wiki/Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li> <li><a class="mw-selflink selflink">Errors and residuals</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><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" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a 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:Regression_bar" title="Template:Regression bar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Regression_bar" title="Template talk:Regression bar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Regression_bar" title="Special:EditPage/Template:Regression bar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimization</a>, <b>errors</b> and <b>residuals</b> are two closely related and easily confused measures of the <a href="/wiki/Deviation_(statistics)" title="Deviation (statistics)">deviation</a> of an <a href="/wiki/Observed_value" class="mw-redirect" title="Observed value">observed value</a> of an <a href="/wiki/Elementary_event" title="Elementary event">element</a> of a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">statistical sample</a> from its "<a href="/wiki/True_value" class="mw-redirect" title="True value">true value</a>" (not necessarily observable). The <b>error</b> of an <a href="/wiki/Observation" title="Observation">observation</a> is the deviation of the observed value from the true value of a quantity of interest (for example, a <a href="/wiki/Population_mean" class="mw-redirect" title="Population mean">population mean</a>). The <b>residual</b> is the difference between the observed value and the <i><a href="/wiki/Estimation" title="Estimation">estimated</a></i> value of the quantity of interest (for example, a <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a>). The distinction is most important in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, where the concepts are sometimes called the <b>regression errors</b> and <b>regression residuals</b> and where they lead to the concept of <a href="/wiki/Studentized_residual" title="Studentized residual">studentized residuals</a>. In <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, "errors" are also called <b>disturbances</b>.<sup id="cite_ref-Kennedy_2008_p._576_1-0" class="reference"><a href="#cite_note-Kennedy_2008_p._576-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wooldridge_2019_p._57_2-0" class="reference"><a href="#cite_note-Wooldridge_2019_p._57-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Das_2019_p._7_3-0" class="reference"><a href="#cite_note-Das_2019_p._7-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose there is a series of observations from a <a href="/wiki/Univariate_distribution" title="Univariate distribution">univariate distribution</a> and we want to estimate the <a href="/wiki/Mean" title="Mean">mean</a> of that distribution (the so-called <a href="/wiki/Location_model_(statistics)" class="mw-redirect" title="Location model (statistics)">location model</a>). In this case, the errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean. </p><p>A <b>statistical error</b> (or <b>disturbance</b>) is the amount by which an observation differs from its <a href="/wiki/Expected_value" title="Expected value">expected value</a>, the latter being based on the whole <a href="/wiki/Statistical_population" title="Statistical population">population</a> from which the statistical unit was chosen randomly. For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters. The expected value, being the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean</a> of the entire population, is typically unobservable, and hence the statistical error cannot be observed either. </p><p>A <b>residual</b> (or fitting deviation), on the other hand, is an observable <i>estimate</i> of the unobservable statistical error. Consider the previous example with men's heights and suppose we have a random sample of <i>n</i> people. The <i><a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a></i> could serve as a good estimator of the <i>population</i> mean. Then we have: </p> <ul><li>The difference between the height of each man in the sample and the unobservable <i>population</i> mean is a <i>statistical error</i>, whereas</li> <li>The difference between the height of each man in the sample and the observable <i>sample</i> mean is a <i>residual</i>.</li></ul> <p>Note that, because of the definition of the sample mean, the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily <i>not <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a></i>. The statistical errors, on the other hand, are independent, and their sum within the random sample is <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> not zero. </p><p>One can standardize statistical errors (especially of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>) in a <a href="/wiki/Z-score" class="mw-redirect" title="Z-score">z-score</a> (or "standard score"), and standardize residuals in a <a href="/wiki/T-statistic" title="T-statistic"><i>t</i>-statistic</a>, or more generally <a href="/wiki/Studentized_residuals" class="mw-redirect" title="Studentized residuals">studentized residuals</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_univariate_distributions">In univariate distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=2" title="Edit section: In univariate distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we assume a normally distributed population with mean μ and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> σ, and choose individuals independently, then we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,X_{n}\sim N\left(\mu ,\sigma ^{2}\right)\,}"> <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> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n}\sim N\left(\mu ,\sigma ^{2}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ab55ea429cb225df2b7f1c0ac9834d007dfbe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.186ex; height:3.343ex;" alt="{\displaystyle X_{1},\dots ,X_{n}\sim N\left(\mu ,\sigma ^{2}\right)\,}"></span></dd></dl> <p>and the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">sample mean</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}={X_{1}+\cdots +X_{n} \over n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}={X_{1}+\cdots +X_{n} \over n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cb1f624d4284a67824c73ccce64933b9823110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.591ex; height:5.176ex;" alt="{\displaystyle {\overline {X}}={X_{1}+\cdots +X_{n} \over n}}"></span></dd></dl> <p>is a random variable distributed such that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∼</mo> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18ab4cfa1ecc1e16b9d82a36ab2ce9d2b0ef311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.792ex; height:6.343ex;" alt="{\displaystyle {\overline {X}}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right).}"></span></dd></dl> <p>The <i>statistical errors</i> are then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}=X_{i}-\mu ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}=X_{i}-\mu ,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21922fac588f8575935a22418ac103af9671e63d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.982ex; height:2.676ex;" alt="{\displaystyle e_{i}=X_{i}-\mu ,\,}"></span></dd></dl> <p>with <a href="/wiki/Expected_Value" class="mw-redirect" title="Expected Value">expected</a> values of zero,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> whereas the <i>residuals</i> are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}=X_{i}-{\overline {X}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}=X_{i}-{\overline {X}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07eafb01c7e0764f4815e15a2ecc8285336c22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.289ex; height:3.343ex;" alt="{\displaystyle r_{i}=X_{i}-{\overline {X}}.}"></span></dd></dl> <p>The sum of squares of the <b>statistical errors</b>, divided by <i>σ</i><sup>2</sup>, has a <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <i>n</i> <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degrees of freedom</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}e_{i}^{2}\sim \chi _{n}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>∼</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}e_{i}^{2}\sim \chi _{n}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8022e0c14e2ec5f73afcdabe43b2edd0f4efc03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.907ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}e_{i}^{2}\sim \chi _{n}^{2}.}"></span></dd></dl> <p>However, this quantity is not observable as the population mean is unknown. The sum of squares of the <b>residuals</b>, on the other hand, is observable. The quotient of that sum by σ<sup>2</sup> has a chi-squared distribution with only <i>n</i> − 1 degrees of freedom: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}r_{i}^{2}\sim \chi _{n-1}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>∼</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}r_{i}^{2}\sim \chi _{n-1}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f156f97ae7cd6bbbb186c8e47782c31d04d5793d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.972ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}r_{i}^{2}\sim \chi _{n-1}^{2}.}"></span></dd></dl> <p>This difference between <i>n</i> and <i>n</i> − 1 degrees of freedom results in <a href="/wiki/Bessel%27s_correction" title="Bessel's correction">Bessel's correction</a> for the estimation of <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> of a population with unknown mean and unknown variance. No correction is necessary if the population mean is known. </p> <div class="mw-heading mw-heading3"><h3 id="Remark">Remark</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=3" title="Edit section: Remark"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is remarkable that the <a href="/wiki/Squared_deviations" class="mw-redirect" title="Squared deviations">sum of squares of the residuals</a> and the sample mean can be shown to be independent of each other, using, e.g. <a href="/wiki/Basu%27s_theorem" title="Basu's theorem">Basu's theorem</a>. That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {{\overline {X}}_{n}-\mu _{0}}{S_{n}/{\sqrt {n}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {{\overline {X}}_{n}-\mu _{0}}{S_{n}/{\sqrt {n}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f1481877e74d51b3122bbc63baefe3bde195d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.864ex; height:7.176ex;" alt="{\displaystyle T={\frac {{\overline {X}}_{n}-\mu _{0}}{S_{n}/{\sqrt {n}}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}_{n}-\mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}-\mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0e8ac9d45ec33bebf2026359e9a4cef369a033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.646ex; height:3.509ex;" alt="{\displaystyle {\overline {X}}_{n}-\mu _{0}}"></span> represents the errors, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span> represents the sample standard deviation for a sample of size <i>n</i>, and unknown <i>σ</i>, and the denominator term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}/{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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 S_{n}/{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5396ff5c47efaf10b53fbe26f009b660e5ae3c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.136ex; height:3.009ex;" alt="{\displaystyle S_{n}/{\sqrt {n}}}"></span> accounts for the standard deviation of the errors according to:<sup id="cite_ref-modernintro_5-0" class="reference"><a href="#cite_note-modernintro-5"><span class="cite-bracket">[</span>5<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 \operatorname {Var} \left({\overline {X}}_{n}\right)={\frac {\sigma ^{2}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} \left({\overline {X}}_{n}\right)={\frac {\sigma ^{2}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1737d21a7124203b2b3fa6a855a2ca2c12beaf52" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.262ex; height:5.676ex;" alt="{\displaystyle \operatorname {Var} \left({\overline {X}}_{n}\right)={\frac {\sigma ^{2}}{n}}}"></span> </p><p>The <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> of the numerator and the denominator separately depend on the value of the unobservable population standard deviation <i>σ</i>, but <i>σ</i> appears in both the numerator and the denominator and cancels. That is fortunate because it means that even though we do not know <i>σ</i>, we know the probability distribution of this quotient: it has a <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t-distribution</a> with <i>n</i> − 1 degrees of freedom. We can therefore use this quotient to find a <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for <i>μ</i>. This t-statistic can be interpreted as "the number of standard errors away from the regression line."<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Regressions">Regressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=4" title="Edit section: Regressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, the distinction between <i>errors</i> and <i>residuals</i> is subtle and important, and leads to the concept of <a href="/wiki/Studentized_residual" title="Studentized residual">studentized residuals</a>. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the <i>fitted</i> function are the residuals. If the linear model is applicable, a scatterplot of residuals plotted against the independent variable should be random about zero with no trend to the residuals.<sup id="cite_ref-modernintro_5-1" class="reference"><a href="#cite_note-modernintro-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> If the data exhibit a trend, the regression model is likely incorrect; for example, the true function may be a quadratic or higher order polynomial. If they are random, or have no trend, but "fan out" - they exhibit a phenomenon called <a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">heteroscedasticity</a>. If all of the residuals are equal, or do not fan out, they exhibit <a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">homoscedasticity</a>. </p><p>However, a terminological difference arises in the expression <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a> (MSE). The mean squared error of a regression is a number computed from the sum of squares of the computed <i>residuals</i>, and not of the unobservable <i>errors</i>. If that sum of squares is divided by <i>n</i>, the number of observations, the result is the mean of the squared residuals. Since this is a <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">biased</a> estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by <i>df</i> = <i>n</i> − <i>p</i> − 1, instead of <i>n</i>, where <i>df</i> is the number of <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degrees of freedom</a> (<i>n</i> minus the number of parameters (excluding the intercept) p being estimated - 1). This forms an unbiased estimate of the variance of the unobserved errors, and is called the mean squared error.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Another method to calculate the mean square of error when analyzing the variance of linear regression using a technique like that used in <a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a> (they are the same because ANOVA is a type of regression), the sum of squares of the residuals (aka sum of squares of the error) is divided by the degrees of freedom (where the degrees of freedom equal <i>n</i> − <i>p</i> − 1, where <i>p</i> is the number of parameters estimated in the model (one for each variable in the regression equation, not including the intercept)). One can then also calculate the mean square of the model by dividing the sum of squares of the model minus the degrees of freedom, which is just the number of parameters. Then the F value can be calculated by dividing the mean square of the model by the mean square of the error, and we can then determine significance (which is why you want the mean squares to begin with.).<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><p>However, because of the behavior of the process of regression, the <i>distributions</i> of residuals at different data points (of the input variable) may vary <i>even if</i> the errors themselves are identically distributed. Concretely, in a <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a> where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be <i>higher</i> than the variability of residuals at the ends of the domain:<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> linear regressions fit endpoints better than the middle. This is also reflected in the <a href="/wiki/Influence_function_(statistics)" class="mw-redirect" title="Influence function (statistics)">influence functions</a> of various data points on the <a href="/wiki/Regression_coefficient" class="mw-redirect" title="Regression coefficient">regression coefficients</a>: endpoints have more influence. </p><p>Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of <i>residuals,</i> which is called <a href="/wiki/Studentizing" class="mw-redirect" title="Studentizing">studentizing</a>. This is particularly important in the case of detecting <a href="/wiki/Outliers" class="mw-redirect" title="Outliers">outliers</a>, where the case in question is somehow different from the others in a dataset. For example, a large residual may be expected in the middle of the domain, but considered an outlier at the end of the domain. </p> <div class="mw-heading mw-heading2"><h2 id="Other_uses_of_the_word_"error"_in_statistics"><span id="Other_uses_of_the_word_.22error.22_in_statistics"></span>Other uses of the word "error" in statistics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=5" title="Edit section: Other uses of the word "error" in statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">Bias (statistics)</a></div> <p>The use of the term "error" as discussed in the sections above is in the sense of a deviation of a value from a hypothetical unobserved value. At least two other uses also occur in statistics, both referring to observable <a href="/wiki/Prediction_error" class="mw-redirect" title="Prediction error">prediction errors</a>: </p><p>The <i><a href="/wiki/Mean_square_error" class="mw-redirect" title="Mean square error"><i>mean squared error</i></a></i> (MSE) refers to the amount by which the values predicted by an estimator differ from the quantities being estimated (typically outside the sample from which the model was estimated). The <i><a href="/wiki/Root_mean_square_deviation" title="Root mean square deviation">root mean square error</a></i> (RMSE) is the square-root of MSE. The <i>sum of squares of errors</i> (SSE) is the MSE multiplied by the sample size. </p><p><i><a href="/wiki/Sum_of_squares_of_residuals" class="mw-redirect" title="Sum of squares of residuals">Sum of squares of residuals</a></i> (SSR) is the sum of the squares of the deviations of the actual values from the predicted values, within the sample used for estimation. This is the basis for the <a href="/wiki/Least_squares" title="Least squares">least squares</a> estimate, where the regression coefficients are chosen such that the SSR is minimal (i.e. its derivative is zero). </p><p>Likewise, the <i><a href="/wiki/Sum_of_absolute_errors" class="mw-redirect" title="Sum of absolute errors">sum of absolute errors</a></i> (SAE) is the sum of the absolute values of the residuals, which is minimized in the <a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">least absolute deviations</a> approach to regression. </p><p>The <b>mean error</b> (ME) is the bias. The <i>mean residual</i> (MR) is always zero for least-squares estimators. </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=Errors_and_residuals&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .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{clear:left;float:left;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, 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class="mw-redirect" title="Consensus forecasts">Consensus forecasts</a></li> <li><a href="/wiki/Error_detection_and_correction" title="Error detection and correction">Error detection and correction</a></li> <li><a href="/wiki/Explained_sum_of_squares" title="Explained sum of squares">Explained sum of squares</a></li> <li><a href="/wiki/Innovation_(signal_processing)" title="Innovation (signal processing)">Innovation (signal processing)</a></li> <li><a href="/wiki/Lack-of-fit_sum_of_squares" title="Lack-of-fit sum of squares">Lack-of-fit sum of squares</a></li> <li><a href="/wiki/Margin_of_error" title="Margin of error">Margin of error</a></li> <li><a href="/wiki/Mean_absolute_error" title="Mean absolute error">Mean absolute error</a></li> <li><a href="/wiki/Observational_error" title="Observational error">Observational error</a></li> <li><a href="/wiki/Propagation_of_error" class="mw-redirect" title="Propagation of error">Propagation of error</a></li> <li><a href="/wiki/Probable_error" title="Probable error">Probable error</a></li> <li><a href="/wiki/Random_and_systematic_errors" class="mw-redirect" title="Random and systematic errors">Random and systematic errors</a></li> <li><a href="/wiki/Reduced_chi-squared_statistic" title="Reduced chi-squared statistic">Reduced chi-squared statistic</a></li> <li><a href="/wiki/Regression_dilution" title="Regression dilution">Regression dilution</a></li> <li><a href="/wiki/Root_mean_square_deviation" title="Root mean square deviation">Root mean square deviation</a></li> <li><a href="/wiki/Sampling_error" title="Sampling error">Sampling error</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">Type I and type II errors</a></li></ul> </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=Errors_and_residuals&action=edit&section=7" title="Edit section: References"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Kennedy_2008_p._576-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kennedy_2008_p._576_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKennedy2008" class="citation book cs1">Kennedy, P. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=69MDEAAAQBAJ&pg=PA576"><i>A Guide to Econometrics</i></a>. Wiley. p. 576. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4051-8257-7" title="Special:BookSources/978-1-4051-8257-7"><bdi>978-1-4051-8257-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+Econometrics&rft.pages=576&rft.pub=Wiley&rft.date=2008&rft.isbn=978-1-4051-8257-7&rft.aulast=Kennedy&rft.aufirst=P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D69MDEAAAQBAJ%26pg%3DPA576&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-Wooldridge_2019_p._57-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wooldridge_2019_p._57_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWooldridge2019" class="citation book cs1">Wooldridge, J.M. (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TONhEAAAQBAJ&pg=PA57"><i>Introductory Econometrics: A Modern Approach</i></a>. Cengage Learning. p. 57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-337-67133-0" title="Special:BookSources/978-1-337-67133-0"><bdi>978-1-337-67133-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Econometrics%3A+A+Modern+Approach&rft.pages=57&rft.pub=Cengage+Learning&rft.date=2019&rft.isbn=978-1-337-67133-0&rft.aulast=Wooldridge&rft.aufirst=J.M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTONhEAAAQBAJ%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-Das_2019_p._7-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Das_2019_p._7_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDas2019" class="citation book cs1">Das, P. (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rK6tDwAAQBAJ&pg=PA7"><i>Econometrics in Theory and Practice: Analysis of Cross Section, Time Series and Panel Data with Stata 15.1</i></a>. Springer Singapore. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-329-019-8" title="Special:BookSources/978-981-329-019-8"><bdi>978-981-329-019-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econometrics+in+Theory+and+Practice%3A+Analysis+of+Cross+Section%2C+Time+Series+and+Panel+Data+with+Stata+15.1&rft.pages=7&rft.pub=Springer+Singapore&rft.date=2019&rft.isbn=978-981-329-019-8&rft.aulast=Das&rft.aufirst=P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrK6tDwAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWetherill,_G._Barrie.1981" class="citation book cs1">Wetherill, G. Barrie. (1981). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/intermediatestat0000weth"><i>Intermediate statistical methods</i></a></span>. London: Chapman and Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-16440-X" title="Special:BookSources/0-412-16440-X"><bdi>0-412-16440-X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/7779780">7779780</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intermediate+statistical+methods&rft.place=London&rft.pub=Chapman+and+Hall&rft.date=1981&rft_id=info%3Aoclcnum%2F7779780&rft.isbn=0-412-16440-X&rft.au=Wetherill%2C+G.+Barrie.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintermediatestat0000weth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-modernintro-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-modernintro_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-modernintro_5-1"><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="CITEREFFrederik_Michel_DekkingCornelis_KraaikampHendrik_Paul_LopuhaäLudolf_Erwin_Meester2005" class="citation book cs1">Frederik Michel Dekking; Cornelis Kraaikamp; Hendrik Paul Lopuhaä; Ludolf Erwin Meester (2005-06-15). <i>A modern introduction to probability and statistics : understanding why and how</i>. London: Springer London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-896-1" title="Special:BookSources/978-1-85233-896-1"><bdi>978-1-85233-896-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/262680588">262680588</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+introduction+to+probability+and+statistics+%3A+understanding+why+and+how&rft.place=London&rft.pub=Springer+London&rft.date=2005-06-15&rft_id=info%3Aoclcnum%2F262680588&rft.isbn=978-1-85233-896-1&rft.au=Frederik+Michel+Dekking&rft.au=Cornelis+Kraaikamp&rft.au=Hendrik+Paul+Lopuha%C3%A4&rft.au=Ludolf+Erwin+Meester&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_BruceAndrew_Bruce2017" class="citation book cs1">Peter Bruce; Andrew Bruce (2017-05-10). <i>Practical statistics for data scientists : 50 essential concepts</i> (First ed.). Sebastopol, CA: O'Reilly Media Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4919-5296-2" title="Special:BookSources/978-1-4919-5296-2"><bdi>978-1-4919-5296-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/987251007">987251007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+statistics+for+data+scientists+%3A+50+essential+concepts&rft.place=Sebastopol%2C+CA&rft.edition=First&rft.pub=O%27Reilly+Media+Inc&rft.date=2017-05-10&rft_id=info%3Aoclcnum%2F987251007&rft.isbn=978-1-4919-5296-2&rft.au=Peter+Bruce&rft.au=Andrew+Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteelTorrie1960" class="citation book cs1">Steel, Robert G. D.; Torrie, James H. (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesproce00stee"><i>Principles and Procedures of Statistics, with Special Reference to Biological Sciences</i></a></span>. McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesproce00stee/page/288">288</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+and+Procedures+of+Statistics%2C+with+Special+Reference+to+Biological+Sciences&rft.pages=288&rft.pub=McGraw-Hill&rft.date=1960&rft.aulast=Steel&rft.aufirst=Robert+G.+D.&rft.au=Torrie%2C+James+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesproce00stee&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" 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="CITEREFZelterman2010" class="citation book cs1">Zelterman, Daniel (2010). <i>Applied linear models with SAS</i> (Online-Ausg. ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521761598" title="Special:BookSources/9780521761598"><bdi>9780521761598</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+linear+models+with+SAS&rft.place=Cambridge&rft.edition=Online-Ausg.&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=9780521761598&rft.aulast=Zelterman&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" 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://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_OpenIntro_Statistics_(Diez_et_al)./7%3A_Introduction_to_Linear_Regression/7.3%3A_Types_of_Outliers_in_Linear_Regression">"7.3: Types of Outliers in Linear Regression"</a>. <i>Statistics LibreTexts</i>. 2013-11-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Statistics+LibreTexts&rft.atitle=7.3%3A+Types+of+Outliers+in+Linear+Regression&rft.date=2013-11-21&rft_id=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_OpenIntro_Statistics_%28Diez_et_al%29.%2F7%253A_Introduction_to_Linear_Regression%2F7.3%253A_Types_of_Outliers_in_Linear_Regression&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errors_and_residuals&action=edit&section=8" title="Edit section: Further reading"><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 id="CITEREFCookWeisberg1982" class="citation book cs1">Cook, R. Dennis; Weisberg, Sanford (1982). <a rel="nofollow" class="external text" href="http://www.stat.umn.edu/rir/"><i>Residuals and Influence in Regression</i></a> (Repr. ed.). New York: <a href="/wiki/Chapman_and_Hall" class="mw-redirect" title="Chapman and Hall">Chapman and Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/041224280X" title="Special:BookSources/041224280X"><bdi>041224280X</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">23 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=Residuals+and+Influence+in+Regression.&rft.place=New+York&rft.edition=Repr.&rft.pub=Chapman+and+Hall&rft.date=1982&rft.isbn=041224280X&rft.aulast=Cook&rft.aufirst=R.+Dennis&rft.au=Weisberg%2C+Sanford&rft_id=http%3A%2F%2Fwww.stat.umn.edu%2Frir%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxSnell1968" class="citation journal cs1"><a href="/wiki/David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, David R.</a>; <a href="/wiki/Joyce_Snell" title="Joyce Snell">Snell, E. Joyce</a> (1968). "A general definition of residuals". <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" class="mw-redirect" title="Journal of the Royal Statistical Society, Series B">Journal of the Royal Statistical Society, Series B</a></i>. <b>30</b> (2): 248–275. <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/2984505">2984505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=A+general+definition+of+residuals&rft.volume=30&rft.issue=2&rft.pages=248-275&rft.date=1968&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2984505%23id-name%3DJSTOR&rft.aulast=Cox&rft.aufirst=David+R.&rft.au=Snell%2C+E.+Joyce&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisberg1985" class="citation book cs1">Weisberg, Sanford (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yRrvAAAAMAAJ&q=editions:UMM1U2yvYVUC"><i>Applied Linear Regression</i></a> (2nd ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471879572" title="Special:BookSources/9780471879572"><bdi>9780471879572</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">23 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=Applied+Linear+Regression&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1985&rft.isbn=9780471879572&rft.aulast=Weisberg&rft.aufirst=Sanford&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyRrvAAAAMAAJ%26q%3Deditions%3AUMM1U2yvYVUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></li> <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=Errors,_theory_of">"Errors, theory of"</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=Errors%2C+theory+of&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DErrors%2C_theory_of&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrors+and+residuals" class="Z3988"></span></li></ul> <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=Errors_and_residuals&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Least_squares_and_regression_analysis" title="Template:Least squares and regression analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Least_squares_and_regression_analysis" title="Template talk:Least squares and regression analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Least_squares_and_regression_analysis" title="Special:EditPage/Template:Least squares and regression analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Least_squares_and_regression_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Least_squares" title="Least squares">Least squares</a> and <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_statistics" title="Computational statistics">Computational statistics</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/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Linear_least_squares_(mathematics)" class="mw-redirect" title="Linear least squares (mathematics)">Linear least squares</a></li> <li><a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear least squares</a></li> <li><a href="/wiki/Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted least squares</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation and dependence</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/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> (<a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's rho</a></li> <li><a href="/wiki/Kendall_tau_rank_correlation_coefficient" class="mw-redirect" title="Kendall tau rank correlation coefficient">Kendall's tau</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></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</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/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/Partial_least_squares_regression" title="Partial least squares regression">Partial least squares</a></li> <li><a href="/wiki/Total_least_squares" title="Total least squares">Total least squares</a></li> <li><a href="/wiki/Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Regression as a <br /><a href="/wiki/Statistical_model" title="Statistical model">statistical model</a></th><td class="navbox-list-with-group 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:1%"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</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/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/Generalized_least_squares" title="Generalized least squares">Generalized least squares</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Predictor structure</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/Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li> <li><a href="/wiki/Growth_curve_(statistics)" title="Growth curve (statistics)">Growth curve (statistics)</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented regression</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-standard</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/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/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-normal errors</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/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li> <li><a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Decomposition of variance</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/Analysis_of_variance" title="Analysis of variance">Analysis of variance</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 AOV</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Model exploration</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/Stepwise_regression" title="Stepwise regression">Stepwise regression</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Mallows%27s_Cp" title="Mallows's Cp">Mallows's <i>C<sub>p</sub></i></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> <li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</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/Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li> <li><a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">Errors and residuals</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Minimum_mean-square_error" class="mw-redirect" title="Minimum mean-square error">Minimum mean-square error</a></li> <li><a href="/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem" title="Frisch–Waugh–Lovell theorem">Frisch–Waugh–Lovell theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</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/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian design</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical</a> <a href="/wiki/Approximation_theory" title="Approximation theory">approximation</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/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a></li> <li><a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">Orthogonal polynomials</a></li> <li><a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a></li> <li><a href="/wiki/Chebyshev_nodes" title="Chebyshev nodes">Chebyshev nodes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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/Curve_fitting" title="Curve fitting">Curve fitting</a></li> <li><a href="/wiki/Calibration_curve" title="Calibration curve">Calibration curve</a></li> <li><a href="/wiki/Numerical_smoothing_and_differentiation" class="mw-redirect" title="Numerical smoothing and differentiation">Numerical smoothing and differentiation</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li> <li><a href="/wiki/Moving_least_squares" title="Moving least squares">Moving least squares</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Category:Regression_analysis" title="Category:Regression analysis">Regression analysis category</a></li> <li><a href="/wiki/Category:Statistics" title="Category:Statistics">Statistics category</a></li> <li><span class="nowrap"><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/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-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/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Statistics outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Statistics topics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a 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_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier 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="Applications" 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 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