Effect size - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Effect size - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"d81c16d7-19f9-455a-b4ab-f8cbd895074f","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Effect_size","wgTitle":"Effect size","wgCurRevisionId":1258181819,"wgRevisionId":1258181819,"wgArticleId":437276,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikipedia articles needing page number citations from August 2016","Articles with short description","Short description matches Wikidata","Articles needing cleanup from May 2011","All pages needing cleanup","Cleanup tagged articles with a reason field from May 2011","Wikipedia pages needing cleanup from May 2011","Wikipedia articles that are too technical from February 2014","All articles that are too technical","Articles with multiple maintenance issues","All accuracy disputes", "Articles with disputed statements from May 2024","Webarchive template wayback links","Effect size","Clinical research","Psychometrics","Statistical hypothesis testing","Clinical trials","Meta-analysis","Medical statistics","Mathematical and quantitative methods (economics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Effect_size","wgRelevantArticleId":437276,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false, "wgWMEPageLength":50000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1287978","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles": "ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Effect size - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Effect_size"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Effect_size&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Effect_size"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Effect_size rootpage-Effect_size skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Effect+size" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Effect+size" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Effect+size" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Effect+size" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Overview</span> </div> </a> <button aria-controls="toc-Overview-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Overview subsection</span> </button> <ul id="toc-Overview-sublist" class="vector-toc-list"> <li id="toc-Population_and_sample_effect_sizes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Population_and_sample_effect_sizes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Population and sample effect sizes</span> </div> </a> <ul id="toc-Population_and_sample_effect_sizes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_test_statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_test_statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Relationship to test statistics</span> </div> </a> <ul id="toc-Relationship_to_test_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standardized_and_unstandardized_effect_sizes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standardized_and_unstandardized_effect_sizes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Standardized and unstandardized effect sizes</span> </div> </a> <ul id="toc-Standardized_and_unstandardized_effect_sizes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interpretation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Interpretation</span> </div> </a> <ul id="toc-Interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Correlation_family:_Effect_sizes_based_on_"variance_explained"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Correlation_family:_Effect_sizes_based_on_"variance_explained""> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Correlation family: Effect sizes based on "variance explained"</span> </div> </a> <ul id="toc-Correlation_family:_Effect_sizes_based_on_"variance_explained"-sublist" class="vector-toc-list"> <li id="toc-Pearson_r_or_correlation_coefficient" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pearson_r_or_correlation_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Pearson <i>r</i> or correlation coefficient</span> </div> </a> <ul id="toc-Pearson_r_or_correlation_coefficient-sublist" class="vector-toc-list"> <li id="toc-Coefficient_of_determination_(r2_or_R2)" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Coefficient_of_determination_(r2_or_R2)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.1</span> <span>Coefficient of determination (<i>r</i><sup>2</sup> or <i>R</i><sup>2</sup>)</span> </div> </a> <ul id="toc-Coefficient_of_determination_(r2_or_R2)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eta-squared_(η2)" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Eta-squared_(η2)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.2</span> <span>Eta-squared (<i>η</i><sup>2</sup>)</span> </div> </a> <ul id="toc-Eta-squared_(η2)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Omega-squared_(ω2)" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Omega-squared_(ω2)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.3</span> <span>Omega-squared (<i>ω</i><sup>2</sup>)</span> </div> </a> <ul id="toc-Omega-squared_(ω2)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cohen's_f2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_f2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Cohen's <i>f</i><sup>2</sup></span> </div> </a> <ul id="toc-Cohen's_f2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohen's_q" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_q"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.3</span> <span>Cohen's <i>q</i></span> </div> </a> <ul id="toc-Cohen's_q-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Difference_family:_Effect_sizes_based_on_differences_between_means" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Difference_family:_Effect_sizes_based_on_differences_between_means"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Difference family: Effect sizes based on differences between means</span> </div> </a> <ul id="toc-Difference_family:_Effect_sizes_based_on_differences_between_means-sublist" class="vector-toc-list"> <li id="toc-Standardized_mean_difference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Standardized_mean_difference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Standardized mean difference</span> </div> </a> <ul id="toc-Standardized_mean_difference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohen's_d" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_d"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Cohen's <i>d</i></span> </div> </a> <ul id="toc-Cohen's_d-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Glass'_Δ" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Glass'_Δ"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Glass' Δ</span> </div> </a> <ul id="toc-Glass'_Δ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hedges'_g" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hedges'_g"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Hedges' <i>g</i></span> </div> </a> <ul id="toc-Hedges'_g-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ψ,_root-mean-square_standardized_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ψ,_root-mean-square_standardized_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.5</span> <span>Ψ, root-mean-square standardized effect</span> </div> </a> <ul id="toc-Ψ,_root-mean-square_standardized_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distribution_of_effect_sizes_based_on_means" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Distribution_of_effect_sizes_based_on_means"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.6</span> <span>Distribution of effect sizes based on means</span> </div> </a> <ul id="toc-Distribution_of_effect_sizes_based_on_means-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_metrics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other_metrics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.7</span> <span>Other metrics</span> </div> </a> <ul id="toc-Other_metrics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Categorical_family:_Effect_sizes_for_associations_among_categorical_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Categorical_family:_Effect_sizes_for_associations_among_categorical_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Categorical family: Effect sizes for associations among categorical variables</span> </div> </a> <ul id="toc-Categorical_family:_Effect_sizes_for_associations_among_categorical_variables-sublist" class="vector-toc-list"> <li id="toc-Cohen's_omega_(ω)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_omega_(ω)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Cohen's omega (<i>ω</i>)</span> </div> </a> <ul id="toc-Cohen's_omega_(ω)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odds_ratio" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Odds_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Odds ratio</span> </div> </a> <ul id="toc-Odds_ratio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relative_risk" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relative_risk"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.3</span> <span>Relative risk</span> </div> </a> <ul id="toc-Relative_risk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Risk_difference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Risk_difference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.4</span> <span>Risk difference</span> </div> </a> <ul id="toc-Risk_difference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohen's_h" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_h"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.5</span> <span>Cohen's <i>h</i></span> </div> </a> <ul id="toc-Cohen's_h-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_of_superiority" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Probability_of_superiority"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.6</span> <span>Probability of superiority</span> </div> </a> <ul id="toc-Probability_of_superiority-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_size_for_ordinal_data" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Effect_size_for_ordinal_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.7</span> <span>Effect size for ordinal data</span> </div> </a> <ul id="toc-Effect_size_for_ordinal_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohen's_g" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cohen's_g"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.8</span> <span>Cohen's g</span> </div> </a> <ul id="toc-Cohen's_g-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Confidence_intervals_by_means_of_noncentrality_parameters" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Confidence_intervals_by_means_of_noncentrality_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Confidence intervals by means of noncentrality parameters</span> </div> </a> <button aria-controls="toc-Confidence_intervals_by_means_of_noncentrality_parameters-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Confidence intervals by means of noncentrality parameters subsection</span> </button> <ul id="toc-Confidence_intervals_by_means_of_noncentrality_parameters-sublist" class="vector-toc-list"> <li id="toc-t-test_for_mean_difference_of_single_group_or_two_related_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#t-test_for_mean_difference_of_single_group_or_two_related_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span><i>t</i>-test for mean difference of single group or two related groups</span> </div> </a> <ul id="toc-t-test_for_mean_difference_of_single_group_or_two_related_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-t-test_for_mean_difference_between_two_independent_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#t-test_for_mean_difference_between_two_independent_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span><i>t</i>-test for mean difference between two independent groups</span> </div> </a> <ul id="toc-t-test_for_mean_difference_between_two_independent_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-One-way_ANOVA_test_for_mean_difference_across_multiple_independent_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One-way_ANOVA_test_for_mean_difference_across_multiple_independent_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>One-way ANOVA test for mean difference across multiple independent groups</span> </div> </a> <ul id="toc-One-way_ANOVA_test_for_mean_difference_across_multiple_independent_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Effect size</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 18 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-18" 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">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85_%D8%A7%D9%84%D8%A3%D8%AB%D8%B1" title="حجم الأثر – Arabic" lang="ar" hreflang="ar" data-title="حجم الأثر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Velikost_%C3%BA%C4%8Dinku" title="Velikost účinku – Czech" lang="cs" hreflang="cs" data-title="Velikost účinku" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Effektst%C3%A4rke" title="Effektstärke – German" lang="de" hreflang="de" data-title="Effektstärke" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tama%C3%B1o_del_efecto" title="Tamaño del efecto – Spanish" lang="es" hreflang="es" data-title="Tamaño del efecto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Efektuaren_tamaina" title="Efektuaren tamaina – Basque" lang="eu" hreflang="eu" data-title="Efektuaren tamaina" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AF%D8%A7%D8%B2%D9%87_%D8%AA%D8%A3%D8%AB%DB%8C%D8%B1" title="اندازه تأثیر – Persian" lang="fa" hreflang="fa" data-title="اندازه تأثیر" 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/Taille_d%27effet" title="Taille d'effet – French" lang="fr" hreflang="fr" data-title="Taille d'effet" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ukuran_efek" title="Ukuran efek – Indonesian" lang="id" hreflang="id" data-title="Ukuran efek" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%95%D7%93%D7%9C_%D7%90%D7%A4%D7%A7%D7%98" title="גודל אפקט – Hebrew" lang="he" hreflang="he" data-title="גודל אפקט" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hat%C3%A1snagys%C3%A1g" title="Hatásnagyság – Hungarian" lang="hu" hreflang="hu" data-title="Hatásnagyság" 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/Ukuran_kesan" title="Ukuran kesan – Malay" lang="ms" hreflang="ms" data-title="Ukuran kesan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Effectgrootte" title="Effectgrootte – Dutch" lang="nl" hreflang="nl" data-title="Effectgrootte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wielko%C5%9B%C4%87_efektu" title="Wielkość efektu – Polish" lang="pl" hreflang="pl" data-title="Wielkość efektu" 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/Tamanho_do_efeito" title="Tamanho do efeito – Portuguese" lang="pt" hreflang="pt" data-title="Tamanho do efeito" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/%C3%89f%C3%A9k_ukuran" title="Éfék ukuran – Sundanese" lang="su" hreflang="su" data-title="Éfék ukuran" 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/Effektstorlek" title="Effektstorlek – Swedish" lang="sv" hreflang="sv" data-title="Effektstorlek" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%88%E6%87%89%E5%80%BC" title="效應值 – Cantonese" lang="yue" hreflang="yue" data-title="效應值" 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/%E6%95%88%E5%BA%94%E5%80%BC" title="效应值 – Chinese" lang="zh" hreflang="zh" data-title="效应值" 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/Q1287978#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Effect_size" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Effect_size" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Effect_size"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Effect_size&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Effect_size&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Effect_size"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Effect_size&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Effect_size&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Effect_size" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Effect_size" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Effect_size&oldid=1258181819" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Effect_size&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Effect_size&id=1258181819&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEffect_size"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEffect_size"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Effect_size&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Effect_size&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-wikiversity mw-list-item"><a href="https://en.wikiversity.org/wiki/Effect_size" hreflang="en"><span>Wikiversity</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1287978" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Statistical measure of the magnitude of a phenomenon</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1248332772">.mw-parser-output .multiple-issues-text{width:95%;margin:0.2em 0}.mw-parser-output .multiple-issues-text>.mw-collapsible-content{margin-top:0.3em}.mw-parser-output .compact-ambox .ambox{border:none;border-collapse:collapse;background-color:transparent;margin:0 0 0 1.6em!important;padding:0!important;width:auto;display:block}body.mediawiki .mw-parser-output .compact-ambox .ambox.mbox-small-left{font-size:100%;width:auto;margin:0}.mw-parser-output .compact-ambox .ambox .mbox-text{padding:0!important;margin:0!important}.mw-parser-output .compact-ambox .ambox .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:disc}body.skin-minerva .mw-parser-output .multiple-issues-text>.mw-collapsible-toggle,.mw-parser-output .compact-ambox .ambox .mbox-image,.mw-parser-output .compact-ambox .ambox .mbox-imageright,.mw-parser-output .compact-ambox .ambox .mbox-empty-cell,.mw-parser-output .compact-ambox .hide-when-compact{display:none}</style><table class="box-Multiple_issues plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Effect_size" title="Special:EditPage/Effect size">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:Effect_size" title="Talk:Effect size">talk page</a></b>. <small><i>(<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove these messages</a>)</i></small> <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>Math notation uses different symbols to represent the same quantities in similar formulas.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Effect_size" title="Special:EditPage/Effect size">improve this article</a> if you can.</span> <span class="date-container"><i>(<span class="date">May 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Effect_size&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">February 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, an <b>effect size</b> is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of <a href="/wiki/Data" title="Data">data</a>, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value.<sup id="cite_ref-Kelley2012_1-0" class="reference"><a href="#cite_note-Kelley2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Examples of effect sizes include the <a href="/wiki/Correlation" title="Correlation">correlation</a> between two variables,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> coefficient in a regression, the <a href="/wiki/Mean_(statistics)" class="mw-redirect" title="Mean (statistics)">mean</a> difference, or the risk of a particular event (such as a heart attack) happening. Effect sizes are a complement tool for <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">statistical hypothesis testing</a>, and play an important role in <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">power</a> analyses to assess the sample size required for new experiments.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Effect size are fundamental in <a href="/wiki/Meta-analysis" title="Meta-analysis">meta-analyses</a> which aim to provide the combined effect size based on data from multiple studies. The cluster of data-analysis methods concerning effect sizes is referred to as <a href="/wiki/Estimation_statistics" title="Estimation statistics">estimation statistics</a>. </p><p>Effect size is an essential component when evaluating the strength of a statistical claim, and it is the first item (magnitude) in the <a href="/wiki/MAGIC_criteria" title="MAGIC criteria">MAGIC criteria</a>. The <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> of the effect size is of critical importance, since it indicates how much uncertainty is included in the measurement. A standard deviation that is too large will make the measurement nearly meaningless. In meta-analysis, where the purpose is to combine multiple effect sizes, the uncertainty in the effect size is used to weigh effect sizes, so that large studies are considered more important than small studies. The uncertainty in the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (<i>N</i>), or the number of observations (<i>n</i>) in each group. </p><p>Reporting effect sizes or estimates thereof (effect estimate [EE], estimate of effect) is considered good practice when presenting empirical research findings in many fields.<sup id="cite_ref-Wilkinson1999_4-0" class="reference"><a href="#cite_note-Wilkinson1999-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nakagawa2007_5-0" class="reference"><a href="#cite_note-Nakagawa2007-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The reporting of effect sizes facilitates the interpretation of the importance of a research result, in contrast to its <a href="/wiki/Statistical_significance" title="Statistical significance">statistical significance</a>.<sup id="cite_ref-Ellis2010_6-0" class="reference"><a href="#cite_note-Ellis2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Effect sizes are particularly prominent in <a href="/wiki/Social_science" title="Social science">social science</a> and in <a href="/wiki/Medical_research" title="Medical research">medical research</a> (where size of <a href="/wiki/Average_treatment_effect" title="Average treatment effect">treatment effect</a> is important). </p><p>Effect sizes may be measured in relative or absolute terms. In relative effect sizes, two groups are directly compared with each other, as in <a href="/wiki/Odds_ratio" title="Odds ratio">odds ratios</a> and <a href="/wiki/Relative_risk" title="Relative risk">relative risks</a>. For absolute effect sizes, a larger <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> always indicates a stronger effect. Many types of measurements can be expressed as either absolute or relative, and these can be used together because they convey different information. A prominent task force in the psychology research community made the following recommendation: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Always present effect sizes for primary outcomes...If the units of measurement are meaningful on a practical level (e.g., number of cigarettes smoked per day), then we usually prefer an unstandardized measure (regression coefficient or mean difference) to a standardized measure (<i>r</i> or <i>d</i>).<sup id="cite_ref-Wilkinson1999_4-1" class="reference"><a href="#cite_note-Wilkinson1999-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p></blockquote> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Population_and_sample_effect_sizes">Population and sample effect sizes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=2" title="Edit section: Population and sample effect sizes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As in <a href="/wiki/Statistical_estimation" class="mw-redirect" title="Statistical estimation">statistical estimation</a>, the true effect size is distinguished from the observed effect size. For example, to measure the risk of disease in a population (the population effect size) one can measure the risk within a sample of that population (the sample effect size). Conventions for describing true and observed effect sizes follow standard statistical practices—one common approach is to use Greek letters like ρ [rho] to denote population parameters and Latin letters like <i>r</i> to denote the corresponding statistic. Alternatively, a "hat" can be placed over the population parameter to denote the statistic, e.g. with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71ec0653c9ec1cad5e168085772c88e293fedef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.376ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}}"></span> being the estimate of the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>. </p><p>As in any statistical setting, effect sizes are estimated with <a href="/wiki/Sampling_error" title="Sampling error">sampling error</a>, and may be biased unless the effect size estimator that is used is appropriate for the manner in which the data were <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampled</a> and the manner in which the measurements were made. An example of this is <a href="/wiki/Publication_bias" title="Publication bias">publication bias</a>, which occurs when scientists report results only when the estimated effect sizes are large or are statistically significant. As a result, if many researchers carry out studies with low statistical power, the reported effect sizes will tend to be larger than the true (population) effects, if any.<sup id="cite_ref-Brand2008_7-0" class="reference"><a href="#cite_note-Brand2008-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Another example where effect sizes may be distorted is in a multiple-trial experiment, where the effect size calculation is based on the averaged or aggregated response across the trials.<sup id="cite_ref-Brand2011_8-0" class="reference"><a href="#cite_note-Brand2011-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Smaller studies sometimes show different, often larger, effect sizes than larger studies. This phenomenon is known as the small-study effect, which may signal publication bias.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_test_statistics">Relationship to test statistics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=3" title="Edit section: Relationship to test statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sample-based effect sizes are distinguished from <a href="/wiki/Test_statistic" title="Test statistic">test statistics</a> used in hypothesis testing, in that they estimate the strength (magnitude) of, for example, an apparent relationship, rather than assigning a <a href="/wiki/Statistical_significance" title="Statistical significance">significance</a> level reflecting whether the magnitude of the relationship observed could be due to chance. The effect size does not directly determine the significance level, or vice versa. Given a sufficiently large sample size, a non-null statistical comparison will always show a statistically significant result unless the population effect size is exactly zero (and even there it will show statistical significance at the rate of the Type I error used). For example, a sample <a href="/wiki/Pearson_correlation" class="mw-redirect" title="Pearson correlation">Pearson correlation</a> coefficient of 0.01 is statistically significant if the sample size is 1000. Reporting only the significant <a href="/wiki/P-value" title="P-value"><i>p</i>-value</a> from this analysis could be misleading if a correlation of 0.01 is too small to be of interest in a particular application. </p> <div class="mw-heading mw-heading3"><h3 id="Standardized_and_unstandardized_effect_sizes">Standardized and unstandardized effect sizes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=4" title="Edit section: Standardized and unstandardized effect sizes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i>effect size</i> can refer to a standardized measure of effect (such as <i>r</i>, <a href="/wiki/Cohen%27s_d" class="mw-redirect" title="Cohen's d">Cohen's <i>d</i></a>, or the <a href="/wiki/Odds_ratio" title="Odds ratio">odds ratio</a>), or to an unstandardized measure (e.g., the difference between group means or the unstandardized regression coefficients). Standardized effect size measures are typically used when: </p> <ul><li>the metrics of variables being studied do not have intrinsic meaning (e.g., a score on a personality test on an arbitrary scale),</li> <li>results from multiple studies are being combined,</li> <li>some or all of the studies use different scales, or</li> <li>it is desired to convey the size of an effect relative to the variability in the population.</li></ul> <p>In meta-analyses, standardized effect sizes are used as a common measure that can be calculated for different studies and then combined into an overall summary. </p> <div class="mw-heading mw-heading2"><h2 id="Interpretation">Interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=5" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Whether an effect size should be interpreted as small, medium, or large depends on its substantive context and its operational definition. Cohen's conventional criteria <i>small</i>, <i>medium</i>, or <i>big</i><sup id="cite_ref-CohenJ1988Statistical_10-0" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> are near ubiquitous across many fields, although Cohen<sup id="cite_ref-CohenJ1988Statistical_10-1" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> cautioned: </p> <blockquote><p>"The terms 'small,' 'medium,' and 'large' are relative, not only to each other, but to the area of behavioral science or even more particularly to the specific content and research method being employed in any given investigation....In the face of this relativity, there is a certain risk inherent in offering conventional operational definitions for these terms for use in power analysis in as diverse a field of inquiry as behavioral science. This risk is nevertheless accepted in the belief that more is to be gained than lost by supplying a common conventional frame of reference which is recommended for use only when no better basis for estimating the ES index is available." (p. 25)</p></blockquote> <p>In the two sample layout, Sawilowsky <sup id="cite_ref-Sawilowsky2009_11-0" class="reference"><a href="#cite_note-Sawilowsky2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> concluded "Based on current research findings in the applied literature, it seems appropriate to revise the rules of thumb for effect sizes," keeping in mind Cohen's cautions, and expanded the descriptions to include <i>very small</i>, <i>very large</i>, and <i>huge</i>. The same de facto standards could be developed for other layouts. </p><p>Lenth <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> noted for a "medium" effect size, "you'll choose the same <i>n</i> regardless of the accuracy or reliability of your instrument, or the narrowness or diversity of your subjects. Clearly, important considerations are being ignored here. Researchers should interpret the substantive significance of their results by grounding them in a meaningful context or by quantifying their contribution to knowledge, and Cohen's effect size descriptions can be helpful as a starting point."<sup id="cite_ref-Ellis2010_6-1" class="reference"><a href="#cite_note-Ellis2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Similarly, a U.S. Dept of Education sponsored report said "The widespread indiscriminate use of Cohen’s generic small, medium, and large effect size values to characterize effect sizes in domains to which his normative values do not apply is thus likewise inappropriate and misleading."<sup id="cite_ref-Lipsey_13-0" class="reference"><a href="#cite_note-Lipsey-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>They suggested that "appropriate norms are those based on distributions of effect sizes for comparable outcome measures from comparable interventions targeted on comparable samples." Thus if a study in a field where most interventions are tiny yielded a small effect (by Cohen's criteria), these new criteria would call it "large". In a related point, see <a href="/wiki/Abelson%27s_paradox" title="Abelson's paradox">Abelson's paradox</a> and Sawilowsky's paradox.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=6" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>About 50 to 100 different measures of effect size are known. Many effect sizes of different types can be converted to other types, as many estimate the separation of two distributions, so are mathematically related. For example, a correlation coefficient can be converted to a Cohen's d and vice versa. </p> <div class="mw-heading mw-heading3"><h3 id="Correlation_family:_Effect_sizes_based_on_"variance_explained""><span id="Correlation_family:_Effect_sizes_based_on_.22variance_explained.22"></span>Correlation family: Effect sizes based on "variance explained"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=7" title="Edit section: Correlation family: Effect sizes based on "variance explained""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These effect sizes estimate the amount of the variance within an experiment that is "explained" or "accounted for" by the experiment's model (<a href="/wiki/Explained_variation" title="Explained variation">Explained variation</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Pearson_r_or_correlation_coefficient">Pearson <i>r</i> or correlation coefficient</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=8" title="Edit section: Pearson r or correlation coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson's correlation</a>, often denoted <i>r</i> and introduced by <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a>, is widely used as an <i>effect size</i> when paired quantitative data are available; for instance if one were studying the relationship between birth weight and longevity. The correlation coefficient can also be used when the data are binary. Pearson's <i>r</i> can vary in magnitude from −1 to 1, with −1 indicating a perfect negative linear relation, 1 indicating a perfect positive linear relation, and 0 indicating no linear relation between two variables. <a href="/wiki/Jacob_Cohen_(statistician)" title="Jacob Cohen (statistician)">Cohen</a> gives the following guidelines for the social sciences:<sup id="cite_ref-CohenJ1988Statistical_10-2" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CohenJ1992_17-0" class="reference"><a href="#cite_note-CohenJ1992-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th>Effect size</th> <th><i>r</i> </th></tr> <tr> <td>Small</td> <td>0.10 </td></tr> <tr> <td>Medium</td> <td>0.30 </td></tr> <tr> <td>Large</td> <td>0.50 </td></tr></tbody></table> <div class="mw-heading mw-heading5"><h5 id="Coefficient_of_determination_(r2_or_R2)"><span id="Coefficient_of_determination_.28r2_or_R2.29"></span>Coefficient of determination (<i>r</i><sup>2</sup> or <i>R</i><sup>2</sup>)</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=9" title="Edit section: Coefficient of determination (r2 or R2)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A related <i>effect size</i> is <i>r</i><sup>2</sup>, the <a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a> (also referred to as <i>R</i><sup>2</sup> or "<i>r</i>-squared"), calculated as the square of the Pearson correlation <i>r</i>. In the case of paired data, this is a measure of the proportion of variance shared by the two variables, and varies from 0 to 1. For example, with an <i>r</i> of 0.21 the coefficient of determination is 0.0441, meaning that 4.4% of the variance of either variable is shared with the other variable. The <i>r</i><sup>2</sup> is always positive, so does not convey the direction of the correlation between the two variables. </p> <div class="mw-heading mw-heading5"><h5 id="Eta-squared_(η2)"><span id="Eta-squared_.28.CE.B72.29"></span>Eta-squared (<i>η</i><sup>2</sup>)</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=10" title="Edit section: Eta-squared (η2)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors, making it analogous to the <i>r</i><sup>2</sup>. Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample). This estimate shares the weakness with <i>r</i><sup>2</sup> that each additional variable will automatically increase the value of <i>η</i><sup>2</sup>. In addition, it measures the variance explained of the sample, not the population, meaning that it will always overestimate the effect size, although the bias grows smaller as the sample grows larger. <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 \eta ^{2}={\frac {SS_{\text{Treatment}}}{SS_{\text{Total}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Treatment</mtext> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Total</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ^{2}={\frac {SS_{\text{Treatment}}}{SS_{\text{Total}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243c3821fe2f747641f38606db845fad558d1a51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.641ex; height:5.843ex;" alt="{\displaystyle \eta ^{2}={\frac {SS_{\text{Treatment}}}{SS_{\text{Total}}}}.}"></span> </p> <div class="mw-heading mw-heading5"><h5 id="Omega-squared_(ω2)"><span id="Omega-squared_.28.CF.892.29"></span>Omega-squared (<i>ω</i><sup>2</sup>)</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=11" title="Edit section: Omega-squared (ω2)"><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/Coefficient_of_determination#Adjusted_R2" title="Coefficient of determination">Adjusted <i>R</i><sup>2</sup></a></div> <p>A less biased estimator of the variance explained in the population is <i>ω</i><sup>2</sup><sup id="cite_ref-Tabachnick_2007,_p._55_18-0" class="reference"><a href="#cite_note-Tabachnick_2007,_p._55-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <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 \omega ^{2}={\frac {{\text{SS}}_{\text{treatment}}-df_{\text{treatment}}\cdot {\text{MS}}_{\text{error}}}{{\text{SS}}_{\text{total}}+{\text{MS}}_{\text{error}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>treatment</mtext> </mrow> </msub> <mo>−</mo> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>treatment</mtext> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>MS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>error</mtext> </mrow> </msub> </mrow> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>MS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>error</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}={\frac {{\text{SS}}_{\text{treatment}}-df_{\text{treatment}}\cdot {\text{MS}}_{\text{error}}}{{\text{SS}}_{\text{total}}+{\text{MS}}_{\text{error}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add2e850c6270674856b7ef66d9a38716024995c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.404ex; height:5.843ex;" alt="{\displaystyle \omega ^{2}={\frac {{\text{SS}}_{\text{treatment}}-df_{\text{treatment}}\cdot {\text{MS}}_{\text{error}}}{{\text{SS}}_{\text{total}}+{\text{MS}}_{\text{error}}}}.}"></span> </p><p>This form of the formula is limited to between-subjects analysis with equal sample sizes in all cells.<sup id="cite_ref-Tabachnick_2007,_p._55_18-1" class="reference"><a href="#cite_note-Tabachnick_2007,_p._55-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Since it is less biased (although not <i>un</i>biased), <i>ω</i><sup>2</sup> is preferable to η<sup>2</sup>; however, it can be more inconvenient to calculate for complex analyses. A generalized form of the estimator has been published for between-subjects and within-subjects analysis, repeated measure, mixed design, and randomized block design experiments.<sup id="cite_ref-OlejnikAlgina_19-0" class="reference"><a href="#cite_note-OlejnikAlgina-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In addition, methods to calculate partial <i>ω</i><sup>2</sup> for individual factors and combined factors in designs with up to three independent variables have been published.<sup id="cite_ref-OlejnikAlgina_19-1" class="reference"><a href="#cite_note-OlejnikAlgina-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_f2"><span id="Cohen.27s_f2"></span>Cohen's <i>f</i><sup>2</sup></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=12" title="Edit section: Cohen's f2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cohen's <i>f</i><sup>2</sup> is one of several effect size measures to use in the context of an <a href="/wiki/F-test" title="F-test">F-test</a> for <a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a> or <a href="/wiki/Multiple_regression" class="mw-redirect" title="Multiple regression">multiple regression</a>. Its amount of bias (overestimation of the effect size for the ANOVA) depends on the bias of its underlying measurement of variance explained (e.g., <i>R</i><sup>2</sup>, <i>η</i><sup>2</sup>, <i>ω</i><sup>2</sup>). </p><p>The <i>f</i><sup>2</sup> effect size measure for multiple regression is defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}={R^{2} \over 1-R^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}={R^{2} \over 1-R^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2a991a17400251eb2cbca282631c12ace483fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.13ex; height:6.176ex;" alt="{\displaystyle f^{2}={R^{2} \over 1-R^{2}}}"></span> where <i>R</i><sup>2</sup> is the <a href="/wiki/Squared_multiple_correlation" class="mw-redirect" title="Squared multiple correlation">squared multiple correlation</a>. </p><p>Likewise, <i>f</i><sup>2</sup> can be defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}={\eta ^{2} \over 1-\eta ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}={\eta ^{2} \over 1-\eta ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03548070caeb6f883518a4cfa33cc8c52b503fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.54ex; height:6.343ex;" alt="{\displaystyle f^{2}={\eta ^{2} \over 1-\eta ^{2}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}={\omega ^{2} \over 1-\omega ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}={\omega ^{2} \over 1-\omega ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70972f9e1724fc839862e50249a1d7ab4810b68d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.812ex; height:6.176ex;" alt="{\displaystyle f^{2}={\omega ^{2} \over 1-\omega ^{2}}}"></span> for models described by those effect size measures.<sup id="cite_ref-Steiger2004_20-0" class="reference"><a href="#cite_note-Steiger2004-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81efa211d9d04493b68b24bf9843d1967ae22cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:3.009ex;" alt="{\displaystyle f^{2}}"></span> effect size measure for sequential multiple regression and also common for <a href="/wiki/Partial_least_squares_path_modeling" title="Partial least squares path modeling">PLS modeling</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> is defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}={R_{AB}^{2}-R_{A}^{2} \over 1-R_{AB}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}={R_{AB}^{2}-R_{A}^{2} \over 1-R_{AB}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b8b861d546bb8f4c926d2a5348232d26d2107c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.855ex; height:7.176ex;" alt="{\displaystyle f^{2}={R_{AB}^{2}-R_{A}^{2} \over 1-R_{AB}^{2}}}"></span> where <i>R</i><sup>2</sup><sub><i>A</i></sub> is the variance accounted for by a set of one or more independent variables <i>A</i>, and <i>R</i><sup>2</sup><sub><i>AB</i></sub> is the combined variance accounted for by <i>A</i> and another set of one or more independent variables of interest <i>B</i>. By convention, <i>f</i><sup>2</sup> effect sizes of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.1^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.1^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48da5e5ed5945bc58f8de896e332cf76457793cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.026ex; height:2.676ex;" alt="{\displaystyle 0.1^{2}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.25^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0.25</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.25^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde2a60ccc7cf1a8d2a9d5edb314eded863d465f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.188ex; height:2.676ex;" alt="{\displaystyle 0.25^{2}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.4^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0.4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.4^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0ce9da3efe6b7d9da92543d7136ac0de0d8db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.026ex; height:2.676ex;" alt="{\displaystyle 0.4^{2}}"></span> are termed <i>small</i>, <i>medium</i>, and <i>large</i>, respectively.<sup id="cite_ref-CohenJ1988Statistical_10-3" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Cohen's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"></span> can also be found for factorial analysis of variance (ANOVA) working backwards, using: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{\text{effect}}={\sqrt {(F_{\text{effect}}df_{\text{effect}}/N)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>effect</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>effect</mtext> </mrow> </msub> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>effect</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{\text{effect}}={\sqrt {(F_{\text{effect}}df_{\text{effect}}/N)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be574c62ee7ee1f67141ff745c242fde5e9cafe3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.859ex; height:4.843ex;" alt="{\displaystyle {\hat {f}}_{\text{effect}}={\sqrt {(F_{\text{effect}}df_{\text{effect}}/N)}}.}"></span> </p><p>In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81efa211d9d04493b68b24bf9843d1967ae22cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:3.009ex;" alt="{\displaystyle f^{2}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {SS(\mu _{1},\mu _{2},\dots ,\mu _{K})} \over {K\times \sigma ^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>×</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>,</mo> </mrow> </mfrac> </mrow> <annotation encoding="application/x-tex">{\displaystyle {SS(\mu _{1},\mu _{2},\dots ,\mu _{K})} \over {K\times \sigma ^{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c40cdb19dcb087b62b090bc6be5b097ab9685d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.863ex; height:6.343ex;" alt="{\displaystyle {SS(\mu _{1},\mu _{2},\dots ,\mu _{K})} \over {K\times \sigma ^{2}},}"></span> wherein <i>μ</i><sub><i>j</i></sub> denotes the population mean within the <i>j</i><sup>th</sup> group of the total <i>K</i> groups, and <i>σ</i> the equivalent population standard deviations within each groups. <i>SS</i> is the <a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">sum of squares</a> in ANOVA. </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_q"><span id="Cohen.27s_q"></span>Cohen's <i>q</i></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=13" title="Edit section: Cohen's q"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another measure that is used with correlation differences is Cohen's q. This is the difference between two Fisher transformed Pearson regression coefficients. In symbols this is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {1}{2}}\log {\frac {1+r_{1}}{1-r_{1}}}-{\frac {1}{2}}\log {\frac {1+r_{2}}{1-r_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\frac {1}{2}}\log {\frac {1+r_{1}}{1-r_{1}}}-{\frac {1}{2}}\log {\frac {1+r_{2}}{1-r_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ecb6f031b0e47926428de5db868ee2554f7fae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.381ex; height:5.509ex;" alt="{\displaystyle q={\frac {1}{2}}\log {\frac {1+r_{1}}{1-r_{1}}}-{\frac {1}{2}}\log {\frac {1+r_{2}}{1-r_{2}}}}"></span> </p><p>where <i>r</i><sub>1</sub> and <i>r</i><sub>2</sub> are the regressions being compared. The expected value of <i>q</i> is zero and its variance is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} (q)={\frac {1}{N_{1}-3}}+{\frac {1}{N_{2}-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} (q)={\frac {1}{N_{1}-3}}+{\frac {1}{N_{2}-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598dce40e5d1539e5cbdde8a00856477c69410ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.638ex; height:5.509ex;" alt="{\displaystyle \operatorname {var} (q)={\frac {1}{N_{1}-3}}+{\frac {1}{N_{2}-3}}}"></span> where <i>N</i><sub>1</sub> and <i>N</i><sub>2</sub> are the number of data points in the first and second regression respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Difference_family:_Effect_sizes_based_on_differences_between_means">Difference family: Effect sizes based on differences between means</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=14" title="Edit section: Difference family: Effect sizes based on differences between means"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The raw effect size pertaining to a comparison of two groups is inherently calculated as the differences between the two means. However, to facilitate interpretation it is common to standardise the effect size; various conventions for statistical standardisation are presented below. </p> <div class="mw-heading mw-heading4"><h4 id="Standardized_mean_difference">Standardized mean difference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=15" title="Edit section: Standardized mean difference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cohens_d_4panel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Cohens_d_4panel.svg/220px-Cohens_d_4panel.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Cohens_d_4panel.svg/330px-Cohens_d_4panel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Cohens_d_4panel.svg/440px-Cohens_d_4panel.svg.png 2x" data-file-width="225" data-file-height="225" /></a><figcaption>Plots of Gaussian densities illustrating various values of Cohen's d.</figcaption></figure> <p>A (population) effect size <i>θ</i> based on means usually considers the standardized mean difference (SMD) between two populations<sup id="cite_ref-HedgesL1985Statistical_22-0" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 78">: 78 </span></sup> <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 \theta ={\frac {\mu _{1}-\mu _{2}}{\sigma }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {\mu _{1}-\mu _{2}}{\sigma }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68df9f56b628749ae89e029b13ef3461b20b294" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.424ex; height:5.176ex;" alt="{\displaystyle \theta ={\frac {\mu _{1}-\mu _{2}}{\sigma }},}"></span> where <i>μ</i><sub>1</sub> is the mean for one population, <i>μ</i><sub>2</sub> is the mean for the other population, and σ is a <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> based on either or both populations. </p><p>In the practical setting the population values are typically not known and must be estimated from sample statistics. The several versions of effect sizes based on means differ with respect to which statistics are used. </p><p>This form for the effect size resembles the computation for a <a href="/wiki/T-test" class="mw-redirect" title="T-test"><i>t</i>-test</a> statistic, with the critical difference that the <i>t</i>-test statistic includes a factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}"></span>. This means that for a given effect size, the significance level increases with the sample size. Unlike the <i>t</i>-test statistic, the effect size aims to estimate a population <a href="/wiki/Parameter" title="Parameter">parameter</a> and is not affected by the sample size. </p><p>SMD values of 0.2 to 0.5 are considered small, 0.5 to 0.8 are considered medium, and greater than 0.8 are considered large.<sup id="cite_ref-Andrade2020_23-0" class="reference"><a href="#cite_note-Andrade2020-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_d"><span id="Cohen.27s_d"></span>Cohen's <i>d</i> <span class="anchor" id="Cohen's_d"></span></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=16" title="Edit section: Cohen's d"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cohen's <i>d</i> is defined as the difference between two means divided by a standard deviation for the data, i.e. <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 d={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>s</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b32c6a5cd69c69c2e57b33caec052a1c7c1425" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.406ex; height:5.009ex;" alt="{\displaystyle d={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s}}.}"></span> </p><p><a href="/wiki/Jacob_Cohen_(statistician)" title="Jacob Cohen (statistician)">Jacob Cohen</a> defined <i>s</i>, the <a href="/wiki/Pooled_standard_deviation" class="mw-redirect" title="Pooled standard deviation">pooled standard deviation</a>, as (for two independent samples):<sup id="cite_ref-CohenJ1988Statistical_10-4" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 67">: 67 </span></sup> <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 s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d62c6e845b54d7935f196e49aaee70a52a1bed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.001ex; height:7.676ex;" alt="{\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}}"></span> where the variance for one of the groups is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}^{2}={\frac {1}{n_{1}-1}}\sum _{i=1}^{n_{1}}(x_{1,i}-{\bar {x}}_{1})^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1}^{2}={\frac {1}{n_{1}-1}}\sum _{i=1}^{n_{1}}(x_{1,i}-{\bar {x}}_{1})^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91de0697a70ab67cb8c2a30386b2fa6826462681" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.417ex; height:7.009ex;" alt="{\displaystyle s_{1}^{2}={\frac {1}{n_{1}-1}}\sum _{i=1}^{n_{1}}(x_{1,i}-{\bar {x}}_{1})^{2},}"></span> and similarly for the other group. </p><p>The table below contains descriptors for magnitudes of <i>d</i> = 0.01 to 2.0, as initially suggested by Cohen (who warned against the values becoming de facto standards, urging flexibility of interpretation) and expanded by Sawilowsky.<sup id="cite_ref-Sawilowsky2009_11-1" class="reference"><a href="#cite_note-Sawilowsky2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th><i>Effect size</i></th> <th><i>d</i></th> <th>Reference </th></tr> <tr> <td>Very small</td> <td align="left">0.01</td> <td><sup id="cite_ref-Sawilowsky2009_11-2" class="reference"><a href="#cite_note-Sawilowsky2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Small</td> <td align="left">0.20</td> <td><sup id="cite_ref-CohenJ1988Statistical_10-5" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Medium</td> <td align="left">0.50</td> <td><sup id="cite_ref-CohenJ1988Statistical_10-6" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Large</td> <td align="left">0.80</td> <td><sup id="cite_ref-CohenJ1988Statistical_10-7" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Very large</td> <td align="left">1.20</td> <td><sup id="cite_ref-Sawilowsky2009_11-3" class="reference"><a href="#cite_note-Sawilowsky2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Huge</td> <td align="left">2.0</td> <td><sup id="cite_ref-Sawilowsky2009_11-4" class="reference"><a href="#cite_note-Sawilowsky2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </td></tr> </tbody></table> <p>Other authors choose a slightly different computation of the standard deviation when referring to "Cohen's <i>d</i>" where the denominator is without "-2"<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 14">: 14 </span></sup> <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 s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b896128743d871f7f5a8890a0d0d981d805b05cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.001ex; height:7.676ex;" alt="{\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}}}}}"></span> This definition of "Cohen's <i>d</i>" is termed the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimator by Hedges and Olkin,<sup id="cite_ref-HedgesL1985Statistical_22-1" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and it is related to Hedges' <i>g</i> by a scaling factor (see below). </p><p>With two paired samples, we look at the distribution of the difference scores. In that case, <i>s</i> is the standard deviation of this distribution of difference scores. This creates the following relationship between the t-statistic to test for a difference in the means of the two groups and Cohen's <i>d</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SE}}}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\frac {\text{SD}}{\sqrt {N}}}}={\frac {{\sqrt {N}}({\bar {X}}_{1}-{\bar {X}}_{2})}{SD}}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mtext>SE</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mfrac> <mtext>SD</mtext> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>S</mi> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SE}}}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\frac {\text{SD}}{\sqrt {N}}}}={\frac {{\sqrt {N}}({\bar {X}}_{1}-{\bar {X}}_{2})}{SD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765c0698ef7b3f61de1809c48fe2beac0d29106d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:45.179ex; height:8.343ex;" alt="{\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SE}}}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\frac {\text{SD}}{\sqrt {N}}}}={\frac {{\sqrt {N}}({\bar {X}}_{1}-{\bar {X}}_{2})}{SD}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SD}}}={\frac {t}{\sqrt {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mtext>SD</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SD}}}={\frac {t}{\sqrt {N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebf5211f1a1fbe419d052d3b21e524d49b332ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.993ex; height:6.676ex;" alt="{\displaystyle d={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SD}}}={\frac {t}{\sqrt {N}}}}"></span> </p><p>Cohen's <i>d</i> is frequently used in <a href="/wiki/Estimating_sample_sizes" class="mw-redirect" title="Estimating sample sizes">estimating sample sizes</a> for statistical testing. A lower Cohen's <i>d</i> indicates the necessity of larger sample sizes, and vice versa, as can subsequently be determined together with the additional parameters of desired <a href="/wiki/Significance_level" class="mw-redirect" title="Significance level">significance level</a> and <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">statistical power</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>For paired samples Cohen suggests that the d calculated is actually a d', which does not provide the correct answer to obtain the power of the test, and that before looking the values up in the tables provided, it should be corrected for r as in the following formula:<sup id="cite_ref-FOOTNOTECohen198849_27-0" class="reference"><a href="#cite_note-FOOTNOTECohen198849-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> <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 d={\frac {d'}{\sqrt {1-r}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mo>′</mo> </msup> <msqrt> <mn>1</mn> <mo>−</mo> <mi>r</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {d'}{\sqrt {1-r}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a569f5cc10b6ccb6e26ace79a51d1dbc34250d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.785ex; height:6.509ex;" alt="{\displaystyle d={\frac {d'}{\sqrt {1-r}}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Glass'_Δ"><span id="Glass.27_.CE.94"></span>Glass' Δ</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=17" title="Edit section: Glass' Δ"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1976, <a href="/wiki/Gene_V._Glass" title="Gene V. Glass">Gene V. Glass</a> proposed an estimator of the effect size that uses only the standard deviation of the second group<sup id="cite_ref-HedgesL1985Statistical_22-2" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 78">: 78 </span></sup> <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 \Delta ={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80256603a33177d3dd8f6a6fe151e3a46c840ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.479ex; height:5.343ex;" alt="{\displaystyle \Delta ={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s_{2}}}}"></span> </p><p>The second group may be regarded as a control group, and Glass argued that if several treatments were compared to the control group it would be better to use just the standard deviation computed from the control group, so that effect sizes would not differ under equal means and different variances. </p><p>Under a correct assumption of equal population variances a pooled estimate for <i>σ</i> is more precise. </p> <div class="mw-heading mw-heading4"><h4 id="Hedges'_g"><span id="Hedges.27_g"></span>Hedges' <i>g</i></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=18" title="Edit section: Hedges' g"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hedges' <i>g</i>, suggested by <a href="/wiki/Larry_Hedges" class="mw-redirect" title="Larry Hedges">Larry Hedges</a> in 1981,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> is like the other measures based on a standardized difference<sup id="cite_ref-HedgesL1985Statistical_22-3" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 79">: 79 </span></sup> <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 g={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0165f5adf3abd44ea8acb553d59d36c6aad11f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.659ex; height:5.009ex;" alt="{\displaystyle g={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s^{*}}}}"></span> where the pooled standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dd30e91ecca56ddf4ee71bf82b506b3249a5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle s^{*}}"></span> is computed as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{*}={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{*}={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85edc17662938d24542a29a296cf86d35af83a4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.702ex; height:7.676ex;" alt="{\displaystyle s^{*}={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}.}"></span> </p><p>However, as an <a href="/wiki/Estimator" title="Estimator">estimator</a> for the population effect size <i>θ</i> it is <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">biased</a>. Nevertheless, this bias can be approximately corrected through multiplication by a factor <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 g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx \,\left(1-{\frac {3}{4(n_{1}+n_{2})-9}}\right)\,\,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>g</mi> <mspace width="thinmathspace" /> <mo>≈</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>9</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx \,\left(1-{\frac {3}{4(n_{1}+n_{2})-9}}\right)\,\,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592fb8b7ddd66c8c1158102bc77fbcc618979da3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.306ex; height:6.343ex;" alt="{\displaystyle g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx \,\left(1-{\frac {3}{4(n_{1}+n_{2})-9}}\right)\,\,g}"></span> Hedges and Olkin refer to this less-biased estimator <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 g^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b02af7a00ce649e878c0276b16174d438e31a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.676ex;" alt="{\displaystyle g^{*}}"></span> as <i>d</i>,<sup id="cite_ref-HedgesL1985Statistical_22-4" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> but it is not the same as Cohen's <i>d</i>. The exact form for the correction factor <i>J</i>() involves the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a><sup id="cite_ref-HedgesL1985Statistical_22-5" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 104">: 104 </span></sup> <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 J(a)={\frac {\Gamma (a/2)}{{\sqrt {a/2\,}}\,\Gamma ((a-1)/2)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(a)={\frac {\Gamma (a/2)}{{\sqrt {a/2\,}}\,\Gamma ((a-1)/2)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e373e37bdfaaf47e94cdf88e51cd29ce298ff4b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.374ex; height:7.009ex;" alt="{\displaystyle J(a)={\frac {\Gamma (a/2)}{{\sqrt {a/2\,}}\,\Gamma ((a-1)/2)}}.}"></span> There are also multilevel variants of Hedges' g, e.g., for use in cluster randomised controlled trials (CRTs).<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> CRTs involve randomising clusters, such as schools or classrooms, to different conditions and are frequently used in education research. </p> <div class="mw-heading mw-heading4"><h4 id="Ψ,_root-mean-square_standardized_effect"><span id=".CE.A8.2C_root-mean-square_standardized_effect"></span>Ψ, root-mean-square standardized effect</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=19" title="Edit section: Ψ, root-mean-square standardized effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A similar effect size estimator for multiple comparisons (e.g., <a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a>) is the Ψ root-mean-square standardized effect:<sup id="cite_ref-Steiger2004_20-1" class="reference"><a href="#cite_note-Steiger2004-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <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 \Psi ={\sqrt {{\frac {1}{k-1}}\cdot \sum _{j=1}^{k}\left({\frac {\mu _{j}-\mu }{\sigma }}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi ={\sqrt {{\frac {1}{k-1}}\cdot \sum _{j=1}^{k}\left({\frac {\mu _{j}-\mu }{\sigma }}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fc7afe48e548e0c219015dfbaaf5874668c799" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.697ex; height:8.343ex;" alt="{\displaystyle \Psi ={\sqrt {{\frac {1}{k-1}}\cdot \sum _{j=1}^{k}\left({\frac {\mu _{j}-\mu }{\sigma }}\right)^{2}}}}"></span> where <i>k</i> is the number of groups in the comparisons. </p><p>This essentially presents the omnibus difference of the entire model adjusted by the root mean square, analogous to <i>d</i> or <i>g</i>. </p><p>In addition, a generalization for multi-factorial designs has been provided.<sup id="cite_ref-Steiger2004_20-2" class="reference"><a href="#cite_note-Steiger2004-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Distribution_of_effect_sizes_based_on_means">Distribution of effect sizes based on means</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=20" title="Edit section: Distribution of effect sizes based on means"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Provided that the data is <a href="/wiki/Gaussian" class="mw-redirect" title="Gaussian">Gaussian</a> distributed a scaled Hedges' <i>g</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd8db4b03f8846efc99fc6c5a4573eee50653de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.435ex; height:3.343ex;" alt="{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\,g}"></span>, follows a <a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">noncentral <i>t</i>-distribution</a> with the <a href="/wiki/Noncentrality_parameter" class="mw-redirect" title="Noncentrality parameter">noncentrality parameter</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d97de24fcb9e4eb11b0c4f583b5dc91ac5b8d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.022ex; height:3.343ex;" alt="{\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\theta }"></span> and <span class="texhtml">(<i>n</i><sub>1</sub> + <i>n</i><sub>2</sub> − 2)</span> degrees of freedom. Likewise, the scaled Glass' Δ is distributed with <span class="texhtml"><i>n</i><sub>2</sub> − 1</span> degrees of freedom. </p><p>From the distribution it is possible to compute the <a href="/wiki/Expected_value" title="Expected value">expectation</a> and variance of the effect sizes. </p><p>In some cases large sample approximations for the variance are used. One suggestion for the variance of Hedges' unbiased estimator is<sup id="cite_ref-HedgesL1985Statistical_22-6" class="reference"><a href="#cite_note-HedgesL1985Statistical-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <sup class="reference nowrap"><span title="Page: 86">: 86 </span></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}^{2}(g^{*})={\frac {n_{1}+n_{2}}{n_{1}n_{2}}}+{\frac {(g^{*})^{2}}{2(n_{1}+n_{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{2}(g^{*})={\frac {n_{1}+n_{2}}{n_{1}n_{2}}}+{\frac {(g^{*})^{2}}{2(n_{1}+n_{2})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25cf0ac6ad5a56ef795af130a4d972d76f1bcdae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.072ex; height:6.676ex;" alt="{\displaystyle {\hat {\sigma }}^{2}(g^{*})={\frac {n_{1}+n_{2}}{n_{1}n_{2}}}+{\frac {(g^{*})^{2}}{2(n_{1}+n_{2})}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Other_metrics">Other metrics</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=21" title="Edit section: Other metrics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a> (D) is a multivariate generalization of Cohen's d, which takes into account the relationships between the variables.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Categorical_family:_Effect_sizes_for_associations_among_categorical_variables">Categorical family: Effect sizes for associations among categorical variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=22" title="Edit section: Categorical family: Effect sizes for associations among categorical variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right"> <tbody><tr> <td align="center"> <p> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\sqrt {\frac {\chi ^{2}}{N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\sqrt {\frac {\chi ^{2}}{N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d747a6818f1049a725eed39262ca0199237a72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.288ex; height:7.509ex;" alt="{\displaystyle \varphi ={\sqrt {\frac {\chi ^{2}}{N}}}}"></span>  </p> </td> <td align="center"> <p> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{c}={\sqrt {\frac {\chi ^{2}}{N(k-1)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{c}={\sqrt {\frac {\chi ^{2}}{N(k-1)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1422a3bb737d916797904bf59e287b2b417b5b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.81ex; height:7.676ex;" alt="{\displaystyle \varphi _{c}={\sqrt {\frac {\chi ^{2}}{N(k-1)}}}}"></span>  </p> </td></tr> <tr> <th>Phi (<i>φ</i>) </th> <th>Cramér's <i>V</i> (<i>φ</i><sub><i>c</i></sub>) </th></tr></tbody></table> <p>Commonly used measures of association for the <a href="/wiki/Chi-squared_test" title="Chi-squared test">chi-squared test</a> are the <a href="/wiki/Phi_coefficient" title="Phi coefficient">Phi coefficient</a> and <a href="/wiki/Harald_Cram%C3%A9r" title="Harald Cramér">Cramér</a>'s <a href="/wiki/Cram%C3%A9r%27s_V_(statistics)" class="mw-redirect" title="Cramér's V (statistics)">V</a> (sometimes referred to as Cramér's phi and denoted as <i>φ</i><sub><i>c</i></sub>). Phi is related to the <a href="/wiki/Point-biserial_correlation_coefficient" title="Point-biserial correlation coefficient">point-biserial correlation coefficient</a> and Cohen's <i>d</i> and estimates the extent of the relationship between two variables (2 × 2).<sup id="cite_ref-Ref_31-0" class="reference"><a href="#cite_note-Ref_-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Cramér's V may be used with variables having more than two levels. </p><p>Phi can be computed by finding the square root of the chi-squared statistic divided by the sample size. </p><p>Similarly, Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (<i>k</i> is the smaller of the number of rows <i>r</i> or columns <i>c</i>). </p><p>φ<sub><i>c</i></sub> is the intercorrelation of the two discrete variables<sup id="cite_ref-Ref_a_32-0" class="reference"><a href="#cite_note-Ref_a-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> and may be computed for any value of <i>r</i> or <i>c</i>. However, as chi-squared values tend to increase with the number of cells, the greater the difference between <i>r</i> and <i>c</i>, the more likely V will tend to 1 without strong evidence of a meaningful correlation. </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_omega_(ω)"><span id="Cohen.27s_omega_.28.CF.89.29"></span>Cohen's omega (<i>ω</i>)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=23" title="Edit section: Cohen's omega (ω)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another measure of effect size used for chi-squared tests is Cohen's omega (<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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>). This is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {\sum _{i=1}^{m}{\frac {(p_{1i}-p_{0i})^{2}}{p_{0i}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\sum _{i=1}^{m}{\frac {(p_{1i}-p_{0i})^{2}}{p_{0i}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9f1d4b9d5a838850688b24b9c77fac8e94c5db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.732ex; height:7.509ex;" alt="{\displaystyle \omega ={\sqrt {\sum _{i=1}^{m}{\frac {(p_{1i}-p_{0i})^{2}}{p_{0i}}}}}}"></span> where <i>p</i><sub>0<i>i</i></sub> is the proportion of the <i>i</i><sup>th</sup> cell under <i>H</i><sub>0</sub>, <i>p</i><sub>1<i>i</i></sub> is the proportion of the <i>i</i><sup>th</sup> cell under <i>H</i><sub>1</sub> and <i>m</i> is the number of cells. </p><p>In <i>Statistical Power Analysis for the Behavioral Sciences</i> (1988, pp.224-225), Cohen gives the following general guideline for interpreting omega (see table below), but warns against its "possible inaptness in any given substantive context" and advises to use context-relevant judgment instead. </p> <table class="wikitable"> <tbody><tr> <th><i>Effect Size</i></th> <th><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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> </th></tr> <tr> <td>Small</td> <td>0.10 </td></tr> <tr> <td>Medium</td> <td>0.30 </td></tr> <tr> <td>Large</td> <td>0.50 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Odds_ratio">Odds ratio</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=24" title="Edit section: Odds ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Odds_ratio" title="Odds ratio">odds ratio</a> (OR) is another useful effect size. It is appropriate when the research question focuses on the degree of association between two <a href="/wiki/Binary_data" title="Binary data">binary variables</a>. For example, consider a study of spelling ability. In a control group, two students pass the class for every one who fails, so the odds of passing are two to one (or 2/1 = 2). In the treatment group, six students pass for every one who fails, so the odds of passing are six to one (or 6/1 = 6). The effect size can be computed by noting that the odds of passing in the treatment group are three times higher than in the control group (because 6 divided by 2 is 3). Therefore, the odds ratio is 3. Odds ratio statistics are on a different scale than Cohen's <i>d</i>, so this '3' is not comparable to a Cohen's <i>d</i> of 3. </p> <div class="mw-heading mw-heading4"><h4 id="Relative_risk">Relative risk</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=25" title="Edit section: Relative risk"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Relative_risk" title="Relative risk">relative risk</a> (RR), also called <b>risk ratio</b>, is simply the risk (probability) of an event relative to some independent variable. This measure of effect size differs from the odds ratio in that it compares <i>probabilities</i> instead of <i>odds</i>, but asymptotically approaches the latter for small probabilities. Using the example above, the <i>probabilities</i> for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively. The effect size can be computed the same as above, but using the probabilities instead. Therefore, the relative risk is 1.28. Since rather large probabilities of passing were used, there is a large difference between relative risk and odds ratio. Had <i>failure</i> (a smaller probability) been used as the event (rather than <i>passing</i>), the difference between the two measures of effect size would not be so great. </p><p>While both measures are useful, they have different statistical uses. In medical research, the <a href="/wiki/Odds_ratio" title="Odds ratio">odds ratio</a> is commonly used for <a href="/wiki/Case-control_study" class="mw-redirect" title="Case-control study">case-control studies</a>, as odds, but not probabilities, are usually estimated.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Relative risk is commonly used in <a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">randomized controlled trials</a> and <a href="/wiki/Cohort_study" title="Cohort study">cohort studies</a>, but relative risk contributes to overestimations of the effectiveness of interventions.<sup id="cite_ref-Stegenga2015_34-0" class="reference"><a href="#cite_note-Stegenga2015-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Risk_difference">Risk difference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=26" title="Edit section: Risk difference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Risk_difference" title="Risk difference">risk difference</a> (RD), sometimes called absolute risk reduction, is simply the difference in risk (probability) of an event between two groups. It is a useful measure in experimental research, since RD tells you the extent to which an experimental interventions changes the probability of an event or outcome. Using the example above, the probabilities for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively, and so the RD effect size is 0.86 − 0.67 = 0.19 (or 19%). RD is the superior measure for assessing effectiveness of interventions.<sup id="cite_ref-Stegenga2015_34-1" class="reference"><a href="#cite_note-Stegenga2015-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_h"><span id="Cohen.27s_h"></span>Cohen's <i>h</i></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=27" title="Edit section: Cohen's h"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cohen%27s_h" title="Cohen's h">Cohen's h</a></div> <p>One measure used in power analysis when comparing two independent proportions is Cohen's <i>h</i>. This is defined as follows <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 h=2(\arcsin {\sqrt {p_{1}}}-\arcsin {\sqrt {p_{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=2(\arcsin {\sqrt {p_{1}}}-\arcsin {\sqrt {p_{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0f7f1d221ccc53e30222f650a398c662f8252e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.267ex; height:3.176ex;" alt="{\displaystyle h=2(\arcsin {\sqrt {p_{1}}}-\arcsin {\sqrt {p_{2}}})}"></span> where <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub> are the proportions of the two samples being compared and arcsin is the arcsine transformation. </p> <div class="mw-heading mw-heading4"><h4 id="Probability_of_superiority">Probability of superiority</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=28" title="Edit section: Probability of superiority"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Probability_of_superiority" title="Probability of superiority">Probability of superiority</a></div> <p>To more easily describe the meaning of an effect size to people outside statistics, the common language effect size, as the name implies, was designed to communicate it in plain English. It is used to describe a difference between two groups and was proposed, as well as named, by Kenneth McGraw and S. P. Wong in 1992.<sup id="cite_ref-McGraw_KO,_Wong_SP_1992_361–365_35-0" class="reference"><a href="#cite_note-McGraw_KO,_Wong_SP_1992_361–365-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> They used the following example (about heights of men and women): "in any random pairing of young adult males and females, the probability of the male being taller than the female is .92, or in simpler terms yet, in 92 out of 100 blind dates among young adults, the male will be taller than the female",<sup id="cite_ref-McGraw_KO,_Wong_SP_1992_361–365_35-1" class="reference"><a href="#cite_note-McGraw_KO,_Wong_SP_1992_361–365-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> when describing the population value of the common language effect size. </p> <div class="mw-heading mw-heading4"><h4 id="Effect_size_for_ordinal_data">Effect size for ordinal data</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=29" title="Edit section: Effect size for ordinal data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Cliff's delta</b> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, originally developed by <a href="/wiki/Norman_Cliff" title="Norman Cliff">Norman Cliff</a> for use with ordinal data,<sup id="cite_ref-Cliff1993_36-0" class="reference"><a href="#cite_note-Cliff1993-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="I'm at least 80% sure this is just a weird name for Kendall's tau. (May 2024)">dubious</span></a> – <a href="/wiki/Talk:Effect_size#Dubious" title="Talk:Effect size">discuss</a></i>]</sup> is a measure of how often the values in one distribution are larger than the values in a second distribution. Crucially, it does not require any assumptions about the shape or spread of the two distributions. </p><p>The sample estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is given by: <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 d={\frac {\sum _{i,j}[x_{i}>x_{j}]-[x_{i}<x_{j}]}{mn}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {\sum _{i,j}[x_{i}>x_{j}]-[x_{i}<x_{j}]}{mn}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d161ce320bc658346872de788497eced043c8f2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.901ex; height:6.009ex;" alt="{\displaystyle d={\frac {\sum _{i,j}[x_{i}>x_{j}]-[x_{i}<x_{j}]}{mn}}}"></span> where the two distributions are of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> with items <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span>, respectively, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\cdot ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>⋅</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\cdot ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b4e841c71afe1890198191aab15bc225bbc0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.941ex; height:2.843ex;" alt="{\displaystyle [\cdot ]}"></span> is the <a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson bracket</a>, which is 1 when the contents are true and 0 when false. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is linearly related to the <a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Mann–Whitney U statistic</a>; however, it captures the direction of the difference in its sign. Given the Mann–Whitney <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {2U}{mn}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>U</mi> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {2U}{mn}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d0a17c6f41473e6a0420f1cf4cb838369db13a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.588ex; height:5.176ex;" alt="{\displaystyle d={\frac {2U}{mn}}-1}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Cohen's_g"><span id="Cohen.27s_g"></span>Cohen's g</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=30" title="Edit section: Cohen's g"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of simplest effect sizes for measuring how much a proportion differs from 50% is Cohen's g<sup id="cite_ref-CohenJ1988Statistical_10-8" class="reference"><a href="#cite_note-CohenJ1988Statistical-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 147">: 147 </span></sup>. It measures how much a proportion differs from 50%. For example, if 85.2% of arrests for car theft are males, then effect size of sex on arrest when measured with Cohen's g is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=0.852-0.5=0.352}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>0.852</mn> <mo>−</mo> <mn>0.5</mn> <mo>=</mo> <mn>0.352</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=0.852-0.5=0.352}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a0b2f229b63ede1e18265710d3ce323afe5ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.718ex; height:2.509ex;" alt="{\displaystyle g=0.852-0.5=0.352}"></span>. In general: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=P-0.50{\text{ or }}0.50-P\quad ({\text{directional}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>P</mi> <mo>−</mo> <mn>0.50</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mn>0.50</mn> <mo>−</mo> <mi>P</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>directional</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=P-0.50{\text{ or }}0.50-P\quad ({\text{directional}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3801150fac2beb685400d35eaa2d18bc80d295dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.4ex; height:2.843ex;" alt="{\displaystyle g=P-0.50{\text{ or }}0.50-P\quad ({\text{directional}}),}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=|P-0.50|\quad ({\text{nondirectional}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mo>−</mo> <mn>0.50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="1em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>nondirectional</mtext> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=|P-0.50|\quad ({\text{nondirectional}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06789a5d7a1a7ee39cdc98d5992652ae51278e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.486ex; height:2.843ex;" alt="{\displaystyle g=|P-0.50|\quad ({\text{nondirectional}}).}"></span> </p><p>Units of Cohen's g are more intuitive (proportion) than in some other effect sizes. It is sometime used in combination with <a href="/wiki/Binomial_test" title="Binomial test">Binomial test</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Confidence_intervals_by_means_of_noncentrality_parameters">Confidence intervals by means of noncentrality parameters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=31" title="Edit section: Confidence intervals by means of noncentrality parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Confidence intervals of standardized effect sizes, especially Cohen's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397fcfcbb193baab76d57c315c2897b494c914d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle {d}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81efa211d9d04493b68b24bf9843d1967ae22cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:3.009ex;" alt="{\displaystyle f^{2}}"></span>, rely on the calculation of confidence intervals of <a href="/wiki/Noncentrality_parameter" class="mw-redirect" title="Noncentrality parameter">noncentrality parameters</a> (<i>ncp</i>). A common approach to construct the confidence interval of <i>ncp</i> is to find the critical <i>ncp</i> values to fit the observed statistic to tail <a href="/wiki/Quantile" title="Quantile">quantiles</a> <i>α</i>/2 and (1 − <i>α</i>/2). The SAS and R-package MBESS provides functions to find critical values of <i>ncp</i>. </p> <div class="mw-heading mw-heading3"><h3 id="t-test_for_mean_difference_of_single_group_or_two_related_groups"><i>t</i>-test for mean difference of single group or two related groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=32" title="Edit section: t-test for mean difference of single group or two related groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a single group, <i>M</i> denotes the sample mean, <i>μ</i> the population mean, <i>SD</i> the sample's standard deviation, <i>σ</i> the population's standard deviation, and <i>n</i> is the sample size of the group. The <i>t</i> value is used to test the hypothesis on the difference between the mean and a baseline <i>μ</i><sub>baseline</sub>. Usually, <i>μ</i><sub>baseline</sub> is zero. In the case of two related groups, the single group is constructed by the differences in pair of samples, while <i>SD</i> and <i>σ</i> denote the sample's and population's standard deviations of differences rather than within original two groups. <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 t:={\frac {M-\mu _{\text{baseline}}}{\text{SE}}}={\frac {M-\mu _{\text{baseline}}}{{\text{SD}}/{\sqrt {n}}}}={\frac {{\sqrt {n}}\left({\frac {M-\mu }{\sigma }}\right)+{\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}{\frac {\text{SD}}{\sigma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mtext>SE</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SD</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mfrac> <mtext>SD</mtext> <mi>σ</mi> </mfrac> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t:={\frac {M-\mu _{\text{baseline}}}{\text{SE}}}={\frac {M-\mu _{\text{baseline}}}{{\text{SD}}/{\sqrt {n}}}}={\frac {{\sqrt {n}}\left({\frac {M-\mu }{\sigma }}\right)+{\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}{\frac {\text{SD}}{\sigma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6852970a7f89afa1a7535ea76c7961e1170d25fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:67.994ex; height:9.009ex;" alt="{\displaystyle t:={\frac {M-\mu _{\text{baseline}}}{\text{SE}}}={\frac {M-\mu _{\text{baseline}}}{{\text{SD}}/{\sqrt {n}}}}={\frac {{\sqrt {n}}\left({\frac {M-\mu }{\sigma }}\right)+{\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}{\frac {\text{SD}}{\sigma }}}}"></span> <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 ncp={\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>c</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ncp={\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd881c29223e39a824bf480d9b9554c07d340795" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.193ex; height:6.176ex;" alt="{\displaystyle ncp={\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}"></span> and Cohen's <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 d:={\frac {M-\mu _{\text{baseline}}}{\text{SD}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mtext>SD</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d:={\frac {M-\mu _{\text{baseline}}}{\text{SD}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e582cb1078a2c5c27798de1343b0846756de8d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.386ex; height:5.509ex;" alt="{\displaystyle d:={\frac {M-\mu _{\text{baseline}}}{\text{SD}}}}"></span> </p><p>is the point estimate of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu -\mu _{\text{baseline}}}{\sigma }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>baseline</mtext> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu -\mu _{\text{baseline}}}{\sigma }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25831a7bedb5bc63ebda0e6ec3eb6c91fa5acfbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.032ex; height:5.176ex;" alt="{\displaystyle {\frac {\mu -\mu _{\text{baseline}}}{\sigma }}.}"></span> </p><p>So, </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 {\tilde {d}}={\frac {ncp}{\sqrt {n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>c</mi> <mi>p</mi> </mrow> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {d}}={\frac {ncp}{\sqrt {n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84d115a99b348e26c8a229c8399f986047b1d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.734ex; height:5.843ex;" alt="{\displaystyle {\tilde {d}}={\frac {ncp}{\sqrt {n}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="t-test_for_mean_difference_between_two_independent_groups"><i>t</i>-test for mean difference between two independent groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=33" title="Edit section: t-test for mean difference between two independent groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>n</i><sub>1</sub> or <i>n</i><sub>2</sub> are the respective sample sizes. <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 t:={\frac {M_{1}-M_{2}}{{\text{SD}}_{\text{within}}/{\sqrt {\frac {2*n_{1}n_{2}}{n_{1}+n_{2}}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SD</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t:={\frac {M_{1}-M_{2}}{{\text{SD}}_{\text{within}}/{\sqrt {\frac {2*n_{1}n_{2}}{n_{1}+n_{2}}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e77de2ec7f5b32bf5875a1daed1f6a9cd9e7cf17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.539ex; height:8.009ex;" alt="{\displaystyle t:={\frac {M_{1}-M_{2}}{{\text{SD}}_{\text{within}}/{\sqrt {\frac {2*n_{1}n_{2}}{n_{1}+n_{2}}}}}},}"></span> </p><p>wherein <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SD}}_{\text{within}}:={\sqrt {\frac {{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}}}={\sqrt {\frac {(n_{1}-1){\text{SD}}_{1}^{2}+(n_{2}-1){\text{SD}}_{2}^{2}}{n_{1}+n_{2}-2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SD</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>SD</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>SD</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SD}}_{\text{within}}:={\sqrt {\frac {{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}}}={\sqrt {\frac {(n_{1}-1){\text{SD}}_{1}^{2}+(n_{2}-1){\text{SD}}_{2}^{2}}{n_{1}+n_{2}-2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf35b20517f0a82a8f3cd57119b949435180b04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.673ex; height:8.009ex;" alt="{\displaystyle {\text{SD}}_{\text{within}}:={\sqrt {\frac {{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}}}={\sqrt {\frac {(n_{1}-1){\text{SD}}_{1}^{2}+(n_{2}-1){\text{SD}}_{2}^{2}}{n_{1}+n_{2}-2}}}.}"></span> <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 ncp={\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}{\frac {\mu _{1}-\mu _{2}}{\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>c</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ncp={\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}{\frac {\mu _{1}-\mu _{2}}{\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac52e15c9b1fc81a9b273086f7b7037df2d20a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.156ex; height:6.343ex;" alt="{\displaystyle ncp={\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}{\frac {\mu _{1}-\mu _{2}}{\sigma }}}"></span> </p><p>and Cohen's <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 d:={\frac {M_{1}-M_{2}}{SD_{\text{within}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d:={\frac {M_{1}-M_{2}}{SD_{\text{within}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788e3f54517c327a569fe327362a1a40d6ec471e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.254ex; height:5.676ex;" alt="{\displaystyle d:={\frac {M_{1}-M_{2}}{SD_{\text{within}}}}}"></span> is the point estimate of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu _{1}-\mu _{2}}{\sigma }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{1}-\mu _{2}}{\sigma }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3189f810005856476613296eb92d9f47d7c6a6f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.235ex; height:5.176ex;" alt="{\displaystyle {\frac {\mu _{1}-\mu _{2}}{\sigma }}.}"></span> </p><p>So, <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 {\tilde {d}}={\frac {ncp}{\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>c</mi> <mi>p</mi> </mrow> <msqrt> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {d}}={\frac {ncp}{\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13974f80c5a4fd320c371536d6eccd73d0f09359" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:14.237ex; height:7.676ex;" alt="{\displaystyle {\tilde {d}}={\frac {ncp}{\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="One-way_ANOVA_test_for_mean_difference_across_multiple_independent_groups">One-way ANOVA test for mean difference across multiple independent groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Effect_size&action=edit&section=34" title="Edit section: One-way ANOVA test for mean difference across multiple independent groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One-way ANOVA test applies <a href="/wiki/Noncentral_F_distribution" class="mw-redirect" title="Noncentral F distribution">noncentral F distribution</a>. While with a given population standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>, the same test question applies <a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral chi-squared distribution</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:={\frac {{\frac {{\text{SS}}_{\text{between}}}{\sigma ^{2}}}/{\text{df}}_{\text{between}}}{{\frac {{\text{SS}}_{\text{within}}}{\sigma ^{2}}}/{\text{df}}_{\text{within}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>between</mtext> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>between</mtext> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>within</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:={\frac {{\frac {{\text{SS}}_{\text{between}}}{\sigma ^{2}}}/{\text{df}}_{\text{between}}}{{\frac {{\text{SS}}_{\text{within}}}{\sigma ^{2}}}/{\text{df}}_{\text{within}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d654eac83f230b8f81449dc37eb996244bbf81db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:23.138ex; height:9.509ex;" alt="{\displaystyle F:={\frac {{\frac {{\text{SS}}_{\text{between}}}{\sigma ^{2}}}/{\text{df}}_{\text{between}}}{{\frac {{\text{SS}}_{\text{within}}}{\sigma ^{2}}}/{\text{df}}_{\text{within}}}}}"></span> </p><p>For each <i>j</i>-th sample within <i>i</i>-th group <i>X</i><sub><i>i</i>,<i>j</i></sub>, denote <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}(X_{i,j}):={\frac {\sum _{w=1}^{n_{i}}X_{i,w}}{n_{i}}};\;\mu _{i}(X_{i,j}):=\mu _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>;</mo> <mspace width="thickmathspace" /> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}(X_{i,j}):={\frac {\sum _{w=1}^{n_{i}}X_{i,w}}{n_{i}}};\;\mu _{i}(X_{i,j}):=\mu _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cb286692185e97dffd03b9a14f5c600651bc99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.154ex; height:6.176ex;" alt="{\displaystyle M_{i}(X_{i,j}):={\frac {\sum _{w=1}^{n_{i}}X_{i,w}}{n_{i}}};\;\mu _{i}(X_{i,j}):=\mu _{i}.}"></span> </p><p>While, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{SS}}_{\text{between}}/\sigma ^{2}&={\frac {{\text{SS}}\left(M_{i}(X_{i,j});i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{\sigma ^{2}}}\\&={\text{SS}}\left({\frac {M_{i}(X_{i,j}-\mu _{i})}{\sigma }}+{\frac {\mu _{i}}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\\&\sim \chi ^{2}\left({\text{df}}=K-1,\;ncp=SS\left({\frac {\mu _{i}(X_{i,j})}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>between</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>;</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>σ</mi> </mfrac> </mrow> <mo>;</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>df</mtext> </mrow> <mo>=</mo> <mi>K</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mi>c</mi> <mi>p</mi> <mo>=</mo> <mi>S</mi> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>;</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{SS}}_{\text{between}}/\sigma ^{2}&={\frac {{\text{SS}}\left(M_{i}(X_{i,j});i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{\sigma ^{2}}}\\&={\text{SS}}\left({\frac {M_{i}(X_{i,j}-\mu _{i})}{\sigma }}+{\frac {\mu _{i}}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\\&\sim \chi ^{2}\left({\text{df}}=K-1,\;ncp=SS\left({\frac {\mu _{i}(X_{i,j})}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248db8565cb29c5a6929c904e9a048b0cfc6c409" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.697ex; margin-bottom: -0.308ex; width:89.414ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\text{SS}}_{\text{between}}/\sigma ^{2}&={\frac {{\text{SS}}\left(M_{i}(X_{i,j});i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{\sigma ^{2}}}\\&={\text{SS}}\left({\frac {M_{i}(X_{i,j}-\mu _{i})}{\sigma }}+{\frac {\mu _{i}}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\\&\sim \chi ^{2}\left({\text{df}}=K-1,\;ncp=SS\left({\frac {\mu _{i}(X_{i,j})}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\right)\end{aligned}}}"></span> </p><p>So, both <i>ncp</i>(<i>s</i>) of <i>F</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0cc9237ec72a1da6d18bc8e7fb24cdda43a49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:3.009ex;" alt="{\displaystyle \chi ^{2}}"></span> equate <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>σ</mi> <mo>;</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c73bd8e87687f645e692f21b209f6361095aca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.182ex; height:3.009ex;" alt="{\displaystyle {\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right).}"></span> </p><p>In case of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=n_{1}=n_{2}=\cdots =n_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>:=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n:=n_{1}=n_{2}=\cdots =n_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb2bff6b501bb5f37608fb81ab043a3a98bb09d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.144ex; height:2.009ex;" alt="{\displaystyle n:=n_{1}=n_{2}=\cdots =n_{K}}"></span> for <i>K</i> independent groups of same size, the total sample size is <i>N</i> := <i>n</i>·<i>K</i>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Cohens }}{\tilde {f}}^{2}:={\frac {{\text{SS}}(\mu _{1},\mu _{2},\dots ,\mu _{K})}{K\cdot \sigma ^{2}}}={\frac {{\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{n\cdot K}}={\frac {ncp}{n\cdot K}}={\frac {ncp}{N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Cohens </mtext> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>K</mi> <mo>⋅</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>SS</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>σ</mi> <mo>;</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>c</mi> <mi>p</mi> </mrow> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>c</mi> <mi>p</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Cohens }}{\tilde {f}}^{2}:={\frac {{\text{SS}}(\mu _{1},\mu _{2},\dots ,\mu _{K})}{K\cdot \sigma ^{2}}}={\frac {{\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{n\cdot K}}={\frac {ncp}{n\cdot K}}={\frac {ncp}{N}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4017bc7cafb9e6c8ad6fca6a375dd53614c75b17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:102.212ex; height:6.009ex;" alt="{\displaystyle {\text{Cohens }}{\tilde {f}}^{2}:={\frac {{\text{SS}}(\mu _{1},\mu _{2},\dots ,\mu _{K})}{K\cdot \sigma ^{2}}}={\frac {{\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{n\cdot K}}={\frac {ncp}{n\cdot K}}={\frac {ncp}{N}}.}"></span> </p><p>The <i>t</i>-test for a pair of independent groups is a special case of one-way ANOVA. Note that the noncentrality parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ncp_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ncp_{F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c1a65eb6ea6febade1d75291d2681f01f8a2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.034ex; height:2.009ex;" alt="{\displaystyle ncp_{F}}"></span> of <i>F</i> is not comparable to the noncentrality parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ncp_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ncp_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dfe27cdac094bb9cb8ebadb71207b2da2d991d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.397ex; height:2.009ex;" alt="{\displaystyle ncp_{t}}"></span> of the corresponding <i>t</i>. Actually, <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 ncp_{F}=ncp_{t}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>c</mi> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ncp_{F}=ncp_{t}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11404ac5dd91ff00d21f3c8efc54d766492e5fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.758ex; height:3.176ex;" alt="{\displaystyle ncp_{F}=ncp_{t}^{2}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}=\left|{\frac {\tilde {d}}{2}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">~</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}=\left|{\frac {\tilde {d}}{2}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7118431ae8bf970880bdb59ce99861fdf18164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:8.509ex; height:6.843ex;" alt="{\displaystyle {\tilde {f}}=\left|{\frac {\tilde {d}}{2}}\right|}"></span>. </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=Effect_size&action=edit&section=35" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Estimation_statistics" title="Estimation statistics">Estimation statistics</a></li> <li><a href="/wiki/Statistical_significance" title="Statistical significance">Statistical significance</a></li> <li><a href="/wiki/Z-factor" title="Z-factor">Z-factor</a>, an alternative measure of effect size</li></ul> <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=Effect_size&action=edit&section=36" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Kelley2012-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kelley2012_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="CITEREFKelleyPreacher2012" class="citation journal cs1">Kelley, Ken; Preacher, Kristopher J. (2012). "On Effect Size". <i>Psychological Methods</i>. <b>17</b> (2): 137–152. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fa0028086">10.1037/a0028086</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22545595">22545595</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:34152884">34152884</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=On+Effect+Size&rft.volume=17&rft.issue=2&rft.pages=137-152&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A34152884%23id-name%3DS2CID&rft_id=info%3Apmid%2F22545595&rft_id=info%3Adoi%2F10.1037%2Fa0028086&rft.aulast=Kelley&rft.aufirst=Ken&rft.au=Preacher%2C+Kristopher+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Rosenthal, Robert, H. Cooper, and L. Hedges. "Parametric measures of effect size." The handbook of research synthesis 621 (1994): 231–244. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0871541635" title="Special:BookSources/978-0871541635">978-0871541635</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen2016" class="citation book cs1">Cohen, J. (2016). "A power primer". In A. E. Kazdin (ed.). <a rel="nofollow" class="external text" href="https://doi.org/10.1037/14805-018"><i>Methodological issues and strategies in clinical research</i></a> (4th ed.). American Psychological Association. pp. 279–284. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F14805-018">10.1037/14805-018</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4338-2091-5" title="Special:BookSources/978-1-4338-2091-5"><bdi>978-1-4338-2091-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+power+primer&rft.btitle=Methodological+issues+and+strategies+in+clinical+research&rft.pages=279-284&rft.edition=4th&rft.pub=American+Psychological+Association&rft.date=2016&rft_id=info%3Adoi%2F10.1037%2F14805-018&rft.isbn=978-1-4338-2091-5&rft.aulast=Cohen&rft.aufirst=J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1037%2F14805-018&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Wilkinson1999-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wilkinson1999_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wilkinson1999_4-1"><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="CITEREFWilkinson1999" class="citation journal cs1">Wilkinson, Leland (1999). "Statistical methods in psychology journals: Guidelines and explanations". <i>American Psychologist</i>. <b>54</b> (8): 594–604. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0003-066X.54.8.594">10.1037/0003-066X.54.8.594</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:428023">428023</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Psychologist&rft.atitle=Statistical+methods+in+psychology+journals%3A+Guidelines+and+explanations&rft.volume=54&rft.issue=8&rft.pages=594-604&rft.date=1999&rft_id=info%3Adoi%2F10.1037%2F0003-066X.54.8.594&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A428023%23id-name%3DS2CID&rft.aulast=Wilkinson&rft.aufirst=Leland&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Nakagawa2007-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nakagawa2007_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNakagawaCuthill,_Innes_C2007" class="citation journal cs1">Nakagawa, Shinichi; Cuthill, Innes C (2007). "Effect size, confidence interval and statistical significance: a practical guide for biologists". <i>Biological Reviews of the Cambridge Philosophical Society</i>. <b>82</b> (4): 591–605. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1469-185X.2007.00027.x">10.1111/j.1469-185X.2007.00027.x</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17944619">17944619</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:615371">615371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biological+Reviews+of+the+Cambridge+Philosophical+Society&rft.atitle=Effect+size%2C+confidence+interval+and+statistical+significance%3A+a+practical+guide+for+biologists&rft.volume=82&rft.issue=4&rft.pages=591-605&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A615371%23id-name%3DS2CID&rft_id=info%3Apmid%2F17944619&rft_id=info%3Adoi%2F10.1111%2Fj.1469-185X.2007.00027.x&rft.aulast=Nakagawa&rft.aufirst=Shinichi&rft.au=Cuthill%2C+Innes+C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Ellis2010-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ellis2010_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ellis2010_6-1"><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="CITEREFEllis2010" class="citation book cs1">Ellis, Paul D. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5obZnfK5pbsC&pg=PP1"><i>The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-14246-5" title="Special:BookSources/978-0-521-14246-5"><bdi>978-0-521-14246-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Essential+Guide+to+Effect+Sizes%3A+Statistical+Power%2C+Meta-Analysis%2C+and+the+Interpretation+of+Research+Results&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-14246-5&rft.aulast=Ellis&rft.aufirst=Paul+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5obZnfK5pbsC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (August 2016)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-Brand2008-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brand2008_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrandBradleyBestStoica2008" class="citation journal cs1">Brand A, Bradley MT, Best LA, Stoica G (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081217175012/http://mtbradley.com/brandbradelybeststoicapdf.pdf">"Accuracy of effect size estimates from published psychological research"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Perceptual_and_Motor_Skills" title="Perceptual and Motor Skills">Perceptual and Motor Skills</a></i>. <b>106</b> (2): 645–649. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2466%2FPMS.106.2.645-649">10.2466/PMS.106.2.645-649</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18556917">18556917</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14340449">14340449</a>. Archived from <a rel="nofollow" class="external text" href="http://mtbradley.com/brandbradelybeststoicapdf.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-12-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-10-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Perceptual+and+Motor+Skills&rft.atitle=Accuracy+of+effect+size+estimates+from+published+psychological+research&rft.volume=106&rft.issue=2&rft.pages=645-649&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14340449%23id-name%3DS2CID&rft_id=info%3Apmid%2F18556917&rft_id=info%3Adoi%2F10.2466%2FPMS.106.2.645-649&rft.aulast=Brand&rft.aufirst=A&rft.au=Bradley%2C+MT&rft.au=Best%2C+LA&rft.au=Stoica%2C+G&rft_id=http%3A%2F%2Fmtbradley.com%2Fbrandbradelybeststoicapdf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Brand2011-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brand2011_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrandBradleyBestStoica2011" class="citation journal cs1">Brand A, Bradley MT, Best LA, Stoica G (2011). <a rel="nofollow" class="external text" href="http://www.ipsychexpts.com/brand_et_al_(2011).pdf">"Multiple trials may yield exaggerated effect size estimates"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/The_Journal_of_General_Psychology" title="The Journal of General Psychology">The Journal of General Psychology</a></i>. <b>138</b> (1): 1–11. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00221309.2010.520360">10.1080/00221309.2010.520360</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21404946">21404946</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:932324">932324</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+General+Psychology&rft.atitle=Multiple+trials+may+yield+exaggerated+effect+size+estimates&rft.volume=138&rft.issue=1&rft.pages=1-11&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A932324%23id-name%3DS2CID&rft_id=info%3Apmid%2F21404946&rft_id=info%3Adoi%2F10.1080%2F00221309.2010.520360&rft.aulast=Brand&rft.aufirst=A&rft.au=Bradley%2C+MT&rft.au=Best%2C+LA&rft.au=Stoica%2C+G&rft_id=http%3A%2F%2Fwww.ipsychexpts.com%2Fbrand_et_al_%282011%29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSterneGavaghanEgger2000" class="citation journal cs1">Sterne, Jonathan A. C.; Gavaghan, David; Egger, Matthias (2000-11-01). <a rel="nofollow" class="external text" href="https://www.jclinepi.com/article/S0895-4356(00)00242-0/abstract">"Publication and related bias in meta-analysis: Power of statistical tests and prevalence in the literature"</a>. <i>Journal of Clinical Epidemiology</i>. <b>53</b> (11): 1119–1129. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0895-4356%2800%2900242-0">10.1016/S0895-4356(00)00242-0</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0895-4356">0895-4356</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11106885">11106885</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+Clinical+Epidemiology&rft.atitle=Publication+and+related+bias+in+meta-analysis%3A+Power+of+statistical+tests+and+prevalence+in+the+literature&rft.volume=53&rft.issue=11&rft.pages=1119-1129&rft.date=2000-11-01&rft.issn=0895-4356&rft_id=info%3Apmid%2F11106885&rft_id=info%3Adoi%2F10.1016%2FS0895-4356%2800%2900242-0&rft.aulast=Sterne&rft.aufirst=Jonathan+A.+C.&rft.au=Gavaghan%2C+David&rft.au=Egger%2C+Matthias&rft_id=https%3A%2F%2Fwww.jclinepi.com%2Farticle%2FS0895-4356%2800%2900242-0%2Fabstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-CohenJ1988Statistical-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-CohenJ1988Statistical_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-CohenJ1988Statistical_10-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1988" class="citation book cs1"><a href="/wiki/Jacob_Cohen_(statistician)" title="Jacob Cohen (statistician)">Cohen, Jacob</a> (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2v9zDAsLvA0C&pg=PP1"><i>Statistical Power Analysis for the Behavioral Sciences</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-134-74270-7" title="Special:BookSources/978-1-134-74270-7"><bdi>978-1-134-74270-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Power+Analysis+for+the+Behavioral+Sciences&rft.pub=Routledge&rft.date=1988&rft.isbn=978-1-134-74270-7&rft.aulast=Cohen&rft.aufirst=Jacob&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2v9zDAsLvA0C%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Sawilowsky2009-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sawilowsky2009_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sawilowsky2009_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Sawilowsky2009_11-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Sawilowsky2009_11-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Sawilowsky2009_11-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawilowsky2009" class="citation journal cs1">Sawilowsky, S (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1257035100">"New effect size rules of thumb"</a>. <i>Journal of Modern Applied Statistical Methods</i>. <b>8</b> (2): 467–474. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1257035100">10.22237/jmasm/1257035100</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Applied+Statistical+Methods&rft.atitle=New+effect+size+rules+of+thumb&rft.volume=8&rft.issue=2&rft.pages=467-474&rft.date=2009&rft_id=info%3Adoi%2F10.22237%2Fjmasm%2F1257035100&rft.aulast=Sawilowsky&rft.aufirst=S&rft_id=https%3A%2F%2Fdoi.org%2F10.22237%252Fjmasm%252F1257035100&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span> <a rel="nofollow" class="external free" href="http://digitalcommons.wayne.edu/jmasm/vol8/iss2/26/">http://digitalcommons.wayne.edu/jmasm/vol8/iss2/26/</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRussell_V._Lenth" class="citation web cs1">Russell V. Lenth. <a rel="nofollow" class="external text" href="http://www.stat.uiowa.edu/~rlenth/Power/">"Java applets for power and sample size"</a>. Division of Mathematical Sciences, the College of Liberal Arts or The University of Iowa<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-10-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Java+applets+for+power+and+sample+size&rft.pub=Division+of+Mathematical+Sciences%2C+the+College+of+Liberal+Arts+or+The+University+of+Iowa&rft.au=Russell+V.+Lenth&rft_id=http%3A%2F%2Fwww.stat.uiowa.edu%2F~rlenth%2FPower%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Lipsey-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lipsey_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipsey,_M.W.2012" class="citation book cs1">Lipsey, M.W.; et al. (2012). <a rel="nofollow" class="external text" href="http://ies.ed.gov/ncser/pubs/20133000/pdf/20133000.pdf"><i>Translating the Statistical Representation of the Effects of Education Interventions Into More Readily Interpretable Forms</i></a> <span class="cs1-format">(PDF)</span>. United States: U.S. Dept of Education, National Center for Special Education Research, Institute of Education Sciences, NCSER 2013–3000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Translating+the+Statistical+Representation+of+the+Effects+of+Education+Interventions+Into+More+Readily+Interpretable+Forms&rft.place=United+States&rft.pub=U.S.+Dept+of+Education%2C+National+Center+for+Special+Education+Research%2C+Institute+of+Education+Sciences%2C+NCSER+2013%E2%80%933000&rft.date=2012&rft.au=Lipsey%2C+M.W.&rft_id=http%3A%2F%2Fies.ed.gov%2Fncser%2Fpubs%2F20133000%2Fpdf%2F20133000.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawilowsky2005" class="citation journal cs1">Sawilowsky, S. S. (2005). <a rel="nofollow" class="external text" href="http://digitalcommons.wayne.edu/coe_tbf/13">"Abelson's paradox and the Michelson-Morley experiment"</a>. <i>Journal of Modern Applied Statistical Methods</i>. <b>4</b> (1): 352. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1114907520">10.22237/jmasm/1114907520</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Applied+Statistical+Methods&rft.atitle=Abelson%27s+paradox+and+the+Michelson-Morley+experiment&rft.volume=4&rft.issue=1&rft.pages=352&rft.date=2005&rft_id=info%3Adoi%2F10.22237%2Fjmasm%2F1114907520&rft.aulast=Sawilowsky&rft.aufirst=S.+S.&rft_id=http%3A%2F%2Fdigitalcommons.wayne.edu%2Fcoe_tbf%2F13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawilowskySawilowskyGrissom2010" class="citation book cs1">Sawilowsky, S.; Sawilowsky, J.; Grissom, R. J. (2010). "Effect Size". In Lovric, M. (ed.). <i>International Encyclopedia of Statistical Science</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Effect+Size&rft.btitle=International+Encyclopedia+of+Statistical+Science&rft.pub=Springer&rft.date=2010&rft.aulast=Sawilowsky&rft.aufirst=S.&rft.au=Sawilowsky%2C+J.&rft.au=Grissom%2C+R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawilowsky2003" class="citation journal cs1">Sawilowsky, S. (2003). <a rel="nofollow" class="external text" href="http://digitalcommons.wayne.edu/coe_tbf/17">"Deconstructing Arguments from the Case Against Hypothesis Testing"</a>. <i>Journal of Modern Applied Statistical Methods</i>. <b>2</b> (2): 467–474. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.22237%2Fjmasm%2F1067645940">10.22237/jmasm/1067645940</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Applied+Statistical+Methods&rft.atitle=Deconstructing+Arguments+from+the+Case+Against+Hypothesis+Testing&rft.volume=2&rft.issue=2&rft.pages=467-474&rft.date=2003&rft_id=info%3Adoi%2F10.22237%2Fjmasm%2F1067645940&rft.aulast=Sawilowsky&rft.aufirst=S.&rft_id=http%3A%2F%2Fdigitalcommons.wayne.edu%2Fcoe_tbf%2F17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-CohenJ1992-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-CohenJ1992_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1992" class="citation journal cs1">Cohen, J (1992). "A power primer". <i>Psychological Bulletin</i>. <b>112</b> (1): 155–159. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0033-2909.112.1.155">10.1037/0033-2909.112.1.155</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19565683">19565683</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=A+power+primer&rft.volume=112&rft.issue=1&rft.pages=155-159&rft.date=1992&rft_id=info%3Adoi%2F10.1037%2F0033-2909.112.1.155&rft_id=info%3Apmid%2F19565683&rft.aulast=Cohen&rft.aufirst=J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Tabachnick_2007,_p._55-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tabachnick_2007,_p._55_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Tabachnick_2007,_p._55_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Tabachnick, B.G. & Fidell, L.S. (2007). Chapter 4: "Cleaning up your act. Screening data prior to analysis", p. 55 In B.G. Tabachnick & L.S. Fidell (Eds.), <i>Using Multivariate Statistics</i>, Fifth Edition. Boston: Pearson Education, Inc. / Allyn and Bacon.</span> </li> <li id="cite_note-OlejnikAlgina-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-OlejnikAlgina_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-OlejnikAlgina_19-1"><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="CITEREFOlejnikAlgina2003" class="citation journal cs1">Olejnik, S.; Algina, J. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100610101507/http://cps.nova.edu/marker/olejnik2003.pdf">"Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs"</a> <span class="cs1-format">(PDF)</span>. <i>Psychological Methods</i>. <b>8</b> (4): 434–447. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F1082-989x.8.4.434">10.1037/1082-989x.8.4.434</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14664681">14664681</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6931663">6931663</a>. Archived from <a rel="nofollow" class="external text" href="http://cps.nova.edu/marker/olejnik2003.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-06-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-10-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=Generalized+Eta+and+Omega+Squared+Statistics%3A+Measures+of+Effect+Size+for+Some+Common+Research+Designs&rft.volume=8&rft.issue=4&rft.pages=434-447&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6931663%23id-name%3DS2CID&rft_id=info%3Apmid%2F14664681&rft_id=info%3Adoi%2F10.1037%2F1082-989x.8.4.434&rft.aulast=Olejnik&rft.aufirst=S.&rft.au=Algina%2C+J.&rft_id=http%3A%2F%2Fcps.nova.edu%2Fmarker%2Folejnik2003.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Steiger2004-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Steiger2004_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Steiger2004_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Steiger2004_20-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteiger2004" class="citation journal cs1">Steiger, J. H. (2004). <a rel="nofollow" class="external text" href="http://www.statpower.net/Steiger%20Biblio/Steiger04.pdf">"Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis"</a> <span class="cs1-format">(PDF)</span>. <i>Psychological Methods</i>. <b>9</b> (2): 164–182. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F1082-989x.9.2.164">10.1037/1082-989x.9.2.164</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15137887">15137887</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=Beyond+the+F+test%3A+Effect+size+confidence+intervals+and+tests+of+close+fit+in+the+analysis+of+variance+and+contrast+analysis&rft.volume=9&rft.issue=2&rft.pages=164-182&rft.date=2004&rft_id=info%3Adoi%2F10.1037%2F1082-989x.9.2.164&rft_id=info%3Apmid%2F15137887&rft.aulast=Steiger&rft.aufirst=J.+H.&rft_id=http%3A%2F%2Fwww.statpower.net%2FSteiger%2520Biblio%2FSteiger04.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Hair, J.; Hult, T. M.; Ringle, C. M. and Sarstedt, M. (2014) <i>A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM)</i>, Sage, pp. 177–178. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1452217440" title="Special:BookSources/1452217440">1452217440</a></span> </li> <li id="cite_note-HedgesL1985Statistical-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-HedgesL1985Statistical_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-HedgesL1985Statistical_22-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarry_V._Hedges_&_Ingram_Olkin1985" class="citation book cs1"><a href="/wiki/Larry_V._Hedges" title="Larry V. Hedges">Larry V. Hedges</a> & <a href="/wiki/Ingram_Olkin" title="Ingram Olkin">Ingram Olkin</a> (1985). <i>Statistical Methods for Meta-Analysis</i>. Orlando: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-336380-0" title="Special:BookSources/978-0-12-336380-0"><bdi>978-0-12-336380-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Methods+for+Meta-Analysis&rft.place=Orlando&rft.pub=Academic+Press&rft.date=1985&rft.isbn=978-0-12-336380-0&rft.au=Larry+V.+Hedges+%26+Ingram+Olkin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Andrade2020-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Andrade2020_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrade2020" class="citation journal cs1">Andrade, Chittaranjan (22 September 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.4088%2FJCP.20f13681">"Mean Difference, Standardized Mean Difference (SMD), and Their Use in Meta-Analysis"</a>. <i>The Journal of Clinical Psychiatry</i>. <b>81</b> (5). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4088%2FJCP.20f13681">10.4088/JCP.20f13681</a></span>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1555-2101">1555-2101</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32965803">32965803</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221865130">221865130</a>. <q>SMD values of 0.2-0.5 are considered small, values of 0.5-0.8 are considered medium, and values > 0.8 are considered large. In psychopharmacology studies that compare independent groups, SMDs that are statistically significant are almost always in the small to medium range. It is rare for large SMDs to be obtained.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Clinical+Psychiatry&rft.atitle=Mean+Difference%2C+Standardized+Mean+Difference+%28SMD%29%2C+and+Their+Use+in+Meta-Analysis&rft.volume=81&rft.issue=5&rft.date=2020-09-22&rft_id=info%3Apmid%2F32965803&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221865130%23id-name%3DS2CID&rft.eissn=1555-2101&rft_id=info%3Adoi%2F10.4088%2FJCP.20f13681&rft.aulast=Andrade&rft.aufirst=Chittaranjan&rft_id=https%3A%2F%2Fdoi.org%2F10.4088%252FJCP.20f13681&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_E._McGrathGregory_J._Meyer2006" class="citation journal cs1">Robert E. McGrath; Gregory J. Meyer (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131008171400/http://www.bobmcgrath.org/Pubs/When_effect_sizes_disagree.pdf">"When Effect Sizes Disagree: The Case of r and d"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Psychological_Methods" title="Psychological Methods">Psychological Methods</a></i>. <b>11</b> (4): 386–401. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.503.754">10.1.1.503.754</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F1082-989x.11.4.386">10.1037/1082-989x.11.4.386</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17154753">17154753</a>. Archived from <a rel="nofollow" class="external text" href="http://www.bobmcgrath.org/Pubs/When_effect_sizes_disagree.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-10-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-07-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=When+Effect+Sizes+Disagree%3A+The+Case+of+r+and+d&rft.volume=11&rft.issue=4&rft.pages=386-401&rft.date=2006&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.503.754%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F17154753&rft_id=info%3Adoi%2F10.1037%2F1082-989x.11.4.386&rft.au=Robert+E.+McGrath&rft.au=Gregory+J.+Meyer&rft_id=http%3A%2F%2Fwww.bobmcgrath.org%2FPubs%2FWhen_effect_sizes_disagree.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartungKnappSinha2008" class="citation book cs1">Hartung, Joachim; Knapp, Guido; Sinha, Bimal K. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JEoNB_2NONQC&pg=PP1"><i>Statistical Meta-Analysis with Applications</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-21096-3" title="Special:BookSources/978-1-118-21096-3"><bdi>978-1-118-21096-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Meta-Analysis+with+Applications&rft.pub=John+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-1-118-21096-3&rft.aulast=Hartung&rft.aufirst=Joachim&rft.au=Knapp%2C+Guido&rft.au=Sinha%2C+Bimal+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJEoNB_2NONQC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenny1987" class="citation book cs1">Kenny, David A. (1987). <a rel="nofollow" class="external text" href="http://davidakenny.net/doc/statbook/chapter_13.pdf">"Chapter 13"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EdqhQgAACAAJ&pg=PP1"><i>Statistics for the Social and Behavioral Sciences</i></a>. Little, Brown. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-316-48915-7" title="Special:BookSources/978-0-316-48915-7"><bdi>978-0-316-48915-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+13&rft.btitle=Statistics+for+the+Social+and+Behavioral+Sciences&rft.pub=Little%2C+Brown&rft.date=1987&rft.isbn=978-0-316-48915-7&rft.aulast=Kenny&rft.aufirst=David+A.&rft_id=http%3A%2F%2Fdavidakenny.net%2Fdoc%2Fstatbook%2Fchapter_13.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECohen198849-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECohen198849_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCohen1988">Cohen 1988</a>, p. 49.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarry_V._Hedges1981" class="citation journal cs1"><a href="/wiki/Larry_V._Hedges" title="Larry V. Hedges">Larry V. Hedges</a> (1981). "Distribution theory for Glass' estimator of effect size and related estimators". <i><a href="/wiki/Journal_of_Educational_Statistics" class="mw-redirect" title="Journal of Educational Statistics">Journal of Educational Statistics</a></i>. <b>6</b> (2): 107–128. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3102%2F10769986006002107">10.3102/10769986006002107</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121719955">121719955</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+Educational+Statistics&rft.atitle=Distribution+theory+for+Glass%27+estimator+of+effect+size+and+related+estimators&rft.volume=6&rft.issue=2&rft.pages=107-128&rft.date=1981&rft_id=info%3Adoi%2F10.3102%2F10769986006002107&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121719955%23id-name%3DS2CID&rft.au=Larry+V.+Hedges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Hedges, L. V. (2011). Effect sizes in three-level cluster-randomized experiments. Journal of Educational and Behavioral Statistics, 36(3), 346-380.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDel_Giudice2013" class="citation journal cs1">Del Giudice, Marco (2013-07-18). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10434404">"Multivariate Misgivings: Is D a Valid Measure of Group and Sex Differences?"</a>. <i>Evolutionary Psychology</i>. <b>11</b> (5): 147470491301100. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F147470491301100511">10.1177/147470491301100511</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10434404">10434404</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Evolutionary+Psychology&rft.atitle=Multivariate+Misgivings%3A+Is+D+a+Valid+Measure+of+Group+and+Sex+Differences%3F&rft.volume=11&rft.issue=5&rft.pages=147470491301100&rft.date=2013-07-18&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10434404%23id-name%3DPMC&rft_id=info%3Adoi%2F10.1177%2F147470491301100511&rft.aulast=Del+Giudice&rft.aufirst=Marco&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10434404&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Ref_-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ref_31-0">^</a></b></span> <span class="reference-text">Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). <a rel="nofollow" class="external text" href="http://www.eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED433353&ERICExtSearch_SearchType_0=no&accno=ED433353">Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula.</a> Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)</span> </li> <li id="cite_note-Ref_a-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ref_a_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSheskin2003" class="citation book cs1">Sheskin, David J. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bmwhcJqq01cC&pg=PP1"><i>Handbook of Parametric and Nonparametric Statistical Procedures</i></a> (Third ed.). CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-3626-8" title="Special:BookSources/978-1-4200-3626-8"><bdi>978-1-4200-3626-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Parametric+and+Nonparametric+Statistical+Procedures&rft.edition=Third&rft.pub=CRC+Press&rft.date=2003&rft.isbn=978-1-4200-3626-8&rft.aulast=Sheskin&rft.aufirst=David+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbmwhcJqq01cC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeeks_J1998" class="citation journal cs1">Deeks J (1998). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1114127">"When can odds ratios mislead? : Odds ratios should be used only in case-control studies and logistic regression analyses"</a>. <i>BMJ</i>. <b>317</b> (7166): 1155–6. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1136%2Fbmj.317.7166.1155a">10.1136/bmj.317.7166.1155a</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1114127">1114127</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9784470">9784470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BMJ&rft.atitle=When+can+odds+ratios+mislead%3F+%3A+Odds+ratios+should+be+used+only+in+case-control+studies+and+logistic+regression+analyses&rft.volume=317&rft.issue=7166&rft.pages=1155-6&rft.date=1998&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1114127%23id-name%3DPMC&rft_id=info%3Apmid%2F9784470&rft_id=info%3Adoi%2F10.1136%2Fbmj.317.7166.1155a&rft.au=Deeks+J&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1114127&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Stegenga2015-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stegenga2015_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stegenga2015_34-1"><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="CITEREFStegenga2015" class="citation journal cs1">Stegenga, J. (2015). <a rel="nofollow" class="external text" href="https://www.academia.edu/16420844">"Measuring Effectiveness"</a>. <i>Studies in History and Philosophy of Biological and Biomedical Sciences</i>. <b>54</b>: 62–71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.shpsc.2015.06.003">10.1016/j.shpsc.2015.06.003</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26199055">26199055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Biological+and+Biomedical+Sciences&rft.atitle=Measuring+Effectiveness&rft.volume=54&rft.pages=62-71&rft.date=2015&rft_id=info%3Adoi%2F10.1016%2Fj.shpsc.2015.06.003&rft_id=info%3Apmid%2F26199055&rft.aulast=Stegenga&rft.aufirst=J.&rft_id=https%3A%2F%2Fwww.academia.edu%2F16420844&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-McGraw_KO,_Wong_SP_1992_361–365-35"><span class="mw-cite-backlink">^ <a href="#cite_ref-McGraw_KO,_Wong_SP_1992_361–365_35-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-McGraw_KO,_Wong_SP_1992_361–365_35-1"><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="CITEREFMcGrawWong1992" class="citation journal cs1">McGraw KO, Wong SP (1992). "A common language effect size statistic". <i><a href="/wiki/Psychological_Bulletin" title="Psychological Bulletin">Psychological Bulletin</a></i>. <b>111</b> (2): 361–365. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0033-2909.111.2.361">10.1037/0033-2909.111.2.361</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=A+common+language+effect+size+statistic&rft.volume=111&rft.issue=2&rft.pages=361-365&rft.date=1992&rft_id=info%3Adoi%2F10.1037%2F0033-2909.111.2.361&rft.aulast=McGraw&rft.aufirst=KO&rft.au=Wong%2C+SP&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> <li id="cite_note-Cliff1993-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cliff1993_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCliff1993" class="citation journal cs1">Cliff, Norman (1993). "Dominance statistics: Ordinal analyses to answer ordinal questions". <i>Psychological Bulletin</i>. <b>114</b> (3): 494–509. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0033-2909.114.3.494">10.1037/0033-2909.114.3.494</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=Dominance+statistics%3A+Ordinal+analyses+to+answer+ordinal+questions&rft.volume=114&rft.issue=3&rft.pages=494-509&rft.date=1993&rft_id=info%3Adoi%2F10.1037%2F0033-2909.114.3.494&rft.aulast=Cliff&rft.aufirst=Norman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></span> </li> </ol></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=Effect_size&action=edit&section=37" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. <a rel="nofollow" class="external text" href="http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED433353">(ERIC Document Reproduction Service No. ED433353)</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBonett2008" class="citation journal cs1">Bonett, D. G. (2008). "Confidence intervals for standardized linear contrasts of means". <i>Psychological Methods</i>. <b>13</b> (2): 99–109. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F1082-989x.13.2.99">10.1037/1082-989x.13.2.99</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18557680">18557680</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=Confidence+intervals+for+standardized+linear+contrasts+of+means&rft.volume=13&rft.issue=2&rft.pages=99-109&rft.date=2008&rft_id=info%3Adoi%2F10.1037%2F1082-989x.13.2.99&rft_id=info%3Apmid%2F18557680&rft.aulast=Bonett&rft.aufirst=D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBonett2009" class="citation journal cs1">Bonett, D. G. (2009). "Estimating standardized linear contrasts of means with desired precision". <i>Psychological Methods</i>. <b>14</b> (1): 1–5. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fa0014270">10.1037/a0014270</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19271844">19271844</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Methods&rft.atitle=Estimating+standardized+linear+contrasts+of+means+with+desired+precision&rft.volume=14&rft.issue=1&rft.pages=1-5&rft.date=2009&rft_id=info%3Adoi%2F10.1037%2Fa0014270&rft_id=info%3Apmid%2F19271844&rft.aulast=Bonett&rft.aufirst=D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrooksDalalNolan2013" class="citation journal cs1">Brooks, M.E.; Dalal, D.K.; Nolan, K.P. (2013). "Are common language effect sizes easier to understand than traditional effect sizes?". <i><a href="/wiki/Journal_of_Applied_Psychology" title="Journal of Applied Psychology">Journal of Applied Psychology</a></i>. <b>99</b> (2): 332–340. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fa0034745">10.1037/a0034745</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24188393">24188393</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+Applied+Psychology&rft.atitle=Are+common+language+effect+sizes+easier+to+understand+than+traditional+effect+sizes%3F&rft.volume=99&rft.issue=2&rft.pages=332-340&rft.date=2013&rft_id=info%3Adoi%2F10.1037%2Fa0034745&rft_id=info%3Apmid%2F24188393&rft.aulast=Brooks&rft.aufirst=M.E.&rft.au=Dalal%2C+D.K.&rft.au=Nolan%2C+K.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCummingFinch2001" class="citation journal cs1">Cumming, G.; Finch, S. (2001). "A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions". <i>Educational and Psychological Measurement</i>. <b>61</b> (4): 530–572. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F0013164401614002">10.1177/0013164401614002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120672914">120672914</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Educational+and+Psychological+Measurement&rft.atitle=A+primer+on+the+understanding%2C+use%2C+and+calculation+of+confidence+intervals+that+are+based+on+central+and+noncentral+distributions&rft.volume=61&rft.issue=4&rft.pages=530-572&rft.date=2001&rft_id=info%3Adoi%2F10.1177%2F0013164401614002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120672914%23id-name%3DS2CID&rft.aulast=Cumming&rft.aufirst=G.&rft.au=Finch%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKelley2007" class="citation journal cs1">Kelley, K (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v020.i08">"Confidence intervals for standardized effect sizes: Theory, application, and implementation"</a>. <i>Journal of Statistical Software</i>. <b>20</b> (8): 1–24. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v020.i08">10.18637/jss.v020.i08</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Software&rft.atitle=Confidence+intervals+for+standardized+effect+sizes%3A+Theory%2C+application%2C+and+implementation&rft.volume=20&rft.issue=8&rft.pages=1-24&rft.date=2007&rft_id=info%3Adoi%2F10.18637%2Fjss.v020.i08&rft.aulast=Kelley&rft.aufirst=K&rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v020.i08&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEffect+size" class="Z3988"></span></li> <li>Lipsey, M. W., & Wilson, D. B. (2001). <i>Practical meta-analysis</i>. Sage: Thousand Oaks, CA.</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=Effect_size&action=edit&section=38" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Special:Search/Effect_size" class="extiw" title="v:Special:Search/Effect size">Effect size</a></b></i></div></div> </div> <p><b>Further explanations</b> </p> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110927074709/http://www.uccs.edu/~faculty/lbecker/es.htm">Effect Size (ES)</a></li> <li><a rel="nofollow" class="external text" href="http://effectsizefaq.com/">EffectSizeFAQ.com</a></li> <li><a rel="nofollow" class="external text" href="https://www.estimationstats.com/#/">EstimationStats.com</a> Web app for generating effect-size plots.</li> <li><a rel="nofollow" class="external text" href="http://davidmlane.com/hyperstat/effect_size.html">Measuring Effect Size</a></li> <li><a rel="nofollow" class="external text" href="http://www.tqmp.org/Content/vol05-1/p025/p025.pdf">Computing and Interpreting Effect size Measures with ViSta</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141227181333/http://www.tqmp.org/Content/vol05-1/p025/p025.pdf">Archived</a> 2014-12-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://CRAN.R-project.org/package=effsize">effsize package for the R Project for Statistical Computing </a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></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"><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: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 uncollapsed 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 class="mw-selflink selflink">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier 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 class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li> <li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Geostatistics" title="Geostatistics">Geostatistics</a></li> <li><a href="/wiki/Kriging" title="Kriging">Kriging</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Category:Statistics" title="Category:Statistics">Category</a></b></li> <li><b><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></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw" title="commons:Category:Statistics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Statistics" title="Wikipedia:WikiProject Statistics">WikiProject</a></b></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="Design_of_experiments" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse 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:Experimental_design" title="Template:Experimental design"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Experimental_design" title="Template talk:Experimental design"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Experimental_design" title="Special:EditPage/Template:Experimental design"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Design_of_experiments" style="font-size:114%;margin:0 4em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Scientific_method" title="Scientific method">Scientific<br />method</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/Experiment" title="Experiment">Scientific experiment</a></li> <li><a href="/wiki/Design_of_experiments" title="Design of experiments">Statistical design</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Control</a></li> <li><a href="/wiki/Internal_validity" title="Internal validity">Internal</a> and <a href="/wiki/External_validity" title="External validity">external</a> <a href="/wiki/Validity_(statistics)" title="Validity (statistics)">validity</a></li> <li><a href="/wiki/Experimental_unit" class="mw-redirect" title="Experimental unit">Experimental unit</a></li> <li><a href="/wiki/Blind_experiment" class="mw-redirect" title="Blind experiment">Blinding</a></li></ul> <ul><li><b><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></b>: <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomization" title="Randomization">Randomization</a></li> <li><a href="/wiki/Restricted_randomization" title="Restricted randomization">Restricted randomization</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication versus subsampling</a></li> <li><a href="/wiki/Sample_size" class="mw-redirect" title="Sample size">Sample size</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a><br /> and <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">blocking</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><b><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a></b></li> <li><a class="mw-selflink selflink">Effect size</a></li> <li><a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">Contrast</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding</a></li> <li><a href="/wiki/Orthogonality#Statistics,_econometrics,_and_economics" title="Orthogonality">Orthogonality</a></li> <li><b><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></b></li> <li><a href="/wiki/Covariate" class="mw-redirect" title="Covariate">Covariate</a></li> <li><a href="/wiki/Nuisance_variable" title="Nuisance variable">Nuisance variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_model" title="Statistical model">Models</a> <br /> and <a href="/wiki/Statistical_inference" title="Statistical inference">inference</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/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li></ul> <ul><li><b><a href="/wiki/Random_effect" class="mw-redirect" title="Random effect">Random effect</a></b></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed model</a></li> <li><a href="/wiki/Hierarchical_linear_modeling" class="mw-redirect" title="Hierarchical linear modeling">Hierarchical model:</a> <a href="/wiki/Hierarchical_Bayes_model" class="mw-redirect" title="Hierarchical Bayes model">Bayesian</a></li></ul> <ul><li><b><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (Anova)</a></b></li> <li><a href="/wiki/Cochran%27s_theorem" title="Cochran's theorem">Cochran's theorem</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance"><b>Manova</b> (<i>multivariate</i>)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance"><b>Ancova</b> (<i>covariance</i>)</a></li></ul> <ul><li><a href="/wiki/Comparing_means" class="mw-redirect" title="Comparing means">Compare means</a></li> <li><a href="/wiki/Multiple_comparison" class="mw-redirect" title="Multiple comparison">Multiple comparison</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">Designs</a> <br /> <br /><a href="/wiki/Completely_randomized_design" title="Completely randomized design">Completely<br />randomized</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><b><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial</a></b></li> <li><a href="/wiki/Fractional_factorial_design" title="Fractional factorial design">Fractional factorial</a></li> <li><a href="/wiki/Plackett-Burman_design" class="mw-redirect" title="Plackett-Burman design">Plackett–Burman</a></li> <li><a href="/wiki/Taguchi_methods" title="Taguchi methods">Taguchi</a></li></ul> <ul><li><b><a href="/wiki/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></b></li> <li><a href="/wiki/Polynomial_and_rational_function_modeling" title="Polynomial and rational function modeling">Polynomial and rational modeling</a></li> <li><a href="/wiki/Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken</a></li> <li><a href="/wiki/Central_composite_design" title="Central composite design">Central composite</a></li></ul> <ul><li><b><a href="/wiki/Randomized_block_design" class="mw-redirect" title="Randomized block design">Block</a></b></li> <li><a href="/wiki/Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a> (GRBD)</li> <li><a href="/wiki/Latin_square" title="Latin square">Latin square</a></li> <li><a href="/wiki/Graeco-Latin_square" class="mw-redirect" title="Graeco-Latin square">Graeco-Latin square</a></li> <li><a href="/wiki/Orthogonal_array" title="Orthogonal array">Orthogonal array</a></li> <li><a href="/wiki/Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube</a> <br /> <b><a href="/wiki/Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></b></li> <li><a href="/wiki/Crossover_study" title="Crossover study">Crossover study</a></li></ul> <ul><li><b><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></b></li> <li><a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a></li> <li><a href="/wiki/Sequential_probability_ratio_test" title="Sequential probability ratio test">Sequential probability ratio test</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Glossary</a></li> <li><a href="/wiki/Category:Design_of_experiments" title="Category:Design of experiments">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">Statistical outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Statistical topics</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Effect_size&oldid=1258181819">https://en.wikipedia.org/w/index.php?title=Effect_size&oldid=1258181819</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Effect_size" title="Category:Effect size">Effect size</a></li><li><a href="/wiki/Category:Clinical_research" title="Category:Clinical research">Clinical research</a></li><li><a href="/wiki/Category:Psychometrics" title="Category:Psychometrics">Psychometrics</a></li><li><a href="/wiki/Category:Statistical_hypothesis_testing" title="Category:Statistical hypothesis testing">Statistical hypothesis testing</a></li><li><a href="/wiki/Category:Clinical_trials" title="Category:Clinical trials">Clinical trials</a></li><li><a href="/wiki/Category:Meta-analysis" title="Category:Meta-analysis">Meta-analysis</a></li><li><a href="/wiki/Category:Medical_statistics" title="Category:Medical statistics">Medical statistics</a></li><li><a href="/wiki/Category:Mathematical_and_quantitative_methods_(economics)" title="Category:Mathematical and quantitative methods (economics)">Mathematical and quantitative methods (economics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Wikipedia_articles_needing_page_number_citations_from_August_2016" title="Category:Wikipedia articles needing page number citations from August 2016">Wikipedia articles needing page number citations from August 2016</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_cleanup_from_May_2011" title="Category:Articles needing cleanup from May 2011">Articles needing cleanup from May 2011</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Cleanup_tagged_articles_with_a_reason_field_from_May_2011" title="Category:Cleanup tagged articles with a reason field from May 2011">Cleanup tagged articles with a reason field from May 2011</a></li><li><a href="/wiki/Category:Wikipedia_pages_needing_cleanup_from_May_2011" title="Category:Wikipedia pages needing cleanup from May 2011">Wikipedia pages needing cleanup from May 2011</a></li><li><a href="/wiki/Category:Wikipedia_articles_that_are_too_technical_from_February_2014" title="Category:Wikipedia articles that are too technical from February 2014">Wikipedia articles that are too technical from February 2014</a></li><li><a href="/wiki/Category:All_articles_that_are_too_technical" title="Category:All articles that are too technical">All articles that are too technical</a></li><li><a href="/wiki/Category:Articles_with_multiple_maintenance_issues" title="Category:Articles with multiple maintenance issues">Articles with multiple maintenance issues</a></li><li><a href="/wiki/Category:All_accuracy_disputes" title="Category:All accuracy disputes">All accuracy disputes</a></li><li><a href="/wiki/Category:Articles_with_disputed_statements_from_May_2024" title="Category:Articles with disputed statements from May 2024">Articles with disputed statements from May 2024</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 November 2024, at 15:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Effect_size&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-hxktd","wgBackendResponseTime":1548,"wgPageParseReport":{"limitreport":{"cputime":"1.100","walltime":"1.378","ppvisitednodes":{"value":6964,"limit":1000000},"postexpandincludesize":{"value":272938,"limit":2097152},"templateargumentsize":{"value":12115,"limit":2097152},"expansiondepth":{"value":20,"limit":100},"expensivefunctioncount":{"value":12,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":169271,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1003.255 1 -total"," 33.48% 335.901 1 Template:Reflist"," 22.72% 227.931 25 Template:Cite_journal"," 17.16% 172.131 1 Template:Statistics"," 16.87% 169.289 1 Template:Navbox_with_collapsible_groups"," 10.73% 107.660 1 Template:Multiple_issues"," 9.44% 94.707 1 Template:Short_description"," 7.43% 74.534 1 Template:Cleanup"," 7.40% 74.208 12 Template:Main_other"," 6.48% 65.002 8 Template:Rp"]},"scribunto":{"limitreport-timeusage":{"value":"0.597","limit":"10.000"},"limitreport-memusage":{"value":9023162,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAndrade2020\"] = 1,\n [\"CITEREFBonett2008\"] = 1,\n [\"CITEREFBonett2009\"] = 1,\n [\"CITEREFBrandBradleyBestStoica2008\"] = 1,\n [\"CITEREFBrandBradleyBestStoica2011\"] = 1,\n [\"CITEREFBrooksDalalNolan2013\"] = 1,\n [\"CITEREFCliff1993\"] = 1,\n [\"CITEREFCohen1988\"] = 1,\n [\"CITEREFCohen1992\"] = 1,\n [\"CITEREFCohen2016\"] = 1,\n [\"CITEREFCummingFinch2001\"] = 1,\n [\"CITEREFDeeks_J1998\"] = 1,\n [\"CITEREFDel_Giudice2013\"] = 1,\n [\"CITEREFEllis2010\"] = 1,\n [\"CITEREFHartungKnappSinha2008\"] = 1,\n [\"CITEREFKelley2007\"] = 1,\n [\"CITEREFKelleyPreacher2012\"] = 1,\n [\"CITEREFKenny1987\"] = 1,\n [\"CITEREFLarry_V._Hedges1981\"] = 1,\n [\"CITEREFLarry_V._Hedges_\u0026amp;_Ingram_Olkin1985\"] = 1,\n [\"CITEREFLipsey,_M.W.2012\"] = 1,\n [\"CITEREFMcGrawWong1992\"] = 1,\n [\"CITEREFNakagawaCuthill,_Innes_C2007\"] = 1,\n [\"CITEREFOlejnikAlgina2003\"] = 1,\n [\"CITEREFRobert_E._McGrathGregory_J._Meyer2006\"] = 1,\n [\"CITEREFRussell_V._Lenth\"] = 1,\n [\"CITEREFSawilowsky2003\"] = 1,\n [\"CITEREFSawilowsky2005\"] = 1,\n [\"CITEREFSawilowsky2009\"] = 1,\n [\"CITEREFSawilowskySawilowskyGrissom2010\"] = 1,\n [\"CITEREFSheskin2003\"] = 1,\n [\"CITEREFStegenga2015\"] = 1,\n [\"CITEREFSteiger2004\"] = 1,\n [\"CITEREFSterneGavaghanEgger2000\"] = 1,\n [\"CITEREFWilkinson1999\"] = 1,\n [\"Cohen\u0026#039;s_d\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 1,\n [\"Anchor\"] = 1,\n [\"Cite book\"] = 9,\n [\"Cite journal\"] = 25,\n [\"Cite web\"] = 1,\n [\"Cleanup\"] = 1,\n [\"Dubious\"] = 1,\n [\"Experimental design\"] = 1,\n [\"ISBN\"] = 2,\n [\"Main\"] = 2,\n [\"Math\"] = 2,\n [\"Multiple issues\"] = 1,\n [\"Page needed\"] = 1,\n [\"Quotation\"] = 1,\n [\"Reflist\"] = 1,\n [\"Rp\"] = 8,\n [\"See also\"] = 1,\n [\"Sfn\"] = 1,\n [\"Short description\"] = 1,\n [\"Statistics\"] = 1,\n [\"Technical\"] = 1,\n [\"Webarchive\"] = 1,\n [\"Wikiversity\"] = 1,\n}\narticle_whitelist = table#1 {\n}\ntable#1 {\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-hxktd","timestamp":"20241125133423","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Effect size","url":"https:\/\/en.wikipedia.org\/wiki\/Effect_size","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1287978","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1287978","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-01-21T04:03:04Z","dateModified":"2024-11-18T15:49:19Z","headline":"statistical measure of the magnitude of a phenomenon"}</script> </body> </html>

CINXE.COM

Effect size - Wikipedia