Kurtosis - Wikipedia

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class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Excess_kurtosis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Excess_kurtosis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Excess kurtosis</span> </div> </a> <button aria-controls="toc-Excess_kurtosis-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 Excess kurtosis subsection</span> </button> <ul id="toc-Excess_kurtosis-sublist" class="vector-toc-list"> <li id="toc-Mesokurtic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mesokurtic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Mesokurtic</span> </div> </a> <ul id="toc-Mesokurtic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Leptokurtic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Leptokurtic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Leptokurtic</span> </div> </a> <ul id="toc-Leptokurtic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Platykurtic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Platykurtic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Platykurtic</span> </div> </a> <ul id="toc-Platykurtic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graphical_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graphical_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Graphical examples</span> </div> </a> <button aria-controls="toc-Graphical_examples-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 Graphical examples subsection</span> </button> <ul id="toc-Graphical_examples-sublist" class="vector-toc-list"> <li id="toc-The_Pearson_type_VII_family" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Pearson_type_VII_family"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The Pearson type VII family</span> </div> </a> <ul id="toc-The_Pearson_type_VII_family-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_well-known_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_well-known_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Other well-known distributions</span> </div> </a> <ul id="toc-Other_well-known_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sample_kurtosis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sample_kurtosis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sample kurtosis</span> </div> </a> <button aria-controls="toc-Sample_kurtosis-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 Sample kurtosis subsection</span> </button> <ul id="toc-Sample_kurtosis-sublist" class="vector-toc-list"> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-A_natural_but_biased_estimator" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#A_natural_but_biased_estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>A natural but biased estimator</span> </div> </a> <ul id="toc-A_natural_but_biased_estimator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standard_unbiased_estimator" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Standard_unbiased_estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Standard unbiased estimator</span> </div> </a> <ul id="toc-Standard_unbiased_estimator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Upper_bound" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Upper_bound"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Upper bound</span> </div> </a> <ul id="toc-Upper_bound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variance_under_normality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variance_under_normality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Variance under normality</span> </div> </a> <ul id="toc-Variance_under_normality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Kurtosis_convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kurtosis_convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Kurtosis convergence</span> </div> </a> <ul id="toc-Kurtosis_convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seismic_signal_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seismic_signal_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Seismic signal analysis</span> </div> </a> <ul id="toc-Seismic_signal_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weather_prediction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weather_prediction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Weather prediction</span> </div> </a> <ul id="toc-Weather_prediction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_measures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other measures</span> </div> </a> <ul id="toc-Other_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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" 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Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%D8%B7%D8%AD" title="تفرطح – Arabic" lang="ar" hreflang="ar" data-title="تفرطح" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Curtosi" title="Curtosi – Catalan" lang="ca" hreflang="ca" data-title="Curtosi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Koeficient_%C5%A1pi%C4%8Datosti" title="Koeficient špičatosti – Czech" lang="cs" hreflang="cs" data-title="Koeficient špičatosti" 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/W%C3%B6lbung_(Statistik)" title="Wölbung (Statistik) – German" lang="de" hreflang="de" data-title="Wölbung (Statistik)" 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/Curtosis" title="Curtosis – Spanish" lang="es" hreflang="es" data-title="Curtosis" 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/Kurtosi" title="Kurtosi – Basque" lang="eu" hreflang="eu" data-title="Kurtosi" 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/%DA%A9%D8%B4%DB%8C%D8%AF%DA%AF%DB%8C_(%D8%A2%D9%85%D8%A7%D8%B1)" title="کشیدگی (آمار) – Persian" lang="fa" hreflang="fa" data-title="کشیدگی (آمار)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Kurtosis" title="Kurtosis – French" lang="fr" hreflang="fr" data-title="Kurtosis" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Curtose" title="Curtose – Galician" lang="gl" hreflang="gl" data-title="Curtose" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%A8%EB%8F%84" title="첨도 – Korean" lang="ko" hreflang="ko" data-title="첨도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Curtosi" title="Curtosi – Italian" lang="it" hreflang="it" data-title="Curtosi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%91%D7%A0%D7%95%D7%A0%D7%99%D7%95%D7%AA_(%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94)" title="גבנוניות (סטטיסטיקה) – Hebrew" lang="he" hreflang="he" data-title="גבנוניות (סטטיסטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kurtosis" title="Kurtosis – Malay" lang="ms" hreflang="ms" data-title="Kurtosis" 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/Kurtosis" title="Kurtosis – Dutch" lang="nl" hreflang="nl" data-title="Kurtosis" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B0%96%E5%BA%A6" title="尖度 – Japanese" lang="ja" hreflang="ja" data-title="尖度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kurtose" title="Kurtose – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kurtose" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kurtoza" title="Kurtoza – Polish" lang="pl" hreflang="pl" data-title="Kurtoza" 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/Curtose" title="Curtose – Portuguese" lang="pt" hreflang="pt" data-title="Curtose" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Kurtoza_e_tep%C3%ABrt" title="Kurtoza e tepërt – Albanian" lang="sq" hreflang="sq" data-title="Kurtoza e tepërt" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D1%83%D1%80%D1%82%D0%BE%D0%B7%D0%B8%D1%81" title="Куртозис – Serbian" lang="sr" hreflang="sr" data-title="Куртозис" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Kurtosis" title="Kurtosis – Sundanese" lang="su" hreflang="su" data-title="Kurtosis" 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/Kurtosis" title="Kurtosis – Swedish" lang="sv" hreflang="sv" data-title="Kurtosis" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bas%C4%B1kl%C4%B1k" title="Basıklık – Turkish" lang="tr" hreflang="tr" data-title="Basıklık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%99_nh%E1%BB%8Dn_(th%E1%BB%91ng_k%C3%AA)" title="Độ nhọn (thống kê) – Vietnamese" lang="vi" hreflang="vi" data-title="Độ nhọn (thống kê)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B3%B0%E5%BA%A6" title="峰度 – Cantonese" lang="yue" hreflang="yue" data-title="峰度" 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/%E5%B3%B0%E5%BA%A6" title="峰度 – Chinese" lang="zh" hreflang="zh" data-title="峰度" 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/Q287251#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/Kurtosis" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon 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<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">Fourth standardized moment in statistics</div> <p class="mw-empty-elt"> </p><p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>kurtosis</b> (from <a href="/wiki/Greek_language" title="Greek language">Greek</a>: <span lang="el">κυρτός</span>, <i>kyrtos</i> or <i>kurtos</i>, meaning "curved, arching") refers to the degree of “tailedness” in the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of a <a href="/wiki/Real-valued" class="mw-redirect" title="Real-valued">real-valued</a> <a href="/wiki/Random_variable" title="Random variable">random variable</a>. Similar to <a href="/wiki/Skewness" title="Skewness">skewness</a>, kurtosis provides insight into specific characteristics of a distribution. Various methods exist for quantifying kurtosis in theoretical distributions, and corresponding techniques allow estimation based on sample data from a population. It’s important to note that different measures of kurtosis can yield varying <a href="#Interpretation">interpretations</a>. </p><p>The standard measure of a distribution's kurtosis, originating with <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a>,<sup id="cite_ref-Pearson1905_1-0" class="reference"><a href="#cite_note-Pearson1905-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a scaled version of the fourth <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moment</a> of the distribution. This number is related to the tails of the distribution, not its peak;<sup id="cite_ref-Westfall2014_2-0" class="reference"><a href="#cite_note-Westfall2014-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> hence, the sometimes-seen characterization of kurtosis as "<a href="/wiki/Peakedness" class="mw-redirect" title="Peakedness">peakedness</a>" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of <a href="/wiki/Deviation_(statistics)" title="Deviation (statistics)">deviations</a> (or <a href="/wiki/Outlier" title="Outlier">outliers</a>), and not the configuration of data near <a href="/wiki/Mean" title="Mean">the mean</a>. </p><p>Excess kurtosis, typically compared to a value of 0, characterizes the “tailedness” of a distribution. A univariate <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> has an excess kurtosis of 0. Negative excess kurtosis indicates a platykurtic distribution, which doesn’t necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution. For instance, the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> is platykurtic. On the other hand, positive excess kurtosis signifies a leptokurtic distribution. The <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a>, for example, has tails that decay more slowly than a Gaussian, resulting in more outliers. To simplify comparison with the normal distribution, excess kurtosis is calculated as Pearson’s kurtosis minus 3. Some authors and software packages use “kurtosis” to refer specifically to excess kurtosis, but this article distinguishes between the two for clarity. </p><p>Alternative measures of kurtosis are: the <a href="/wiki/L-kurtosis" class="mw-redirect" title="L-kurtosis">L-kurtosis</a>, which is a scaled version of the fourth <a href="/wiki/L-moment" title="L-moment">L-moment</a>; measures based on four population or sample <a href="/wiki/Quantiles" class="mw-redirect" title="Quantiles">quantiles</a>.<sup id="cite_ref-Joanes1998_3-0" class="reference"><a href="#cite_note-Joanes1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> These are analogous to the alternative measures of <a href="/wiki/Skewness" title="Skewness">skewness</a> that are not based on ordinary moments.<sup id="cite_ref-Joanes1998_3-1" class="reference"><a href="#cite_note-Joanes1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pearson_moments">Pearson moments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=1" title="Edit section: Pearson moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The kurtosis is the fourth <a href="/wiki/Standardized_moment" title="Standardized moment">standardized moment</a>, 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 \operatorname {Kurt} [X]=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{4}\right]={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Kurt} [X]=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{4}\right]={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abb6badbf13364972b05d9249962f5ff87aba236" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.914ex; height:7.509ex;" alt="{\displaystyle \operatorname {Kurt} [X]=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{4}\right]={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}},}"></span> where <i>μ</i><sub>4</sub> is the fourth <a href="/wiki/Central_moment" title="Central moment">central moment</a> and <i>σ</i> is the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. Several letters are used in the literature to denote the kurtosis. A very common choice is <i>κ</i>, which is fine as long as it is clear that it does not refer to a <a href="/wiki/Cumulant" title="Cumulant">cumulant</a>. Other choices include <i>γ</i><sub>2</sub>, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. </p><p>The kurtosis is bounded below by the squared <a href="/wiki/Skewness" title="Skewness">skewness</a> plus 1:<sup id="cite_ref-Pearson1916_4-0" class="reference"><a href="#cite_note-Pearson1916-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 432">: 432 </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 {\frac {\mu _{4}}{\sigma ^{4}}}\geq \left({\frac {\mu _{3}}{\sigma ^{3}}}\right)^{2}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>≥</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{4}}{\sigma ^{4}}}\geq \left({\frac {\mu _{3}}{\sigma ^{3}}}\right)^{2}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c2a573209c92ea21d748f145a16419c9ef8b91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.808ex; height:6.509ex;" alt="{\displaystyle {\frac {\mu _{4}}{\sigma ^{4}}}\geq \left({\frac {\mu _{3}}{\sigma ^{3}}}\right)^{2}+1,}"></span> where <i>μ</i><sub>3</sub> is the third <a href="/wiki/Central_moment" title="Central moment">central moment</a>. The lower bound is realized by the <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a>. There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite. </p><p>A reason why some authors favor the excess kurtosis is that cumulants are <a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">extensive</a>. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let <i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub> be independent random variables for which the fourth moment exists, and let <i>Y</i> be the random variable defined by the sum of the <i>X</i><sub><i>i</i></sub>. The excess kurtosis of <i>Y</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Kurt} [Y]-3={\frac {1}{\left(\sum _{j=1}^{n}\sigma _{j}^{\,2}\right)^{2}}}\sum _{i=1}^{n}\sigma _{i}^{\,4}\cdot \left(\operatorname {Kurt} \left[X_{i}\right]-3\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>4</mn> </mrow> </msubsup> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>Kurt</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>]</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Kurt} [Y]-3={\frac {1}{\left(\sum _{j=1}^{n}\sigma _{j}^{\,2}\right)^{2}}}\sum _{i=1}^{n}\sigma _{i}^{\,4}\cdot \left(\operatorname {Kurt} \left[X_{i}\right]-3\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f295e334581a6f264d3ca40bf75df3293fb0f3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:52.735ex; height:8.843ex;" alt="{\displaystyle \operatorname {Kurt} [Y]-3={\frac {1}{\left(\sum _{j=1}^{n}\sigma _{j}^{\,2}\right)^{2}}}\sum _{i=1}^{n}\sigma _{i}^{\,4}\cdot \left(\operatorname {Kurt} \left[X_{i}\right]-3\right),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> is the standard deviation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span>. In particular if all of the <i>X</i><sub><i>i</i></sub> have the same variance, then this simplifies to <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 {Kurt} [Y]-3={1 \over n^{2}}\sum _{i=1}^{n}\left(\operatorname {Kurt} \left[X_{i}\right]-3\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>Kurt</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>]</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Kurt} [Y]-3={1 \over n^{2}}\sum _{i=1}^{n}\left(\operatorname {Kurt} \left[X_{i}\right]-3\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c397ae36f0e23dd0fff0c1137e5dcddc2e9217b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.28ex; height:6.843ex;" alt="{\displaystyle \operatorname {Kurt} [Y]-3={1 \over n^{2}}\sum _{i=1}^{n}\left(\operatorname {Kurt} \left[X_{i}\right]-3\right).}"></span> </p><p>The reason not to subtract 3 is that the bare <a href="/wiki/Moment_(statistics)" class="mw-redirect" title="Moment (statistics)">moment</a> better generalizes to <a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">multivariate distributions</a>, especially when independence is not assumed. The <a href="/wiki/Cokurtosis" title="Cokurtosis">cokurtosis</a> between pairs of variables is an order four <a href="/wiki/Tensor" title="Tensor">tensor</a>. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to "correct" for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> are zero. </p><p>For two random variables, <i>X</i> and <i>Y</i>, not necessarily independent, the kurtosis of the sum, <i>X</i> + <i>Y</i>, is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {Kurt} [X+Y]={1 \over \sigma _{X+Y}^{4}}{\big (}&\sigma _{X}^{4}\operatorname {Kurt} [X]+4\sigma _{X}^{3}\sigma _{Y}\operatorname {Cokurt} [X,X,X,Y]\\&{}+6\sigma _{X}^{2}\sigma _{Y}^{2}\operatorname {Cokurt} [X,X,Y,Y]\\[6pt]&{}+4\sigma _{X}\sigma _{Y}^{3}\operatorname {Cokurt} [X,Y,Y,Y]+\sigma _{Y}^{4}\operatorname {Kurt} [Y]{\big )}.\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 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mn>4</mn> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>Cokurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>6</mn> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>Cokurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>4</mn> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mi>Cokurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mi>Kurt</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Kurt} [X+Y]={1 \over \sigma _{X+Y}^{4}}{\big (}&\sigma _{X}^{4}\operatorname {Kurt} [X]+4\sigma _{X}^{3}\sigma _{Y}\operatorname {Cokurt} [X,X,X,Y]\\&{}+6\sigma _{X}^{2}\sigma _{Y}^{2}\operatorname {Cokurt} [X,X,Y,Y]\\[6pt]&{}+4\sigma _{X}\sigma _{Y}^{3}\operatorname {Cokurt} [X,Y,Y,Y]+\sigma _{Y}^{4}\operatorname {Kurt} [Y]{\big )}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0a7f4889310fb96ed071c23a6d6af959ef500d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:68.819ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Kurt} [X+Y]={1 \over \sigma _{X+Y}^{4}}{\big (}&\sigma _{X}^{4}\operatorname {Kurt} [X]+4\sigma _{X}^{3}\sigma _{Y}\operatorname {Cokurt} [X,X,X,Y]\\&{}+6\sigma _{X}^{2}\sigma _{Y}^{2}\operatorname {Cokurt} [X,X,Y,Y]\\[6pt]&{}+4\sigma _{X}\sigma _{Y}^{3}\operatorname {Cokurt} [X,Y,Y,Y]+\sigma _{Y}^{4}\operatorname {Kurt} [Y]{\big )}.\end{aligned}}}"></span> Note that the fourth-power <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> (1, 4, 6, 4, 1) appear in the above equation. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation">Interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=2" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The interpretation of the Pearson measure of kurtosis (or excess kurtosis) was once debated, but it is now well-established. As noted by Westfall in 2014<sup id="cite_ref-Westfall2014_2-1" class="reference"><a href="#cite_note-Westfall2014-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>, "...<i>its unambiguous interpretation relates to tail extremity.</i> Specifically, it reflects either the presence of existing outliers (for sample kurtosis) or the tendency to produce outliers (for the kurtosis of a probability distribution). The underlying logic is straightforward: Kurtosis represents the average (or <a href="/wiki/Expected_value" title="Expected value">expected value</a>) of standardized data raised to the fourth power. Standardized values less than 1—corresponding to data within one standard deviation of the mean (where the “peak” occurs)—contribute minimally to kurtosis. This is because raising a number less than 1 to the fourth power brings it closer to zero. The meaningful contributors to kurtosis are data values outside the peak region, i.e., the outliers. Therefore, kurtosis primarily measures outliers and provides no information about the central "peak". </p><p>Numerous misconceptions about kurtosis relate to notions of peakedness. One such misconception is that kurtosis measures both the “peakedness” of a distribution and the <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heaviness of its tail</a> .<sup id="cite_ref-Balanda1988_5-0" class="reference"><a href="#cite_note-Balanda1988-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Other incorrect interpretations include notions like “lack of shoulders” (where the “shoulder” refers vaguely to the area between the peak and the tail, or more specifically, the region about one <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> from the mean) or “bimodality.” <sup id="cite_ref-Darlington1970_6-0" class="reference"><a href="#cite_note-Darlington1970-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Balanda and <a href="/wiki/Helen_MacGillivray" title="Helen MacGillivray">MacGillivray</a> argue that the standard definition of kurtosis “poorly captures the kurtosis, peakedness, or tail weight of a distribution.”Instead, they propose a vague definition of kurtosis as the location- and scale-free movement of <a href="/wiki/Probability_mass" class="mw-redirect" title="Probability mass">probability mass</a> from the distribution’s shoulders into its center and tails. <sup id="cite_ref-Balanda1988_5-1" class="reference"><a href="#cite_note-Balanda1988-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Moors'_interpretation"><span id="Moors.27_interpretation"></span>Moors' interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=3" title="Edit section: Moors' interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1986, Moors gave an interpretation of kurtosis.<sup id="cite_ref-Moors1986_7-0" class="reference"><a href="#cite_note-Moors1986-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Let <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 Z={\frac {X-\mu }{\sigma }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={\frac {X-\mu }{\sigma }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a333cf878d9b62884a5113d828d602655e25864" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.484ex; height:5.343ex;" alt="{\displaystyle Z={\frac {X-\mu }{\sigma }},}"></span> where <i>X</i> is a random variable, <i>μ</i> is the mean and <i>σ</i> is the standard deviation. </p><p>Now by definition of the kurtosis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span>, and by the well-known identity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[V^{2}\right]=\operatorname {var} [V]+[E[V]]^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>V</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>V</mi> <mo stretchy="false">]</mo> <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 E\left[V^{2}\right]=\operatorname {var} [V]+[E[V]]^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6782f578cb9d22c509a08ab7d43a21735eba1cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.246ex; height:3.343ex;" alt="{\displaystyle E\left[V^{2}\right]=\operatorname {var} [V]+[E[V]]^{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 \kappa =E\left[Z^{4}\right]=\operatorname {var} \left[Z^{2}\right]+\left[E\left[Z^{2}\right]\right]^{2}=\operatorname {var} \left[Z^{2}\right]+[\operatorname {var} [Z]]^{2}=\operatorname {var} \left[Z^{2}\right]+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>+</mo> <mo stretchy="false">[</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =E\left[Z^{4}\right]=\operatorname {var} \left[Z^{2}\right]+\left[E\left[Z^{2}\right]\right]^{2}=\operatorname {var} \left[Z^{2}\right]+[\operatorname {var} [Z]]^{2}=\operatorname {var} \left[Z^{2}\right]+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14062cd540b9bba4aa7aac49d5e5a5eccf1718e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:73.42ex; height:3.843ex;" alt="{\displaystyle \kappa =E\left[Z^{4}\right]=\operatorname {var} \left[Z^{2}\right]+\left[E\left[Z^{2}\right]\right]^{2}=\operatorname {var} \left[Z^{2}\right]+[\operatorname {var} [Z]]^{2}=\operatorname {var} \left[Z^{2}\right]+1.}"></span> </p><p>The kurtosis can now be seen as a measure of the dispersion of <i>Z</i><sup>2</sup> around its expectation. Alternatively it can be seen to be a measure of the dispersion of <i>Z</i> around +1 and −1. <i>κ</i> attains its minimal value in a symmetric two-point distribution. In terms of the original variable <i>X</i>, the kurtosis is a measure of the dispersion of <i>X</i> around the two values <i>μ</i> ± <i>σ</i>. </p><p>High values of <i>κ</i> arise in two circumstances: </p> <ul><li>where the probability mass is concentrated around the mean and the data-generating process produces occasional values far from the mean</li> <li>where the probability mass is concentrated in the tails of the distribution.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Maximal_entropy">Maximal entropy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=4" title="Edit section: Maximal entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The entropy of a distribution 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 \int p(x)\ln p(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int p(x)\ln p(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff4143c054e233c68cedd2057ed72ecc40f6ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.844ex; height:5.676ex;" alt="{\displaystyle \int p(x)\ln p(x)\,dx}"></span>. </p><p>For any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {R} ^{n},\Sigma \in \mathbb {R} ^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {R} ^{n},\Sigma \in \mathbb {R} ^{n\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3765b4b13b46cf68669381088ddc10352e5fef61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.853ex; height:2.843ex;" alt="{\displaystyle \mu \in \mathbb {R} ^{n},\Sigma \in \mathbb {R} ^{n\times n}}"></span> 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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> positive definite, among all probability distributions on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and covariance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>, the normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(\mu ,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e2c1eb7b083522691f4ecaf7e751b0b967b64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.26ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(\mu ,\Sigma )}"></span> has the largest entropy. </p><p>Since mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and covariance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> are the first two moments, it is natural to consider extension to higher moments. In fact, by <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> method, for any prescribed first n moments, if there exists some probability distribution of form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)\propto e^{\sum _{i}a_{i}x_{i}+\sum _{ij}b_{ij}x_{i}x_{j}+\cdots +\sum _{i_{1}\cdots i_{n}}x_{i_{1}}\cdots x_{i_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⋯</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)\propto e^{\sum _{i}a_{i}x_{i}+\sum _{ij}b_{ij}x_{i}x_{j}+\cdots +\sum _{i_{1}\cdots i_{n}}x_{i_{1}}\cdots x_{i_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48a2cfb64283875d38ae08ff7cbc1286edfa8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:42.073ex; height:3.509ex;" alt="{\displaystyle p(x)\propto e^{\sum _{i}a_{i}x_{i}+\sum _{ij}b_{ij}x_{i}x_{j}+\cdots +\sum _{i_{1}\cdots i_{n}}x_{i_{1}}\cdots x_{i_{n}}}}"></span> that has the prescribed moments (if it is feasible), then it is the maximal entropy distribution under the given constraints.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><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><p>By serial expansion, <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}&\int {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&{\frac {1}{\sqrt {2\pi }}}\int e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&\sum _{k}{\frac {1}{k!}}(-g/4)^{k}(2n+4k-1)!!\\[6pt]={}&(2n-1)!!-{\frac {1}{4}}g(2n+3)!!+O(g^{2})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>g</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>g</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\int {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&{\frac {1}{\sqrt {2\pi }}}\int e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&\sum _{k}{\frac {1}{k!}}(-g/4)^{k}(2n+4k-1)!!\\[6pt]={}&(2n-1)!!-{\frac {1}{4}}g(2n+3)!!+O(g^{2})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c48b4902dc57596c5d3d22c5642c9b6a7b8847" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.576ex; margin-bottom: -0.262ex; width:37.081ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}&\int {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&{\frac {1}{\sqrt {2\pi }}}\int e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]={}&\sum _{k}{\frac {1}{k!}}(-g/4)^{k}(2n+4k-1)!!\\[6pt]={}&(2n-1)!!-{\frac {1}{4}}g(2n+3)!!+O(g^{2})\end{aligned}}}"></span>so if a random variable has probability distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}/Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>g</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}/Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc41346436a93660160e8a503ab18c840ed76bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.552ex; height:4.009ex;" alt="{\displaystyle p(x)=e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}/Z}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> is a normalization constant, then its kurtosis is <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3-6g+O(g^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>−</mo> <mn>6</mn> <mi>g</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3-6g+O(g^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9863882343d988cbf9c3e06d9ca6df249a14d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.877ex; height:3.176ex;" alt="{\displaystyle 3-6g+O(g^{2})}"></span>.</span><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Excess_kurtosis">Excess kurtosis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=5" title="Edit section: Excess kurtosis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>excess kurtosis</i> is defined as kurtosis minus 3. There are 3 distinct regimes as described below. </p> <div class="mw-heading mw-heading3"><h3 id="Mesokurtic">Mesokurtic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=6" title="Edit section: Mesokurtic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Distributions with zero excess kurtosis are called <b>mesokurtic</b>, or <b>mesokurtotic</b>. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its <a href="/wiki/Parameter" title="Parameter">parameters</a>. A few other well-known distributions can be mesokurtic, depending on parameter values: for example, the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> is mesokurtic for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p=1/2\pm {\sqrt {1/12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p=1/2\pm {\sqrt {1/12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49955e8ad0be8da49d39e1792b6f85ef4e9f47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:17.659ex; height:3.343ex;" alt="{\textstyle p=1/2\pm {\sqrt {1/12}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Leptokurtic">Leptokurtic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=7" title="Edit section: Leptokurtic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A distribution with <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> excess kurtosis is called <b>leptokurtic</b>, or <b>leptokurtotic</b>. "Lepto-" means "slender".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In terms of shape, a leptokurtic distribution has <i><a href="/wiki/Fat-tailed_distribution" title="Fat-tailed distribution">fatter tails</a></i>. Examples of leptokurtic distributions include the <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t-distribution</a>, <a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh distribution</a>, <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a>, <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> and the <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a>. Such distributions are sometimes termed <i>super-Gaussian</i>.<sup id="cite_ref-Beneviste1980_12-0" class="reference"><a href="#cite_note-Beneviste1980-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Three_probability_density_functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Three_probability_density_functions.png/220px-Three_probability_density_functions.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Three_probability_density_functions.png/330px-Three_probability_density_functions.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Three_probability_density_functions.png/440px-Three_probability_density_functions.png 2x" data-file-width="1151" data-file-height="862" /></a><figcaption>Three symmetric increasingly leptokurtic probability density functions; their intersections are indicated by vertical lines.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Platykurtic">Platykurtic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=8" title="Edit section: Platykurtic"><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:1909_US_Penny.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/1909_US_Penny.jpg/220px-1909_US_Penny.jpg" decoding="async" width="220" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/1909_US_Penny.jpg/330px-1909_US_Penny.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/1909_US_Penny.jpg/440px-1909_US_Penny.jpg 2x" data-file-width="2200" data-file-height="1110" /></a><figcaption>The <a href="/wiki/Coin_toss" class="mw-redirect" title="Coin toss">coin toss</a> is the most platykurtic distribution</figcaption></figure> <p>A distribution with <a href="/wiki/Negative_number" title="Negative number">negative</a> excess kurtosis is called <b>platykurtic</b>, or <b>platykurtotic</b>. "Platy-" means "broad".<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In terms of shape, a platykurtic distribution has <i>thinner tails</i>. Examples of platykurtic distributions include the <a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">continuous</a> and <a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">discrete uniform distributions</a>, and the <a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">raised cosine distribution</a>. The most platykurtic distribution of all is the <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a> with <i>p</i> = 1/2 (for example the number of times one obtains "heads" when flipping a coin once, a <a href="/wiki/Coin_toss" class="mw-redirect" title="Coin toss">coin toss</a>), for which the excess kurtosis is −2. </p> <div class="mw-heading mw-heading2"><h2 id="Graphical_examples">Graphical examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=9" title="Edit section: Graphical examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_Pearson_type_VII_family">The Pearson type VII family</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=10" title="Edit section: The Pearson type VII family"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pearson_type_VII_distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Pearson_type_VII_distribution_PDF.svg/300px-Pearson_type_VII_distribution_PDF.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Pearson_type_VII_distribution_PDF.svg/450px-Pearson_type_VII_distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Pearson_type_VII_distribution_PDF.svg/600px-Pearson_type_VII_distribution_PDF.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption><a href="/wiki/Probability_density_function" title="Probability density function">pdf</a> for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); and 0 (black)</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pearson_type_VII_distribution_log-PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Pearson_type_VII_distribution_log-PDF.svg/300px-Pearson_type_VII_distribution_log-PDF.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Pearson_type_VII_distribution_log-PDF.svg/450px-Pearson_type_VII_distribution_log-PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Pearson_type_VII_distribution_log-PDF.svg/600px-Pearson_type_VII_distribution_log-PDF.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption>log-pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)</figcaption></figure> <p>The effects of kurtosis are illustrated using a <a href="/wiki/Parametric_family" title="Parametric family">parametric family</a> of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider the <a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson type VII family</a>, which is a special case of the <a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson type IV family</a> restricted to symmetric densities. The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> 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 f(x;a,m)={\frac {\Gamma (m)}{a\,{\sqrt {\pi }}\,\Gamma (m-1/2)}}\left[1+\left({\frac {x}{a}}\right)^{2}\right]^{-m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;a,m)={\frac {\Gamma (m)}{a\,{\sqrt {\pi }}\,\Gamma (m-1/2)}}\left[1+\left({\frac {x}{a}}\right)^{2}\right]^{-m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa28a12d8d7612d4c54447dfabdde07ab6c0c53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.646ex; height:6.843ex;" alt="{\displaystyle f(x;a,m)={\frac {\Gamma (m)}{a\,{\sqrt {\pi }}\,\Gamma (m-1/2)}}\left[1+\left({\frac {x}{a}}\right)^{2}\right]^{-m},}"></span> where <i>a</i> is a <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> and <i>m</i> is a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a>. </p><p>All densities in this family are symmetric. The <i>k</i>th moment exists provided <i>m</i> > (<i>k</i> + 1)/2. For the kurtosis to exist, we require <i>m</i> > 5/2. Then the mean and <a href="/wiki/Skewness" title="Skewness">skewness</a> exist and are both identically zero. Setting <i>a</i><sup>2</sup> = 2<i>m</i> − 3 makes the variance equal to unity. Then the only free parameter is <i>m</i>, which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize 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 m=5/2+3/\gamma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=5/2+3/\gamma _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0a988dabbd1739208683b17264253bb0f38c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.05ex; height:2.843ex;" alt="{\displaystyle m=5/2+3/\gamma _{2}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482832093b568cdc09c3aeaa2585c5fc49100b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{2}}"></span> is the excess kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary non-negative excess kurtosis. The reparameterized density 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 g(x;\gamma _{2})=f\left(x;\;a={\sqrt {2+{\frac {6}{\gamma _{2}}}}},\;m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>;</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x;\gamma _{2})=f\left(x;\;a={\sqrt {2+{\frac {6}{\gamma _{2}}}}},\;m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393908ee203444cddfa72d442009c5f606cb8f81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.537ex; height:7.676ex;" alt="{\displaystyle g(x;\gamma _{2})=f\left(x;\;a={\sqrt {2+{\frac {6}{\gamma _{2}}}}},\;m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}\right).}"></span> </p><p>In the limit as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a462745c5753fda4d2778846e7acef03d64c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:2.343ex;" alt="{\displaystyle \gamma _{2}\to \infty }"></span> one obtains the density <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(x)=3\left(2+x^{2}\right)^{-{\frac {5}{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=3\left(2+x^{2}\right)^{-{\frac {5}{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd6764a32435f750d769cf4d1468192985caddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.694ex; height:4.509ex;" alt="{\displaystyle g(x)=3\left(2+x^{2}\right)^{-{\frac {5}{2}}},}"></span> which is shown as the red curve in the images on the right. </p><p>In the other direction as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed75331d477b4de8dd715d31adafaa39627c0266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.035ex; height:2.676ex;" alt="{\displaystyle \gamma _{2}\to 0}"></span> one obtains the <a href="/wiki/Normal_distribution" title="Normal distribution">standard normal</a> density as the limiting distribution, shown as the black curve. </p><p>In the images on the right, the blue curve represents the density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto g(x;2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto g(x;2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b1dff907091d8f25bb169a64d6fe9d97a1023b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.395ex; height:2.843ex;" alt="{\displaystyle x\mapsto g(x;2)}"></span> with excess kurtosis of 2. The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a <a href="/wiki/Parabola" title="Parabola">parabola</a>. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with excess kurtosis of 2. Between the blue curve and the black are other Pearson type VII densities with <i>γ</i><sub>2</sub> = 1, 1/2, 1/4, 1/8, and 1/16. The red curve again shows the upper limit of the Pearson type VII family, 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 \gamma _{2}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1492724ef96391f2f4aaaccfcb4c65623b3095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:2.176ex;" alt="{\displaystyle \gamma _{2}=\infty }"></span> (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin ("has fat tails"). </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Other_well-known_distributions">Other well-known distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=11" title="Edit section: Other well-known distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Standard_symmetric_pdfs.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Standard_symmetric_pdfs.svg/300px-Standard_symmetric_pdfs.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Standard_symmetric_pdfs.svg/450px-Standard_symmetric_pdfs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Standard_symmetric_pdfs.svg/600px-Standard_symmetric_pdfs.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption><a href="/wiki/Probability_density_function" title="Probability density function">Probability density functions</a> for selected distributions with <a href="/wiki/Expected_value" title="Expected value">mean</a> 0, <a href="/wiki/Variance" title="Variance">variance</a> 1 and different excess kurtosis</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Standard_symmetric_pdfs_logscale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Standard_symmetric_pdfs_logscale.svg/300px-Standard_symmetric_pdfs_logscale.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Standard_symmetric_pdfs_logscale.svg/450px-Standard_symmetric_pdfs_logscale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Standard_symmetric_pdfs_logscale.svg/600px-Standard_symmetric_pdfs_logscale.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption><a href="/wiki/Logarithm" title="Logarithm">Logarithms</a> of <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a> for selected distributions with <a href="/wiki/Expected_value" title="Expected value">mean</a> 0, <a href="/wiki/Variance" title="Variance">variance</a> 1 and different excess kurtosis</figcaption></figure> <p>Several well-known, unimodal, and symmetric distributions from different parametric families are compared here. Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a <a href="/wiki/Linear_scale" title="Linear scale">linear scale</a> and <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic scale</a>: </p> <ul><li>D: <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a>, also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3</li> <li>S: <a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">hyperbolic secant distribution</a>, orange curve, excess kurtosis = 2</li> <li>L: <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a>, green curve, excess kurtosis = 1.2</li> <li>N: <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, black curve (inverted parabola in the log-scale plot), excess kurtosis = 0</li> <li>C: <a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">raised cosine distribution</a>, cyan curve, excess kurtosis = −0.593762...</li> <li>W: <a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle distribution</a>, blue curve, excess kurtosis = −1</li> <li>U: <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a>, magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.</li></ul> <p>Note that in these cases the platykurtic densities have bounded <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a>, whereas the densities with positive or zero excess kurtosis are supported on the whole <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. </p><p>One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support, </p> <ul><li>e.g., <a href="/wiki/Exponential_power_distribution" class="mw-redirect" title="Exponential power distribution">exponential power distributions</a> with sufficiently large shape parameter <i>b</i></li></ul> <p>and there exist leptokurtic densities with finite support. </p> <ul><li>e.g., a distribution that is uniform between −3 and −0.3, between −0.3 and 0.3, and between 0.3 and 3, with the same density in the (−3, −0.3) and (0.3, 3) intervals, but with 20 times more density in the (−0.3, 0.3) interval</li></ul> <p>Also, there exist platykurtic densities with infinite peakedness, </p> <ul><li>e.g., an equal mixture of the <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> with parameters 0.5 and 1 with its reflection about 0.0</li></ul> <p>and there exist leptokurtic densities that appear flat-topped, </p> <ul><li>e.g., a mixture of distribution that is uniform between −1 and 1 with a T(4.0000001) <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t-distribution</a>, with mixing probabilities 0.999 and 0.001.</li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Sample_kurtosis">Sample kurtosis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=12" title="Edit section: Sample kurtosis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=13" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="A_natural_but_biased_estimator">A natural but biased estimator</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=14" title="Edit section: A natural but biased estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> of <i>n</i> values, a <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a> estimator of the population excess kurtosis 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 g_{2}={\frac {m_{4}}{m_{2}^{2}}}-3={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{4}}{\left[{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right]^{2}}}-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}={\frac {m_{4}}{m_{2}^{2}}}-3={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{4}}{\left[{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right]^{2}}}-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30ed8db4466ff3e448b71e729464506ddce7770" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:40.968ex; height:8.009ex;" alt="{\displaystyle g_{2}={\frac {m_{4}}{m_{2}^{2}}}-3={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{4}}{\left[{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right]^{2}}}-3}"></span> where <i>m</i><sub>4</sub> is the fourth sample <a href="/wiki/Moment_about_the_mean" class="mw-redirect" title="Moment about the mean">moment about the mean</a>, <i>m</i><sub>2</sub> is the second sample moment about the mean (that is, the <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a>), <i>x</i><sub><i>i</i></sub> is the <i>i</i><sup>th</sup> value, 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 {\overline {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"></span> is the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a>. </p><p>This formula has the simpler representation, <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_{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{4}-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{4}-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/946d782cd05421dec3120c2f92fb1d070510b7a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.769ex; height:6.843ex;" alt="{\displaystyle g_{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{4}-3}"></span> where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span> values are the standardized data values using the standard deviation defined using <i>n</i> rather than <i>n</i> − 1 in the denominator. </p><p>For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999. </p><p>Then 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 z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span> values are −0.239, −0.225, −0.221, −0.234, −0.230, −0.225, −0.239, −0.230, −0.234, −0.225, −0.230, −0.239, −0.230, −0.230, −0.225, −0.230, −0.216, −0.230, −0.225, 4.359 </p><p>and 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 z_{i}^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bddbb4c78a5e3944cd625c26f88f05948202e19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.145ex; height:3.176ex;" alt="{\displaystyle z_{i}^{4}}"></span> values are 0.003, 0.003, 0.002, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.002, 0.003, 0.003, 360.976. </p><p>The average of these values is 18.05 and the excess kurtosis is thus 18.05 − 3 = 15.05. This example makes it clear that data near the "middle" or "peak" of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure "peakedness". It is simply a measure of the outlier, 999 in this example. </p> <div class="mw-heading mw-heading4"><h4 id="Standard_unbiased_estimator">Standard unbiased estimator</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=15" title="Edit section: Standard unbiased estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a sub-set of samples from a population, the sample excess kurtosis <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"></span> above is a <a href="/wiki/Biased_estimator" class="mw-redirect" title="Biased estimator">biased estimator</a> of the population excess kurtosis. An alternative estimator of the population excess kurtosis, which is unbiased in random samples of a normal distribution, is defined as follows:<sup id="cite_ref-Joanes1998_3-2" class="reference"><a href="#cite_note-Joanes1998-3"><span class="cite-bracket">[</span>3<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 {\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}\\[6pt]&={\frac {n^{2}\,[(n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2}]}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,{\frac {m_{4}}{m_{2}^{2}}}-3\,(n-1)\right]\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,g_{2}+6\right]\\[6pt]&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\right)^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\[6pt]&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>6</mn> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>n</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}\\[6pt]&={\frac {n^{2}\,[(n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2}]}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,{\frac {m_{4}}{m_{2}^{2}}}-3\,(n-1)\right]\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,g_{2}+6\right]\\[6pt]&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\right)^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\[6pt]&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b0b3b5d766ffd8f2c3be0fbe899873b41bb961" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.005ex; width:65.6ex; height:49.176ex;" alt="{\displaystyle {\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}\\[6pt]&={\frac {n^{2}\,[(n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2}]}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,{\frac {m_{4}}{m_{2}^{2}}}-3\,(n-1)\right]\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,g_{2}+6\right]\\[6pt]&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\right)^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\[6pt]&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}}"></span> where <i>k</i><sub>4</sub> is the unique symmetric <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> estimator of the fourth <a href="/wiki/Cumulant" title="Cumulant">cumulant</a>, <i>k</i><sub>2</sub> is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), <i>m</i><sub>4</sub> is the fourth sample moment about the mean, <i>m</i><sub>2</sub> is the second sample moment about the mean, <i>x</i><sub><i>i</i></sub> is the <i>i</i><sup>th</sup> value, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></span> is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645011b0c6933a02f5f7d84624f78220d747427e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}"></span> is the version found in <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Excel</a> and several statistical packages including <a href="/wiki/Minitab" title="Minitab">Minitab</a>, <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a>, and <a href="/wiki/SPSS" title="SPSS">SPSS</a>.<sup id="cite_ref-Doane2011_14-0" class="reference"><a href="#cite_note-Doane2011-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Unfortunately, in nonnormal samples <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645011b0c6933a02f5f7d84624f78220d747427e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}"></span> is itself generally biased. </p> <div class="mw-heading mw-heading3"><h3 id="Upper_bound">Upper bound</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=16" title="Edit section: Upper bound"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An upper bound for the sample kurtosis of <i>n</i> (<i>n</i> > 2) real numbers is<sup id="cite_ref-Sharma2015_15-0" class="reference"><a href="#cite_note-Sharma2015-15"><span class="cite-bracket">[</span>15<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 g_{2}\leq {\frac {1}{2}}{\frac {n-3}{n-2}}g_{1}^{2}+{\frac {n}{2}}-3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <msubsup> <mi>g</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> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}\leq {\frac {1}{2}}{\frac {n-3}{n-2}}g_{1}^{2}+{\frac {n}{2}}-3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141a1012a1556e22c2332169942485f9deb3aca1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.387ex; height:5.343ex;" alt="{\displaystyle g_{2}\leq {\frac {1}{2}}{\frac {n-3}{n-2}}g_{1}^{2}+{\frac {n}{2}}-3.}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}=m_{3}/m_{2}^{3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}=m_{3}/m_{2}^{3/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15cd9f8c353828d20d290215d9d451557f94e916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.257ex; height:3.676ex;" alt="{\displaystyle g_{1}=m_{3}/m_{2}^{3/2}}"></span> is the corresponding sample skewness. </p> <div class="mw-heading mw-heading3"><h3 id="Variance_under_normality">Variance under normality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=17" title="Edit section: Variance under normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The variance of the sample kurtosis of a sample of size <i>n</i> from the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> is<sup id="cite_ref-Fisher1930_16-0" class="reference"><a href="#cite_note-Fisher1930-16"><span class="cite-bracket">[</span>16<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 \operatorname {var} (g_{2})={\frac {24n(n-1)^{2}}{(n-3)(n-2)(n+3)(n+5)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>24</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} (g_{2})={\frac {24n(n-1)^{2}}{(n-3)(n-2)(n+3)(n+5)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7f1b1167bfbb702829f53ad953c59b9aa5f28e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.036ex; height:6.676ex;" alt="{\displaystyle \operatorname {var} (g_{2})={\frac {24n(n-1)^{2}}{(n-3)(n-2)(n+3)(n+5)}}}"></span> </p><p>Stated differently, under the assumption that the underlying random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is normally distributed, it can be shown that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}g_{2}\,\xrightarrow {d} \,{\mathcal {N}}(0,24)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>d</mi> </mpadded> </mover> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>24</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}g_{2}\,\xrightarrow {d} \,{\mathcal {N}}(0,24)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b9f321dbf6a3bf6a58a612ed91ba91d4680e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.311ex; width:18.488ex; height:4.343ex;" alt="{\displaystyle {\sqrt {n}}g_{2}\,\xrightarrow {d} \,{\mathcal {N}}(0,24)}"></span>.<sup id="cite_ref-Kendall1969_17-0" class="reference"><a href="#cite_note-Kendall1969-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: Page number needed">: Page number needed </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=18" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods. </p><p><a href="/wiki/D%27Agostino%27s_K-squared_test" title="D'Agostino's K-squared test">D'Agostino's K-squared test</a> is a <a href="/wiki/Goodness-of-fit" class="mw-redirect" title="Goodness-of-fit">goodness-of-fit</a> <a href="/wiki/Normality_test" title="Normality test">normality test</a> based on a combination of the sample skewness and sample kurtosis, as is the <a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera test</a> for normality. </p><p>For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see <a href="/wiki/Variance#Distribution_of_the_sample_variance" title="Variance">variance</a>. </p><p>Pearson's definition of kurtosis is used as an indicator of intermittency in <a href="/wiki/Turbulence" title="Turbulence">turbulence</a>.<sup id="cite_ref-Sandborn1959_18-0" class="reference"><a href="#cite_note-Sandborn1959-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> It is also used in magnetic resonance imaging to quantify non-Gaussian diffusion.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>A concrete example is the following lemma by He, Zhang, and Zhang:<sup id="cite_ref-He2010_20-0" class="reference"><a href="#cite_note-He2010-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Assume a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has expectation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X]=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X]=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ed977b56d8e513d9eb92193de5454ac545231e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.549ex; height:2.843ex;" alt="{\displaystyle E[X]=\mu }"></span>, variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[(X-\mu )^{2}\right]=\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[(X-\mu )^{2}\right]=\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cedd17629b1a84a72ee82ac046028e7628811148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.671ex; height:3.343ex;" alt="{\displaystyle E\left[(X-\mu )^{2}\right]=\sigma ^{2}}"></span> and kurtosis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\tfrac {1}{\sigma ^{4}}}E\left[(X-\mu )^{4}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\tfrac {1}{\sigma ^{4}}}E\left[(X-\mu )^{4}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba57733216b88c28481cf490a6e2f90d0ab83d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:21.268ex; height:3.843ex;" alt="{\displaystyle \kappa ={\tfrac {1}{\sigma ^{4}}}E\left[(X-\mu )^{4}\right].}"></span> Assume we sample <span class="mwe-math-element"><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={\tfrac {2{\sqrt {3}}+3}{3}}\kappa \log {\tfrac {1}{\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mn>3</mn> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>κ</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>δ</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\tfrac {2{\sqrt {3}}+3}{3}}\kappa \log {\tfrac {1}{\delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f74b237218ae22e5db0358bb21120231d8c4ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.186ex; height:4.343ex;" alt="{\displaystyle n={\tfrac {2{\sqrt {3}}+3}{3}}\kappa \log {\tfrac {1}{\delta }}}"></span> many independent copies. Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr \left[\max _{i=1}^{n}X_{i}\leq \mu \right]\leq \delta \quad {\text{and}}\quad \Pr \left[\min _{i=1}^{n}X_{i}\geq \mu \right]\leq \delta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mi>μ</mi> </mrow> <mo>]</mo> </mrow> <mo>≤</mo> <mi>δ</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>μ</mi> </mrow> <mo>]</mo> </mrow> <mo>≤</mo> <mi>δ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \left[\max _{i=1}^{n}X_{i}\leq \mu \right]\leq \delta \quad {\text{and}}\quad \Pr \left[\min _{i=1}^{n}X_{i}\geq \mu \right]\leq \delta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193e367b4d43bcc0eaae3de498bd2970fd38b98b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.817ex; height:6.176ex;" alt="{\displaystyle \Pr \left[\max _{i=1}^{n}X_{i}\leq \mu \right]\leq \delta \quad {\text{and}}\quad \Pr \left[\min _{i=1}^{n}X_{i}\geq \mu \right]\leq \delta .}"></span> </p><p>This shows that 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 \Theta (\kappa \log {\tfrac {1}{\delta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>δ</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (\kappa \log {\tfrac {1}{\delta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f98ec22c62d7e7470f0dc717d936e06a9e80c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.36ex; height:3.676ex;" alt="{\displaystyle \Theta (\kappa \log {\tfrac {1}{\delta }})}"></span> many samples, we will see one that is above the expectation with probability at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fc18b68a939b8f9eb465e354a64164a1202901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.509ex;" alt="{\displaystyle 1-\delta }"></span>. In other words: If the kurtosis is large, we might see a lot values either all below or above the mean. </p> <div class="mw-heading mw-heading3"><h3 id="Kurtosis_convergence">Kurtosis convergence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=19" title="Edit section: Kurtosis convergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applying <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass filters</a> to <a href="/wiki/Digital_image" title="Digital image">digital images</a>, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed <i>kurtosis convergence</i>, can be used to detect image splicing in <a href="/wiki/Forensic_analysis" class="mw-redirect" title="Forensic analysis">forensic analysis</a>.<sup id="cite_ref-Pan2012_21-0" class="reference"><a href="#cite_note-Pan2012-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Seismic_signal_analysis">Seismic signal analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=20" title="Edit section: Seismic signal analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kurtosis can be used in <a href="/wiki/Geophysics" title="Geophysics">geophysics</a> to distinguish different types of <a href="/wiki/Seismology" title="Seismology">seismic signals</a>. It is particularly effective in differentiating seismic signals generated by human footsteps from other signals.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This is useful in security and surveillance systems that rely on seismic detection. </p> <div class="mw-heading mw-heading3"><h3 id="Weather_prediction">Weather prediction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=21" title="Edit section: Weather prediction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, kurtosis is used to analyze weather data distributions. It helps predict extreme weather events by assessing the probability of outlier values in historical data,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> which is valuable for long-term climate studies and short-term weather forecasting. </p> <div class="mw-heading mw-heading2"><h2 id="Other_measures">Other measures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=22" title="Edit section: Other measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A different measure of "kurtosis" is provided by using <a href="/wiki/L-moment" title="L-moment">L-moments</a> instead of the ordinary moments.<sup id="cite_ref-Hosking1992_24-0" class="reference"><a href="#cite_note-Hosking1992-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hosking2006_25-0" class="reference"><a href="#cite_note-Hosking2006-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </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=Kurtosis&action=edit&section=23" title="Edit section: See also"><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/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Kurtosis" class="extiw" title="commons:Category:Kurtosis">Kurtosis</a></span>.</div></div> </div> <ul><li><a href="/wiki/Kurtosis_risk" title="Kurtosis risk">Kurtosis risk</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy probability distribution</a></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=Kurtosis&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Pearson1905-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pearson1905_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="CITEREFPearson1905" class="citation cs2">Pearson, Karl (1905), "Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson. 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A Rejoinder], <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i>, <b>4</b> (1–2): 169–212, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F4.1-2.169">10.1093/biomet/4.1-2.169</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2331536">2331536</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Das+Fehlergesetz+und+seine+Verallgemeinerungen+durch+Fechner+und+Pearson.+A+Rejoinder&rft.volume=4&rft.issue=1%E2%80%932&rft.pages=169-212&rft.date=1905&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F4.1-2.169&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2331536%23id-name%3DJSTOR&rft.aulast=Pearson&rft.aufirst=Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Westfall2014-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Westfall2014_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Westfall2014_2-1"><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="CITEREFWestfall2014" class="citation cs2">Westfall, Peter H. (2014), "Kurtosis as Peakedness, 1905 - 2014. <i>R.I.P.</i>", <i><a href="/wiki/The_American_Statistician" title="The American Statistician">The American Statistician</a></i>, <b>68</b> (3): 191–195, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2014.917055">10.1080/00031305.2014.917055</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/PMC4321753">4321753</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/25678714">25678714</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Kurtosis+as+Peakedness%2C+1905+-+2014.+R.I.P.&rft.volume=68&rft.issue=3&rft.pages=191-195&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4321753%23id-name%3DPMC&rft_id=info%3Apmid%2F25678714&rft_id=info%3Adoi%2F10.1080%2F00031305.2014.917055&rft.aulast=Westfall&rft.aufirst=Peter+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Joanes1998-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Joanes1998_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Joanes1998_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Joanes1998_3-2"><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="CITEREFJoanesGill1998" class="citation cs2">Joanes, Derrick N.; Gill, Christine A. (1998), "Comparing measures of sample skewness and kurtosis", <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_D" class="mw-redirect" title="Journal of the Royal Statistical Society, Series D">Journal of the Royal Statistical Society, Series D</a></i>, <b>47</b> (1): 183–189, <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%2F1467-9884.00122">10.1111/1467-9884.00122</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2988433">2988433</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+D&rft.atitle=Comparing+measures+of+sample+skewness+and+kurtosis&rft.volume=47&rft.issue=1&rft.pages=183-189&rft.date=1998&rft_id=info%3Adoi%2F10.1111%2F1467-9884.00122&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2988433%23id-name%3DJSTOR&rft.aulast=Joanes&rft.aufirst=Derrick+N.&rft.au=Gill%2C+Christine+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Pearson1916-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pearson1916_4-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPearson1916" class="citation cs2">Pearson, Karl (1916), "Mathematical Contributions to the Theory of Evolution. — XIX. Second Supplement to a Memoir on Skew Variation.", <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society_of_London_A" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London A">Philosophical Transactions of the Royal Society of London A</a></i>, <b>216</b> (546): 429–457, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1916RSPTA.216..429P">1916RSPTA.216..429P</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1916.0009">10.1098/rsta.1916.0009</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/91092">91092</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London+A&rft.atitle=Mathematical+Contributions+to+the+Theory+of+Evolution.+%E2%80%94+XIX.+Second+Supplement+to+a+Memoir+on+Skew+Variation.&rft.volume=216&rft.issue=546&rft.pages=429-457&rft.date=1916&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F91092%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frsta.1916.0009&rft_id=info%3Abibcode%2F1916RSPTA.216..429P&rft.aulast=Pearson&rft.aufirst=Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Balanda1988-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Balanda1988_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Balanda1988_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalandaMacGillivray1988" class="citation cs2">Balanda, Kevin P.; <a href="/wiki/Helen_MacGillivray" title="Helen MacGillivray">MacGillivray, Helen L.</a> (1988), "Kurtosis: A Critical Review", <i>The American Statistician</i>, <b>42</b> (2): 111–119, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2684482">10.2307/2684482</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2684482">2684482</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Kurtosis%3A+A+Critical+Review&rft.volume=42&rft.issue=2&rft.pages=111-119&rft.date=1988&rft_id=info%3Adoi%2F10.2307%2F2684482&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2684482%23id-name%3DJSTOR&rft.aulast=Balanda&rft.aufirst=Kevin+P.&rft.au=MacGillivray%2C+Helen+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Darlington1970-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Darlington1970_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDarlington1970" class="citation cs2">Darlington, Richard B. (1970), "Is Kurtosis Really 'Peakedness'?", <i>The American Statistician</i>, <b>24</b> (2): 19–22, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1970.10478885">10.1080/00031305.1970.10478885</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2681925">2681925</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Is+Kurtosis+Really+%27Peakedness%27%3F&rft.volume=24&rft.issue=2&rft.pages=19-22&rft.date=1970&rft_id=info%3Adoi%2F10.1080%2F00031305.1970.10478885&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2681925%23id-name%3DJSTOR&rft.aulast=Darlington&rft.aufirst=Richard+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Moors1986-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Moors1986_7-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoors1986" class="citation cs2">Moors, J. 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"Bounding probability of small deviation: A fourth moment approach". <i><a href="/wiki/Mathematics_of_Operations_Research" title="Mathematics of Operations Research">Mathematics of Operations Research</a></i>. <b>35</b> (1): 208–232. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmoor.1090.0438">10.1287/moor.1090.0438</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:11298475">11298475</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Operations+Research&rft.atitle=Bounding+probability+of+small+deviation%3A+A+fourth+moment+approach&rft.volume=35&rft.issue=1&rft.pages=208-232&rft.date=2010&rft_id=info%3Adoi%2F10.1287%2Fmoor.1090.0438&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11298475%23id-name%3DS2CID&rft.aulast=He&rft.aufirst=Simai&rft.au=Zhang%2C+Jiawei&rft.au=Zhang%2C+Shuzhong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Pan2012-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pan2012_21-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPanZhangLyu2012" class="citation cs2">Pan, Xunyu; 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Wei, Jianming; Zhao, Junyu; Liu, Haitao; Li, Baoqing; Shen, Jie; Zheng, Chunlei (2008-08-27). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3705491">"The Statistical Meaning of Kurtosis and Its New Application to Identification of Persons Based on Seismic Signals"</a>. <i>Sensors</i>. <b>8</b> (8): 5106–5119. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008Senso...8.5106L">2008Senso...8.5106L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fs8085106">10.3390/s8085106</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1424-8220">1424-8220</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/PMC3705491">3705491</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/27873804">27873804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sensors&rft.atitle=The+Statistical+Meaning+of+Kurtosis+and+Its+New+Application+to+Identification+of+Persons+Based+on+Seismic+Signals&rft.volume=8&rft.issue=8&rft.pages=5106-5119&rft.date=2008-08-27&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3705491%23id-name%3DPMC&rft_id=info%3Abibcode%2F2008Senso...8.5106L&rft_id=info%3Apmid%2F27873804&rft_id=info%3Adoi%2F10.3390%2Fs8085106&rft.issn=1424-8220&rft.aulast=Liang&rft.aufirst=Zhiqiang&rft.au=Wei%2C+Jianming&rft.au=Zhao%2C+Junyu&rft.au=Liu%2C+Haitao&rft.au=Li%2C+Baoqing&rft.au=Shen%2C+Jie&rft.au=Zheng%2C+Chunlei&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3705491&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSupraja2024" class="citation web cs1">Supraja (2024-05-27). <a rel="nofollow" class="external text" href="https://www.analyticsinsight.net/tech-news/kurtosis-in-practice-real-world-applications-and-interpretations">"Kurtosis in Practice: Real-World Applications and Interpretations"</a>. <i>Analytics Insight</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Analytics+Insight&rft.atitle=Kurtosis+in+Practice%3A+Real-World+Applications+and+Interpretations&rft.date=2024-05-27&rft.au=Supraja&rft_id=https%3A%2F%2Fwww.analyticsinsight.net%2Ftech-news%2Fkurtosis-in-practice-real-world-applications-and-interpretations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Hosking1992-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hosking1992_24-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHosking1992" class="citation cs2">Hosking, Jonathan R. M. (1992), "Moments or <i>L</i> moments? An example comparing two measures of distributional shape", <i>The American Statistician</i>, <b>46</b> (3): 186–189, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1992.10475880">10.1080/00031305.1992.10475880</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2685210">2685210</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Moments+or+L+moments%3F+An+example+comparing+two+measures+of+distributional+shape&rft.volume=46&rft.issue=3&rft.pages=186-189&rft.date=1992&rft_id=info%3Adoi%2F10.1080%2F00031305.1992.10475880&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2685210%23id-name%3DJSTOR&rft.aulast=Hosking&rft.aufirst=Jonathan+R.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> <li id="cite_note-Hosking2006-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hosking2006_25-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHosking2006" class="citation cs2">Hosking, Jonathan R. M. (2006), "On the characterization of distributions by their <i>L</i>-moments", <i><a href="/wiki/Journal_of_Statistical_Planning_and_Inference" title="Journal of Statistical Planning and Inference">Journal of Statistical Planning and Inference</a></i>, <b>136</b> (1): 193–198, <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.jspi.2004.06.004">10.1016/j.jspi.2004.06.004</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+Statistical+Planning+and+Inference&rft.atitle=On+the+characterization+of+distributions+by+their+L-moments&rft.volume=136&rft.issue=1&rft.pages=193-198&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2004.06.004&rft.aulast=Hosking&rft.aufirst=Jonathan+R.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=25" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimWhite2003" class="citation journal cs1">Kim, Tae-Hwan; White, Halbert (2003). <a rel="nofollow" class="external text" href="http://escholarship.org/uc/item/7b52v07p">"On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index"</a>. <i>Finance Research Letters</i>. <b>1</b>: 56–70. <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%2FS1544-6123%2803%2900003-5">10.1016/S1544-6123(03)00003-5</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:16913409">16913409</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Finance+Research+Letters&rft.atitle=On+More+Robust+Estimation+of+Skewness+and+Kurtosis%3A+Simulation+and+Application+to+the+S%26P500+Index&rft.volume=1&rft.pages=56-70&rft.date=2003&rft_id=info%3Adoi%2F10.1016%2FS1544-6123%2803%2900003-5&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16913409%23id-name%3DS2CID&rft.aulast=Kim&rft.aufirst=Tae-Hwan&rft.au=White%2C+Halbert&rft_id=http%3A%2F%2Fescholarship.org%2Fuc%2Fitem%2F7b52v07p&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111118123903/http://weber.ucsd.edu/~hwhite/pub_files/hwcv-092.pdf">Alternative source</a> (Comparison of kurtosis estimators)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeierBonett2003" class="citation journal cs1">Seier, E.; Bonett, D.G. (2003). "Two families of kurtosis measures". <i>Metrika</i>. <b>58</b>: 59–70. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs001840200223">10.1007/s001840200223</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:115990880">115990880</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrika&rft.atitle=Two+families+of+kurtosis+measures&rft.volume=58&rft.pages=59-70&rft.date=2003&rft_id=info%3Adoi%2F10.1007%2Fs001840200223&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115990880%23id-name%3DS2CID&rft.aulast=Seier&rft.aufirst=E.&rft.au=Bonett%2C+D.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kurtosis&action=edit&section=26" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <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/Kurtosis" class="extiw" title="v:Special:Search/Kurtosis">Kurtosis</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Excess_coefficient">"Excess coefficient"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Excess+coefficient&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DExcess_coefficient&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKurtosis" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.fxsolver.com/solve/share/RMqwaVp85T_5rbacksPD4g==/">Kurtosis calculator</a></li> <li><a rel="nofollow" class="external text" href="https://archive.today/20121208231710/http://www.wessa.net/skewkurt.wasp">Free Online Software (Calculator)</a> computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests)..</li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/k.html">Kurtosis</a> on the <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathword.html">Earliest known uses of some of the words of mathematics</a></li> <li><a rel="nofollow" class="external text" href="https://faculty.etsu.edu/seier/doc/Kurtosis100years.doc">Celebrating 100 years of Kurtosis</a> a history of the topic, with different measures of kurtosis.</li></ul> <div style="clear:both;" class=""></div> <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 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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 class="mw-selflink selflink">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible 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" 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