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id="toc-Definitions_and_detection-sublist" class="vector-toc-list"> <li id="toc-Peirce's_criterion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Peirce's_criterion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Peirce's criterion</span> </div> </a> <ul id="toc-Peirce's_criterion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tukey's_fences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tukey's_fences"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Tukey's fences</span> </div> </a> <ul id="toc-Tukey's_fences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_anomaly_detection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_anomaly_detection"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>In anomaly detection</span> </div> </a> <ul id="toc-In_anomaly_detection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modified_Thompson_Tau_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modified_Thompson_Tau_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Modified Thompson Tau test</span> </div> </a> <ul id="toc-Modified_Thompson_Tau_test-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Working_with_outliers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Working_with_outliers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Working with outliers</span> </div> </a> <button aria-controls="toc-Working_with_outliers-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 Working with outliers subsection</span> </button> <ul id="toc-Working_with_outliers-sublist" class="vector-toc-list"> <li id="toc-Retention" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Retention"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Retention</span> </div> </a> <ul id="toc-Retention-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exclusion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exclusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Exclusion</span> </div> </a> <ul id="toc-Exclusion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-normal_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-normal_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Non-normal distributions</span> </div> </a> <ul id="toc-Non-normal_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set-membership_uncertainties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set-membership_uncertainties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Set-membership uncertainties</span> </div> </a> <ul id="toc-Set-membership_uncertainties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Alternative models</span> </div> </a> <ul id="toc-Alternative_models-sublist" class="vector-toc-list"> </ul> </li> </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">4</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">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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" 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<div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 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-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Uitskieter" title="Uitskieter – Afrikaans" lang="af" hreflang="af" data-title="Uitskieter" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%8A%D9%85%D8%A9_%D8%B4%D8%A7%D8%B0%D8%A9" title="قيمة شاذة – Arabic" lang="ar" hreflang="ar" data-title="قيمة شاذة" 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/Observaci%C3%B3_at%C3%ADpica" title="Observació atípica – Catalan" lang="ca" hreflang="ca" data-title="Observació atípica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ausrei%C3%9Fer" title="Ausreißer – German" lang="de" hreflang="de" data-title="Ausreißer" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Erind" title="Erind – Estonian" lang="et" hreflang="et" data-title="Erind" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Valor_at%C3%ADpico" title="Valor atípico – Spanish" lang="es" hreflang="es" data-title="Valor atípico" 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/Muturreko_datu" title="Muturreko datu – Basque" lang="eu" hreflang="eu" data-title="Muturreko datu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%D8%AF%D9%87_%D9%BE%D8%B1%D8%AA" title="داده پرت – Persian" lang="fa" hreflang="fa" data-title="داده پرت" 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/Donn%C3%A9e_aberrante" title="Donnée aberrante – French" lang="fr" hreflang="fr" data-title="Donnée aberrante" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pencilan" title="Pencilan – Indonesian" lang="id" hreflang="id" data-title="Pencilan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Outlier" title="Outlier – Italian" lang="it" hreflang="it" data-title="Outlier" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A8%D7%99%D7%92_%D7%97%D7%A9%D7%95%D7%93_%D7%98%D7%A2%D7%95%D7%AA" title="חריג חשוד טעות – Hebrew" lang="he" hreflang="he" data-title="חריג חשוד טעות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kiugr%C3%B3_%C3%A9rt%C3%A9k" title="Kiugró érték – Hungarian" lang="hu" hreflang="hu" data-title="Kiugró érték" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Uitbijter" title="Uitbijter – Dutch" lang="nl" hreflang="nl" data-title="Uitbijter" 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%A4%96%E3%82%8C%E5%80%A4" title="外れ値 – Japanese" lang="ja" hreflang="ja" data-title="外れ値" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ekstremverdi" title="Ekstremverdi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ekstremverdi" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ekstremverdi_i_statistikk" title="Ekstremverdi i statistikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ekstremverdi i statistikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Obserwacja_odstaj%C4%85ca" title="Obserwacja odstająca – Polish" lang="pl" hreflang="pl" data-title="Obserwacja odstająca" 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/Valor_at%C3%ADpico" title="Valor atípico – Portuguese" lang="pt" hreflang="pt" data-title="Valor atípico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8B%D0%B1%D1%80%D0%BE%D1%81_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Выброс (статистика) – Russian" lang="ru" hreflang="ru" data-title="Выброс (статистика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Outlier" title="Outlier – Sundanese" lang="su" hreflang="su" data-title="Outlier" 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/Outlier" title="Outlier – Swedish" lang="sv" hreflang="sv" data-title="Outlier" 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/Ayk%C4%B1r%C4%B1_de%C4%9Fer" title="Aykırı değer – Turkish" lang="tr" hreflang="tr" data-title="Aykırı değer" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%B4_(%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0)" title="Викид (статистика) – Ukrainian" lang="uk" hreflang="uk" data-title="Викид (статистика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90i%E1%BB%83m_b%E1%BA%A5t_th%C6%B0%E1%BB%9Dng_(th%E1%BB%91ng_k%C3%AA)" title="Điểm bất thường (thống kê) – Vietnamese" lang="vi" hreflang="vi" data-title="Điểm bất thường (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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BC%82%E5%B8%B8%E5%80%BC" title="异常值 – Chinese" lang="zh" hreflang="zh" data-title="异常值" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q779824#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Observation far apart from others in statistics and data science</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the statistical term. For other uses, see <a href="/wiki/Outlier_(disambiguation)" class="mw-disambig" title="Outlier (disambiguation)">Outlier (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Michelsonmorley-boxplot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Michelsonmorley-boxplot.svg/220px-Michelsonmorley-boxplot.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Michelsonmorley-boxplot.svg/330px-Michelsonmorley-boxplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Michelsonmorley-boxplot.svg/440px-Michelsonmorley-boxplot.svg.png 2x" data-file-width="432" data-file-height="432" /></a><figcaption>Figure 1. <a href="/wiki/Box_plot" title="Box plot">Box plot</a> of data from the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> displaying four outliers in the middle column, as well as one outlier in the first column.</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, an <b>outlier</b> is a <a href="/wiki/Data_point" class="mw-redirect" title="Data point">data point</a> that differs significantly from other observations.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the <a href="/wiki/Data_set" title="Data set">data set</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. </p><p>Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, <a href="/wiki/Measurement_error" class="mw-redirect" title="Measurement error">measurement error</a>, or that the population has a <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy-tailed distribution</a>. In the case of measurement error, one wishes to discard them or use statistics that are <a href="/wiki/Robust_statistics" title="Robust statistics">robust</a> to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high <a href="/wiki/Skewness" title="Skewness">skewness</a> and that one should be very cautious in using tools or intuitions that assume a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate 'correct trial' versus 'measurement error'; this is modeled by a <a href="/wiki/Mixture_model" title="Mixture model">mixture model</a>. </p><p>In most larger samplings of data, some data points will be further away from the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">sample mean</a> than what is deemed reasonable. This can be due to incidental <a href="/wiki/Systematic_error" class="mw-redirect" title="Systematic error">systematic error</a> or flaws in the <a href="/wiki/Theory" title="Theory">theory</a> that generated an assumed family of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>, or it may be that some observations are far from the center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected (and not due to any anomalous condition). </p><p>Outliers, being the most extreme observations, may include the <a href="/wiki/Sample_maximum" class="mw-redirect" title="Sample maximum">sample maximum</a> or <a href="/wiki/Sample_minimum" class="mw-redirect" title="Sample minimum">sample minimum</a>, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations. </p><p>Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the <a href="/wiki/Average" title="Average">average</a> temperature of 10 objects in a room, and nine of them are between 20 and 25 <a href="/wiki/Degrees_Celsius" class="mw-redirect" title="Degrees Celsius">degrees Celsius</a>, but an oven is at 175 °C, the <a href="/wiki/Median" title="Median">median</a> of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object (but not the temperature in the room) than the mean; naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may indicate data points that belong to a different <a href="/wiki/Statistical_population" title="Statistical population">population</a> than the rest of the <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> set. </p><p><a href="/wiki/Estimator" title="Estimator">Estimators</a> capable of coping with outliers are said to be robust: the median is a robust statistic of <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a>, while the mean is not.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_causes">Occurrence and causes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=1" title="Edit section: Occurrence and causes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a><figcaption>Relative probabilities in a normal distribution</figcaption></figure> <p>In the case of <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a> data, the <a href="/wiki/Three_sigma_rule" class="mw-redirect" title="Three sigma rule">three sigma rule</a> means that roughly 1 in 22 observations will differ by twice the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> or more from the mean, and 1 in 370 will deviate by three times the standard deviation.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> – and not indicate an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times the expected number. </p><p>In general, if the nature of the population distribution is known <i>a priori</i>, it is possible to test if the number of outliers deviate <a href="/wiki/Statistical_significance" title="Statistical significance">significantly</a> from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability <i>p</i>) of a given distribution, the number of outliers will follow a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> with parameter <i>p</i>, which can generally be well-approximated by the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> with λ = <i>pn</i>. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, <i>p</i> is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3. </p> <div class="mw-heading mw-heading3"><h3 id="Causes">Causes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=2" title="Edit section: Causes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transmission or transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end (<a href="/wiki/King_effect" title="King effect">King effect</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Definitions_and_detection">Definitions and detection</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=3" title="Edit section: Definitions and detection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation is an outlier is ultimately a subjective exercise.<sup id="cite_ref-ZimekFilzmoser2018_7-0" class="reference"><a href="#cite_note-ZimekFilzmoser2018-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> There are various methods of outlier detection, some of which are treated as synonymous with novelty detection.<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><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><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><sup id="cite_ref-subspace_12-0" class="reference"><a href="#cite_note-subspace-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Some are graphical such as <a href="/wiki/Normal_probability_plot" title="Normal probability plot">normal probability plots</a>. Others are model-based. <a href="/wiki/Box_plot" title="Box plot">Box plots</a> are a hybrid. </p><p>Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on mean and standard deviation: </p> <ul><li><a href="/wiki/Chauvenet%27s_criterion" title="Chauvenet's criterion">Chauvenet's criterion</a></li> <li><a href="/wiki/Grubbs%27s_test_for_outliers" class="mw-redirect" title="Grubbs's test for outliers">Grubbs's test for outliers</a></li> <li><a href="/wiki/Dixon%27s_Q_test" title="Dixon's Q test">Dixon's <i>Q</i> test</a></li> <li><a href="/wiki/ASTM" class="mw-redirect" title="ASTM">ASTM</a> E178: Standard Practice for Dealing With Outlying Observations<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></li> <li><a href="/wiki/Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a> and <a href="/wiki/Leverage_(statistics)" title="Leverage (statistics)">leverage</a> are often used to detect outliers, especially in the development of linear regression models.</li> <li>Subspace and correlation based techniques for high-dimensional numerical data<sup id="cite_ref-subspace_12-1" class="reference"><a href="#cite_note-subspace-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Peirce's_criterion"><span id="Peirce.27s_criterion"></span>Peirce's criterion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=4" title="Edit section: Peirce's criterion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Peirce%27s_criterion" title="Peirce's criterion">Peirce's criterion</a></div> <blockquote> <p>It is proposed to determine in a series 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from <i>A Manual of Astronomy</i> 2:558 by Chauvenet.) <sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Tukey's_fences"><span id="Tukey.27s_fences"></span>Tukey's fences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=5" title="Edit section: Tukey's fences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other methods flag observations based on measures such as the <a href="/wiki/Interquartile_range" title="Interquartile range">interquartile range</a>. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ea6463cb36d8278ff71214fb4d13127039ae53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b6c2a91a49263a333768fc2ebebdc379ddf5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{3}}"></span> are the lower and upper <a href="/wiki/Quartile" title="Quartile">quartiles</a> respectively, then one could define an outlier to be any observation outside the range: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big [}Q_{1}-k(Q_{3}-Q_{1}),Q_{3}+k(Q_{3}-Q_{1}){\big ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big [}Q_{1}-k(Q_{3}-Q_{1}),Q_{3}+k(Q_{3}-Q_{1}){\big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa0c7280d2bbf322c81de7c0a7d65296a6e5d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.731ex; height:3.176ex;" alt="{\displaystyle {\big [}Q_{1}-k(Q_{3}-Q_{1}),Q_{3}+k(Q_{3}-Q_{1}){\big ]}}"></span></dd></dl> <p>for some nonnegative constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. <a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a> proposed this test, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a8c09d9b3300c5452e640784eac32342586393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.281ex; height:2.176ex;" alt="{\displaystyle k=1.5}"></span> indicates an "outlier", 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 k=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662e06a2436f8a44fec791f5c794621f10dc8f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=3}"></span> indicates data that is "far out".<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_anomaly_detection">In anomaly detection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=6" title="Edit section: In anomaly detection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div> <p>In various domains such as, but not limited to, <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Finance" title="Finance">finance</a>, <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, <a href="/wiki/Manufacturing" title="Manufacturing">manufacturing</a>, <a href="/wiki/Network_science" title="Network science">networking</a> and <a href="/wiki/Data_mining" title="Data mining">data mining</a>, the task of <i>anomaly detection</i> may take other approaches. Some of these may be distance-based<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><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and density-based such as <a href="/wiki/Local_Outlier_Factor" class="mw-redirect" title="Local Outlier Factor">Local Outlier Factor</a> (LOF).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Some approaches may use the distance to the <a href="/wiki/K-nearest_neighbor" class="mw-redirect" title="K-nearest neighbor">k-nearest neighbors</a> to label observations as outliers or non-outliers.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Modified_Thompson_Tau_test">Modified Thompson Tau test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=7" title="Edit section: Modified Thompson Tau test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Studentized_residual#Distribution" title="Studentized residual">Studentized residual § Distribution</a></div> <p>The modified Thompson Tau test<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2016)">citation needed</span></a></i>]</sup> is a method used to determine if an outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set's standard deviation, average and provides a statistically determined rejection zone; thus providing an objective method to determine if a data point is an outlier.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Although intuitively appealing, this method appears to be unpublished (it is ''not'' described in Thompson (1985) so one should use it with caution. (October 2016)">citation needed</span></a></i>]</sup><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> How it works: First, a data set's average is determined. Next the absolute deviation between each data point and the average are determined. Thirdly, a rejection region is determined using the formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Rejection Region}}{=}{\frac {{t_{\alpha /2}}{\left(n-1\right)}}{{\sqrt {n}}{\sqrt {n-2+{t_{\alpha /2}^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Rejection Region</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Rejection Region}}{=}{\frac {{t_{\alpha /2}}{\left(n-1\right)}}{{\sqrt {n}}{\sqrt {n-2+{t_{\alpha /2}^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54853eca96ba880458fbd120a3c2e1a5aff3fee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:37.418ex; height:8.676ex;" alt="{\displaystyle {\text{Rejection Region}}{=}{\frac {{t_{\alpha /2}}{\left(n-1\right)}}{{\sqrt {n}}{\sqrt {n-2+{t_{\alpha /2}^{2}}}}}}}"></span>;</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {t_{\alpha /2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {t_{\alpha /2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdb2a245343dfec3ad8322a03d246e610ef36ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.947ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {t_{\alpha /2}}}"></span> is the critical value from the Student <span class="texhtml mvar" style="font-style:italic;">t</span> distribution with <i>n</i>-2 degrees of freedom, <i>n</i> is the sample size, and s is the sample standard deviation. To determine if a value is an outlier: Calculate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \delta =|(X-mean(X))/s|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>m</mi> <mi>e</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \delta =|(X-mean(X))/s|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ad133e9e86179f845a39a56407c1c3eb1237ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.23ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \delta =|(X-mean(X))/s|}"></span>. If <i>δ</i> > Rejection Region, the data point is an outlier. If <i>δ</i> ≤ Rejection Region, the data point is not an outlier. </p><p>The modified Thompson Tau test is used to find one outlier at a time (largest value of <i>δ</i> is removed if it is an outlier). Meaning, if a data point is found to be an outlier, it is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set. </p><p>Some work has also examined outliers for nominal (or categorical) data. In the context of a set of examples (or instances) in a data set, instance hardness measures the probability that an instance will be misclassified ( <span class="mwe-math-element"><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-p(y|x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p(y|x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169d1e22c9c0ee0eec60e81dca13adb5fd150b35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.114ex; height:2.843ex;" alt="{\displaystyle 1-p(y|x)}"></span> where <span class="texhtml mvar" style="font-style:italic;">y</span> is the assigned class label and <span class="texhtml mvar" style="font-style:italic;">x</span> represent the input attribute value for an instance in the training set <span class="texhtml mvar" style="font-style:italic;">t</span>).<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Ideally, instance hardness would be calculated by summing over the set of all possible hypotheses <span class="texhtml mvar" style="font-style:italic;">H</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}IH(\langle x,y\rangle )&=\sum _{H}(1-p(y,x,h))p(h|t)\\&=\sum _{H}p(h|t)-p(y,x,h)p(h|t)\\&=1-\sum _{H}p(y,x,h)p(h|t).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>I</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}IH(\langle x,y\rangle )&=\sum _{H}(1-p(y,x,h))p(h|t)\\&=\sum _{H}p(h|t)-p(y,x,h)p(h|t)\\&=1-\sum _{H}p(y,x,h)p(h|t).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d652a607e1fd5a9ed0b6d7eaa8e843819c12a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:41.285ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}IH(\langle x,y\rangle )&=\sum _{H}(1-p(y,x,h))p(h|t)\\&=\sum _{H}p(h|t)-p(y,x,h)p(h|t)\\&=1-\sum _{H}p(y,x,h)p(h|t).\end{aligned}}}"></span></dd></dl> <p>Practically, this formulation is unfeasible as <span class="texhtml mvar" style="font-style:italic;">H</span> is potentially infinite and calculating <span class="mwe-math-element"><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(h|t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(h|t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7fe299777b747e92dfb2f83cd5fe1e0469413b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.894ex; height:2.843ex;" alt="{\displaystyle p(h|t)}"></span> is unknown for many algorithms. Thus, instance hardness can be approximated using a diverse subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc27b37f0b7c8530db75f332b00abdce1d65c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.745ex; height:2.176ex;" alt="{\displaystyle L\subset H}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IH_{L}(\langle x,y\rangle )=1-{\frac {1}{|L|}}\sum _{j=1}^{|L|}p(y|x,g_{j}(t,\alpha ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IH_{L}(\langle x,y\rangle )=1-{\frac {1}{|L|}}\sum _{j=1}^{|L|}p(y|x,g_{j}(t,\alpha ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f0eb43b509af606e3372988a9613deb9b8495a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.869ex; height:8.009ex;" alt="{\displaystyle IH_{L}(\langle x,y\rangle )=1-{\frac {1}{|L|}}\sum _{j=1}^{|L|}p(y|x,g_{j}(t,\alpha ))}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{j}(t,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{j}(t,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480149a95d7c306cb4eb1f91dd3ff230f6996179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.189ex; height:3.009ex;" alt="{\displaystyle g_{j}(t,\alpha )}"></span> is the hypothesis induced by learning algorithm <span class="mwe-math-element"><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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc0c1f707e4786c1121db1ba0a608fe85a94e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.019ex; height:2.343ex;" alt="{\displaystyle g_{j}}"></span> trained on training set <span class="texhtml mvar" style="font-style:italic;">t</span> with hyperparameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Instance hardness provides a continuous value for determining if an instance is an outlier instance. </p> <div class="mw-heading mw-heading2"><h2 id="Working_with_outliers">Working with outliers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=8" title="Edit section: Working with outliers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The choice of how to deal with an outlier should depend on the cause. Some estimators are highly sensitive to outliers, notably <a href="/wiki/Estimation_of_covariance_matrices" title="Estimation of covariance matrices">estimation of covariance matrices</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Retention">Retention</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=9" title="Edit section: Retention"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case.<sup id="cite_ref-karch2023_25-0" class="reference"><a href="#cite_note-karch2023-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Instead, one should use a method that is robust to outliers to model or analyze data with naturally occurring outliers.<sup id="cite_ref-karch2023_25-1" class="reference"><a href="#cite_note-karch2023-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Exclusion">Exclusion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=10" title="Edit section: Exclusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When deciding whether to remove an outlier, the cause has to be considered. As mentioned earlier, if the outlier's origin can be attributed to an experimental error, or if it can be otherwise determined that the outlying data point is erroneous, it is generally recommended to remove it.<sup id="cite_ref-karch2023_25-2" class="reference"><a href="#cite_note-karch2023-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bakker2014_26-0" class="reference"><a href="#cite_note-bakker2014-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> However, it is more desirable to correct the erroneous value, if possible. </p><p>Removing a data point solely because it is an outlier, on the other hand, is a controversial practice, often frowned upon by many scientists and science instructors, as it typically invalidates statistical results.<sup id="cite_ref-karch2023_25-3" class="reference"><a href="#cite_note-karch2023-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bakker2014_26-1" class="reference"><a href="#cite_note-bakker2014-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> While mathematical criteria provide an objective and quantitative method for data rejection, they do not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. </p><p>The two common approaches to exclude outliers are <a href="/wiki/Truncation_(statistics)" title="Truncation (statistics)">truncation</a> (or trimming) and <a href="/wiki/Winsorising" class="mw-redirect" title="Winsorising">Winsorising</a>. Trimming discards the outliers whereas Winsorising replaces the outliers with the nearest "nonsuspect" data.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Exclusion can also be a consequence of the measurement process, such as when an experiment is not entirely capable of measuring such extreme values, resulting in <a href="/wiki/Censoring_(statistics)" title="Censoring (statistics)">censored</a> data.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the estimated coefficients, using a measure such as <a href="/wiki/Cook%27s_distance" title="Cook's distance">Cook's distance</a>.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>If a data point (or points) is excluded from the <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>, this should be clearly stated on any subsequent report. </p> <div class="mw-heading mw-heading3"><h3 id="Non-normal_distributions">Non-normal distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=11" title="Edit section: Non-normal distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The possibility should be considered that the underlying distribution of the data is not approximately normal, having "<a href="/wiki/Fat_tails" class="mw-redirect" title="Fat tails">fat tails</a>". For instance, when sampling from a <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> the sample variance increases with the sample size, the sample mean fails to converge as the sample size increases, and outliers are expected at far larger rates than for a normal distribution. Even a slight difference in the fatness of the tails can make a large difference in the expected number of extreme values. </p> <div class="mw-heading mw-heading3"><h3 id="Set-membership_uncertainties">Set-membership uncertainties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=12" title="Edit section: Set-membership uncertainties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Set_estimation" title="Set estimation">set membership approach</a> considers that the uncertainty corresponding to the <i>i</i>th measurement of an unknown random vector <i>x</i> is represented by a set <i>X</i><sub>i</sub> (instead of a probability density function). If no outliers occur, <i>x</i> should belong to the intersection of all <i>X</i><sub>i</sub>'s. When outliers occur, this intersection could be empty, and we should relax a small number of the sets <i>X</i><sub>i</sub> (as small as possible) in order to avoid any inconsistency.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> This can be done using the notion of <i>q</i>-<a href="/wiki/Relaxed_intersection" title="Relaxed intersection">relaxed intersection</a>. As illustrated by the figure, the <i>q</i>-relaxed intersection corresponds to the set of all <i>x</i> which belong to all sets except <i>q</i> of them. Sets <i>X</i><sub>i</sub> that do not intersect the <i>q</i>-relaxed intersection could be suspected to be outliers. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wiki_q_inter_def.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wiki_q_inter_def.jpg/220px-Wiki_q_inter_def.jpg" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wiki_q_inter_def.jpg/330px-Wiki_q_inter_def.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Wiki_q_inter_def.jpg/440px-Wiki_q_inter_def.jpg 2x" data-file-width="1594" data-file-height="1302" /></a><figcaption>Figure 5. <i>q</i>-relaxed intersection of 6 sets for <i>q</i>=2 (red), <i>q</i>=3 (green), <i>q</i>= 4 (blue), <i>q</i>= 5 (yellow).</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Alternative_models">Alternative models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=13" title="Edit section: Alternative models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a <a href="/wiki/Hierarchical_Bayes_model" class="mw-redirect" title="Hierarchical Bayes model">hierarchical Bayes model</a>, or a <a href="/wiki/Mixture_model" title="Mixture model">mixture model</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </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=Outlier&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Anomaly_(natural_sciences)" title="Anomaly (natural sciences)">Anomaly (natural sciences)</a></li> <li><a href="/wiki/Novelty_detection" title="Novelty detection">Novelty detection</a></li> <li><a href="/wiki/Anscombe%27s_quartet" title="Anscombe's quartet">Anscombe's quartet</a></li> <li><a href="/wiki/Data_transformation_(statistics)" title="Data transformation (statistics)">Data transformation (statistics)</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory</a></li> <li><a href="/wiki/Influential_observation" title="Influential observation">Influential observation</a></li> <li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">Random sample consensus</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust regression</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Winsorizing" title="Winsorizing">Winsorizing</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGrubbs1969" class="citation journal cs1">Grubbs, F. E. (February 1969). "Procedures for detecting outlying observations in samples". <i>Technometrics</i>. <b>11</b> (1): 1–21. <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%2F00401706.1969.10490657">10.1080/00401706.1969.10490657</a>. <q>An outlying observation, or "outlier," is one that appears to deviate markedly from other members of the sample in which it occurs.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technometrics&rft.atitle=Procedures+for+detecting+outlying+observations+in+samples&rft.volume=11&rft.issue=1&rft.pages=1-21&rft.date=1969-02&rft_id=info%3Adoi%2F10.1080%2F00401706.1969.10490657&rft.aulast=Grubbs&rft.aufirst=F.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaddala1992" class="citation book cs1"><a href="/wiki/G._S._Maddala" title="G. S. Maddala">Maddala, G. S.</a> (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nBS3AAAAIAAJ&pg=PA89">"Outliers"</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec00madd/page/89"><i>Introduction to Econometrics</i></a> (2nd ed.). New York: MacMillan. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec00madd/page/89">89</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-374545-4" title="Special:BookSources/978-0-02-374545-4"><bdi>978-0-02-374545-4</bdi></a>. <q>An outlier is an observation that is far removed from the rest of the observations.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Outliers&rft.btitle=Introduction+to+Econometrics&rft.place=New+York&rft.pages=89&rft.edition=2nd&rft.pub=MacMillan&rft.date=1992&rft.isbn=978-0-02-374545-4&rft.aulast=Maddala&rft.aufirst=G.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnBS3AAAAIAAJ%26pg%3DPA89&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Pimentel, M. A., Clifton, D. A., Clifton, L., & Tarassenko, L. (2014). A review of novelty detection. Signal Processing, 99, 215-249.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrubbs1969">Grubbs 1969</a>, p. 1 stating "An outlying observation may be merely an extreme manifestation of the random variability inherent in the data. ... On the other hand, an outlying observation may be the result of gross deviation from prescribed experimental procedure or an error in calculating or recording the numerical value."</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Ripley, Brian D. 2004. <a rel="nofollow" class="external text" href="http://www.stats.ox.ac.uk/pub/StatMeth/Robust.pdf">Robust statistics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121021081319/http://www.stats.ox.ac.uk/pub/StatMeth/Robust.pdf">Archived</a> 2012-10-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuanChenKerre2005" class="citation book cs1"><a href="/wiki/Da_Ruan" title="Da Ruan">Ruan, Da</a>; Chen, Guoqing; Kerre, Etienne (2005). Wets, G. (ed.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/intelligentdatam00ruan_742"><i>Intelligent Data Mining: Techniques and Applications</i></a></span>. Studies in Computational Intelligence Vol. 5. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/intelligentdatam00ruan_742/page/n326">318</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-26256-5" title="Special:BookSources/978-3-540-26256-5"><bdi>978-3-540-26256-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intelligent+Data+Mining%3A+Techniques+and+Applications&rft.series=Studies+in+Computational+Intelligence+Vol.+5&rft.pages=318&rft.pub=Springer&rft.date=2005&rft.isbn=978-3-540-26256-5&rft.aulast=Ruan&rft.aufirst=Da&rft.au=Chen%2C+Guoqing&rft.au=Kerre%2C+Etienne&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintelligentdatam00ruan_742&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-ZimekFilzmoser2018-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-ZimekFilzmoser2018_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZimekFilzmoser2018" class="citation journal cs1">Zimek, Arthur; Filzmoser, Peter (2018). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211114121638/https://findresearcher.sdu.dk:8443/ws/files/153197807/There_and_Back_Again.pdf">"There and back again: Outlier detection between statistical reasoning and data mining algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery</i>. <b>8</b> (6): e1280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fwidm.1280">10.1002/widm.1280</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1942-4787">1942-4787</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:53305944">53305944</a>. Archived from <a rel="nofollow" class="external text" href="https://findresearcher.sdu.dk:8443/ws/files/153197807/There_and_Back_Again.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2021-11-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wiley+Interdisciplinary+Reviews%3A+Data+Mining+and+Knowledge+Discovery&rft.atitle=There+and+back+again%3A+Outlier+detection+between+statistical+reasoning+and+data+mining+algorithms&rft.volume=8&rft.issue=6&rft.pages=e1280&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53305944%23id-name%3DS2CID&rft.issn=1942-4787&rft_id=info%3Adoi%2F10.1002%2Fwidm.1280&rft.aulast=Zimek&rft.aufirst=Arthur&rft.au=Filzmoser%2C+Peter&rft_id=https%3A%2F%2Ffindresearcher.sdu.dk%3A8443%2Fws%2Ffiles%2F153197807%2FThere_and_Back_Again.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Pimentel, M. A., Clifton, D. A., Clifton, L., & Tarassenko, L. (2014). A review of novelty detection. Signal Processing, 99, 215-249.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRousseeuwLeroy1996" class="citation cs2"><a href="/wiki/Peter_Rousseeuw" title="Peter Rousseeuw">Rousseeuw, P</a>; Leroy, A. (1996), <a href="/wiki/Robust_Regression_and_Outlier_Detection" title="Robust Regression and Outlier Detection"><i>Robust Regression and Outlier Detection</i></a> (3rd ed.), John Wiley & Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+Regression+and+Outlier+Detection&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1996&rft.aulast=Rousseeuw&rft.aufirst=P&rft.au=Leroy%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgeAustin2004" class="citation cs2">Hodge, Victoria J.; Austin, Jim (2004), "A Survey of Outlier Detection Methodologies", <i>Artificial Intelligence Review</i>, <b>22</b> (2): 85–126, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.1943">10.1.1.109.1943</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AAIRE.0000045502.10941.a9">10.1023/B:AIRE.0000045502.10941.a9</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:3330313">3330313</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Artificial+Intelligence+Review&rft.atitle=A+Survey+of+Outlier+Detection+Methodologies&rft.volume=22&rft.issue=2&rft.pages=85-126&rft.date=2004&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.109.1943%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3330313%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FB%3AAIRE.0000045502.10941.a9&rft.aulast=Hodge&rft.aufirst=Victoria+J.&rft.au=Austin%2C+Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnettLewis1994" class="citation cs2">Barnett, Vic; Lewis, Toby (1994) [1978], <i>Outliers in Statistical Data</i> (3 ed.), Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-93094-5" title="Special:BookSources/978-0-471-93094-5"><bdi>978-0-471-93094-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Outliers+in+Statistical+Data&rft.edition=3&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-93094-5&rft.aulast=Barnett&rft.aufirst=Vic&rft.au=Lewis%2C+Toby&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-subspace-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-subspace_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-subspace_12-1"><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="CITEREFZimekSchubertKriegel2012" class="citation journal cs1">Zimek, A.; Schubert, E.; <a href="/wiki/Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Kriegel, H.-P.</a> (2012). "A survey on unsupervised outlier detection in high-dimensional numerical data". <i>Statistical Analysis and Data Mining</i>. <b>5</b> (5): 363–387. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsam.11161">10.1002/sam.11161</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:6724536">6724536</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Analysis+and+Data+Mining&rft.atitle=A+survey+on+unsupervised+outlier+detection+in+high-dimensional+numerical+data&rft.volume=5&rft.issue=5&rft.pages=363-387&rft.date=2012&rft_id=info%3Adoi%2F10.1002%2Fsam.11161&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6724536%23id-name%3DS2CID&rft.aulast=Zimek&rft.aufirst=A.&rft.au=Schubert%2C+E.&rft.au=Kriegel%2C+H.-P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.nrc.gov/docs/ML1023/ML102371244.pdf">E178: Standard Practice for Dealing With Outlying Observations</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a>, <a rel="nofollow" class="external text" href="http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1852AJ......2..161P;data_type=PDF_HIGH">"Criterion for the Rejection of Doubtful Observations"</a>, <i>Astronomical Journal</i> II 45 (1852) and <a rel="nofollow" class="external text" href="http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1852AJ......2..176P;data_type=PDF_HIGH">Errata to the original paper</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1877–1878" class="citation journal cs1"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (May 1877 – May 1878). "On Peirce's criterion". <i>Proceedings of the American Academy of Arts and Sciences</i>. <b>13</b>: 348–351. <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%2F25138498">10.2307/25138498</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/25138498">25138498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=On+Peirce%27s+criterion&rft.volume=13&rft.pages=348-351&rft.date=1877-05%2F1878-05&rft_id=info%3Adoi%2F10.2307%2F25138498&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25138498%23id-name%3DJSTOR&rft.aulast=Peirce&rft.aufirst=Benjamin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1873" class="citation journal cs1"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles Sanders</a> (1873) [1870]. 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NOAA <a rel="nofollow" class="external text" href="http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0019.PDF#page=215">PDF Eprint</a> (goes to Report p. 200, PDF's p. 215).</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1986" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles Sanders</a> (1986) [1982]. "On the Theory of Errors of Observation". In Kloesel, Christian J. W.; et al. (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/writingsofcharle0002peir/page/140"><i>Writings of Charles S. Peirce: A Chronological Edition</i></a>. Vol. 3, 1872–1878. Bloomington, Indiana: Indiana University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/writingsofcharle0002peir/page/140">140–160</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-253-37201-7" title="Special:BookSources/978-0-253-37201-7"><bdi>978-0-253-37201-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+Theory+of+Errors+of+Observation&rft.btitle=Writings+of+Charles+S.+Peirce%3A+A+Chronological+Edition&rft.place=Bloomington%2C+Indiana&rft.pages=140-160&rft.pub=Indiana+University+Press&rft.date=1986&rft.isbn=978-0-253-37201-7&rft.aulast=Peirce&rft.aufirst=Charles+Sanders&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fwritingsofcharle0002peir%2Fpage%2F140&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span> – Appendix 21, according to the editorial note on page 515</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTukey1977" class="citation book cs1">Tukey, John W (1977). <a rel="nofollow" class="external text" href="https://archive.org/details/exploratorydataa00tuke_0"><i>Exploratory Data Analysis</i></a>. 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"Novelty detection and Neural Network validation". <i>IEE Proceedings - Vision, Image, and Signal Processing</i>. <b>141</b> (4): 217–222. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fip-vis%3A19941330">10.1049/ip-vis:19941330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEE+Proceedings+-+Vision%2C+Image%2C+and+Signal+Processing&rft.atitle=Novelty+detection+and+Neural+Network+validation&rft.volume=141&rft.issue=4&rft.pages=217-222&rft.date=1994-08&rft_id=info%3Adoi%2F10.1049%2Fip-vis%3A19941330&rft.aulast=Bishop&rft.aufirst=C.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outlier&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/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:Outliers" class="extiw" title="commons:Category:Outliers">Outliers</a></span>.</div></div> </div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Outlier"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Renze, John. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Outlier.html">"Outlier"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Outlier&rft.au=Renze%2C+John&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FOutlier.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalakrishnanChilds2001" class="citation cs2">Balakrishnan, N.; Childs, A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Outlier">"Outlier"</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></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Outlier&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Balakrishnan&rft.aufirst=N.&rft.au=Childs%2C+A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOutlier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutlier" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h.htm">Grubbs test</a> described by NIST manual</li></ul> <div class="navbox-styles"><style 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