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population subsection</span> </button> <ul id="toc-Quantiles_of_a_population-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Even-sized_population" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Even-sized_population"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Even-sized population</span> </div> </a> <ul id="toc-Even-sized_population-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odd-sized_population" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Odd-sized_population"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Odd-sized population</span> </div> </a> <ul id="toc-Odd-sized_population-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_to_the_mean" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_the_mean"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Relationship to the mean</span> </div> </a> <ul id="toc-Relationship_to_the_mean-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Estimating_quantiles_from_a_sample" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Estimating_quantiles_from_a_sample"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Estimating quantiles from a sample</span> </div> </a> <ul id="toc-Estimating_quantiles_from_a_sample-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_asymptotic_distribution_of_the_sample_median" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_asymptotic_distribution_of_the_sample_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The asymptotic distribution of the sample median</span> </div> </a> <ul id="toc-The_asymptotic_distribution_of_the_sample_median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximate_quantiles_from_a_stream" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Approximate_quantiles_from_a_stream"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Approximate quantiles from a stream</span> </div> </a> <ul id="toc-Approximate_quantiles_from_a_stream-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discussion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Discussion</span> </div> </a> <ul id="toc-Discussion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_quantifications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_quantifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other quantifications</span> </div> </a> <ul id="toc-Other_quantifications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quantile</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%8B%D0%BB%D1%8C" title="Квантыль – Belarusian" lang="be" hreflang="be" data-title="Квантыль" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Позиционна средна величина – Bulgarian" lang="bg" hreflang="bg" data-title="Позиционна средна величина" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quantil" title="Quantil – Catalan" lang="ca" hreflang="ca" data-title="Quantil" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantil" title="Kvantil – Czech" lang="cs" hreflang="cs" data-title="Kvantil" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fraktil" title="Fraktil – Danish" lang="da" hreflang="da" data-title="Fraktil" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantil_(Wahrscheinlichkeitstheorie)" title="Quantil (Wahrscheinlichkeitstheorie) – German" lang="de" hreflang="de" data-title="Quantil (Wahrscheinlichkeitstheorie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuantil" title="Cuantil – Spanish" lang="es" hreflang="es" data-title="Cuantil" 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/Kuantil" title="Kuantil – Basque" lang="eu" hreflang="eu" data-title="Kuantil" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%DA%A9" title="چندک – Persian" lang="fa" hreflang="fa" data-title="چندک" 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/Quantile" title="Quantile – French" lang="fr" hreflang="fr" data-title="Quantile" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quantile" title="Quantile – Italian" lang="it" hreflang="it" data-title="Quantile" 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%A9%D7%91%D7%A8%D7%95%D7%9F_(%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94)" title="שברון (סטטיסטיקה) – Hebrew" lang="he" hreflang="he" data-title="שברון (סטטיסטיקה)" 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/Kvantilisek" title="Kvantilisek – Hungarian" lang="hu" hreflang="hu" data-title="Kvantilisek" 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/Kwantiel" title="Kwantiel – Dutch" lang="nl" hreflang="nl" data-title="Kwantiel" 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%88%86%E4%BD%8D%E6%95%B0" title="分位数 – Japanese" lang="ja" hreflang="ja" data-title="分位数" 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/Kvantil" title="Kvantil – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantil" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kwantyl" title="Kwantyl – Polish" lang="pl" hreflang="pl" data-title="Kwantyl" 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/Quantil" title="Quantil – Portuguese" lang="pt" hreflang="pt" data-title="Quantil" 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%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B8%D0%BB%D1%8C" title="Квантиль – Russian" lang="ru" hreflang="ru" data-title="Квантиль" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantil" title="Kvantil – Slovenian" lang="sl" hreflang="sl" data-title="Kvantil" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B8%D0%BB" title="Квантил – Serbian" lang="sr" hreflang="sr" data-title="Квантил" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Quantile" title="Quantile – Sundanese" lang="su" hreflang="su" data-title="Quantile" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvantiili" title="Kvantiili – Finnish" lang="fi" hreflang="fi" data-title="Kvantiili" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B9%84%E0%B8%97%E0%B8%A5%E0%B9%8C" title="ควอนไทล์ – Thai" lang="th" hreflang="th" data-title="ควอนไทล์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B8%D0%BB%D1%8C" title="Квантиль – Ukrainian" lang="uk" hreflang="uk" data-title="Квантиль" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%86%E4%BD%8D%E6%95%B0" title="分位数 – Chinese" lang="zh" hreflang="zh" data-title="分位数" 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/Q578714#sitelinks-wikipedia" title="Edit interlanguage links" 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searchaux" style="display:none">Statistical method of dividing data into equal-sized intervals for analysis</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Iqr_with_quantile.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Iqr_with_quantile.png/220px-Iqr_with_quantile.png" decoding="async" width="220" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Iqr_with_quantile.png/330px-Iqr_with_quantile.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Iqr_with_quantile.png/440px-Iqr_with_quantile.png 2x" data-file-width="742" data-file-height="428" /></a><figcaption>Probability density of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, with quantiles shown. The area below the red curve is the same in the intervals <span class="texhtml">(−∞,<i>Q</i><sub>1</sub>)</span>, <span class="texhtml">(<i>Q</i><sub>1</sub>,<i>Q</i><sub>2</sub>)</span>, <span class="texhtml">(<i>Q</i><sub>2</sub>,<i>Q</i><sub>3</sub>)</span>, and <span class="texhtml">(<i>Q</i><sub>3</sub>,+∞)</span>.</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Probability" title="Probability">probability</a>, <b>quantiles</b> are cut points dividing the <a href="/wiki/Range_(statistics)" title="Range (statistics)">range</a> of a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> into continuous intervals with equal probabilities, or dividing the <a href="/wiki/Observation_(statistics)" class="mw-redirect" title="Observation (statistics)">observations</a> in a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as <i><a href="/wiki/Quartile" title="Quartile">quartiles</a></i> (four groups), <i><a href="/wiki/Decile" title="Decile">deciles</a></i> (ten groups), and <i><a href="/wiki/Percentile" title="Percentile">percentiles</a></i> (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points. </p><p><b><span class="texhtml mvar" style="font-style:italic;">q</span></b>-<b>quantiles</b> are values that <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> a <a href="/wiki/Finite_set" title="Finite set">finite set</a> of values into <span class="texhtml mvar" style="font-style:italic;">q</span> <a href="/wiki/Subset" title="Subset">subsets</a> of (nearly) equal sizes. There are <span class="texhtml"><i>q</i> − 1</span> partitions of the <span class="texhtml mvar" style="font-style:italic;">q</span>-quantiles, one for each <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml mvar" style="font-style:italic;">k</span> satisfying <span class="texhtml">0 < <i>k</i> < <i>q</i></span>. In some cases the value of a quantile may not be uniquely determined, as can be the case for the <a href="/wiki/Median" title="Median">median</a> (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> distributions, providing a way to generalize <a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">rank statistics</a> to continuous variables (see <a href="/wiki/Percentile_rank" title="Percentile rank">percentile rank</a>). When the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> is known, the <span class="texhtml mvar" style="font-style:italic;">q</span>-quantiles are the application of the<i> <a href="/wiki/Quantile_function" title="Quantile function">quantile function</a></i> (the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> of the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>) to the values <span class="texhtml">{1/<i>q</i>, 2/<i>q</i>, …, (<i>q</i> − 1)/<i>q</i></span>}. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Quantiles_of_a_population">Quantiles of a population</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=1" title="Edit section: Quantiles of a population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As in the computation of, for example, <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>, the estimation of a quantile depends upon whether one is operating with a <a href="/wiki/Statistical_population" title="Statistical population">statistical population</a> or with a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> drawn from it. For a population, of discrete values or for a continuous population density, the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile is the data value where the cumulative distribution function crosses <span class="texhtml"><i>k</i>/<i>q</i></span>. That is, <span class="texhtml mvar" style="font-style:italic;">x</span> is a <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile for a variable <span class="texhtml mvar" style="font-style:italic;">X</span> if </p> <dl><dd><span class="texhtml">Pr[<i>X</i> < <i>x</i>] ≤ <i>k</i>/<i>q</i></span> or, equivalently, <span class="texhtml">Pr[<i>X</i> ≥ <i>x</i>] ≥ 1 − <i>k</i>/<i>q</i></span></dd></dl> <p>and </p> <dl><dd><span class="texhtml">Pr[<i>X</i> ≤ <i>x</i>] ≥ <i>k</i>/<i>q</i></span>.</dd></dl> <p>For a finite population of <span class="texhtml mvar" style="font-style:italic;">N</span> equally probable values indexed <span class="texhtml">1, …, <i>N</i></span> from lowest to highest, the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile of this population can equivalently be computed via the value of <span class="texhtml mvar" style="font-style:italic;">I<sub>p</sub> = <i>N</i> <i>k</i>/<i>q</i></span>. If <span class="texhtml mvar" style="font-style:italic;">I<sub>p</sub></span> is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile. On the other hand, if <span class="texhtml mvar" style="font-style:italic;">I<sub>p</sub></span> is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see <a href="#Estimating_quantiles_from_a_sample">Estimating quantiles from a sample</a>). </p><p>If, instead of using integers <span class="texhtml mvar" style="font-style:italic;">k</span> and <span class="texhtml mvar" style="font-style:italic;">q</span>, the "<span class="texhtml mvar" style="font-style:italic;">p</span>-quantile" is based on a <a href="/wiki/Real_number" title="Real number">real number</a> <span class="texhtml mvar" style="font-style:italic;">p</span> with <span class="texhtml">0 < <i>p</i> < 1</span> then <span class="texhtml mvar" style="font-style:italic;">p</span> replaces <span class="texhtml"><i>k</i>/<i>q</i></span> in the above formulas. This broader terminology is used when quantiles are used to <a href="/wiki/Quantile-parameterized_distribution" title="Quantile-parameterized distribution">parameterize continuous probability distributions</a>. Moreover, some software programs (including <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a>) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology is an extension beyond traditional statistics definitions. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following two examples use the Nearest Rank definition of quantile with rounding. For an explanation of this definition, see <a href="/wiki/Percentile" title="Percentile">percentiles</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Even-sized_population">Even-sized population</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=3" title="Edit section: Even-sized population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? </p> <table class="wikitable"> <tbody><tr> <th>Quartile </th> <th>Calculation </th> <th>Result </th></tr> <tr> <td>Zeroth quartile </td> <td>Although not universally accepted, one can also speak of the zeroth quartile. This is the minimum value of the set, so the zeroth quartile in this example would be 3. </td> <td>3 </td></tr> <tr> <td>First quartile </td> <td>The rank of the first quartile is 10×(1/4) = 2.5, which rounds up to 3, meaning that 3 is the rank in the population (from least to greatest values) at which approximately 1/4 of the values are less than the value of the first quartile. The third value in the population is 7. </td> <td>7 </td></tr> <tr> <td>Second quartile </td> <td>The rank of the second quartile (same as the median) is 10×(2/4) = 5, which is an integer, while the number of values (10) is an even number, so the average of both the fifth and sixth values is taken—that is (8+10)/2 = 9, though any value from 8 through to 10 could be taken to be the median. </td> <td>9 </td></tr> <tr> <td>Third quartile </td> <td>The rank of the third quartile is 10×(3/4) = 7.5, which rounds up to 8. The eighth value in the population is 15. </td> <td>15 </td></tr> <tr> <td>Fourth quartile </td> <td>Although not universally accepted, one can also speak of the fourth quartile. This is the maximum value of the set, so the fourth quartile in this example would be 20. Under the Nearest Rank definition of quantile, the rank of the fourth quartile is the rank of the biggest number, so the rank of the fourth quartile would be 10. </td> <td>20 </td></tr></tbody></table> <p>So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20. </p> <div class="mw-heading mw-heading4"><h4 id="Odd-sized_population">Odd-sized population</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=4" title="Edit section: Odd-sized population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? </p> <table class="wikitable"> <tbody><tr> <th>Quartile </th> <th>Calculation </th> <th>Result </th></tr> <tr> <td>Zeroth quartile </td> <td>Although not universally accepted, one can also speak of the zeroth quartile. This is the minimum value of the set, so the zeroth quartile in this example would be 3. </td> <td>3 </td></tr> <tr> <td>First quartile </td> <td>The first quartile is determined by 11×(1/4) = 2.75, which rounds up to 3, meaning that 3 is the rank in the population (from least to greatest values) at which approximately 1/4 of the values are less than the value of the first quartile. The third value in the population is 7. </td> <td>7 </td></tr> <tr> <td>Second quartile </td> <td>The second quartile value (same as the median) is determined by 11×(2/4) = 5.5, which rounds up to 6. Therefore, 6 is the rank in the population (from least to greatest values) at which approximately 2/4 of the values are less than the value of the second quartile (or median). The sixth value in the population is 9. </td> <td>9 </td></tr> <tr> <td>Third quartile </td> <td>The third quartile value for the original example above is determined by 11×(3/4) = 8.25, which rounds up to 9. The ninth value in the population is 15. </td> <td>15 </td></tr> <tr> <td>Fourth quartile </td> <td>Although not universally accepted, one can also speak of the fourth quartile. This is the maximum value of the set, so the fourth quartile in this example would be 20. Under the Nearest Rank definition of quantile, the rank of the fourth quartile is the rank of the biggest number, so the rank of the fourth quartile would be 11. </td> <td>20 </td></tr></tbody></table> <p>So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20. </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_the_mean">Relationship to the mean</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=5" title="Edit section: Relationship to the mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>−</mo> <mi>σ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> <mo>≤</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9d6668df93a2d006a18fede8eb355400a097d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.214ex; height:7.509ex;" alt="{\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,}"></span> where <span class="texhtml mvar" style="font-style:italic;">Q(p)</span> is the value of the <span class="texhtml mvar" style="font-style:italic;">p</span>-quantile for <span class="texhtml">0 < <i>p</i> < 1</span> (or equivalently is the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile for <span class="texhtml"><i>p</i> = <i>k</i>/<i>q</i></span>), where <span class="texhtml mvar" style="font-style:italic;">μ</span> is the distribution's <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>, and where <span class="texhtml mvar" style="font-style:italic;">σ</span> is the distribution's <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>.<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> In particular, the median <span class="texhtml">(<i>p</i> = <i>k</i>/<i>q</i> = 1/2)</span> is never more than one standard deviation from the mean. </p><p>The above formula can be used to bound the value <span class="texhtml"><i>μ</i> + <i>zσ</i></span> in terms of quantiles. When <span class="texhtml"><i>z</i> ≥ 0</span>, the value that is <a href="/wiki/Standard_score" title="Standard score"><span class="texhtml"><i>z</i></span> standard deviations above the mean</a> has a lower bound <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>z</mi> <mi>σ</mi> <mo>≥</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <mi>z</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e395c7ce8d8c33f377160da7c43c1fbde0a575" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.293ex; height:6.343ex;" alt="{\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.}"></span> For example, the value that is <span class="texhtml"><i>z</i> = 1</span> standard deviation above the mean is always greater than or equal to <span class="texhtml"><i>Q</i>(<i>p</i> = 0.5)</span>, the median, and the value that is <span class="texhtml"><i>z</i> = 2</span> standard deviations above the mean is always greater than or equal to <span class="texhtml"><i>Q</i>(<i>p</i> = 0.8)</span>, the fourth quintile. </p><p>When <span class="texhtml"><i>z</i> ≤ 0</span>, there is instead an upper bound <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>z</mi> <mi>σ</mi> <mo>≤</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <mi>z</mi> <mo>≤</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ff40d3d5c635387b2fedc458f9086315399260" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.293ex; height:6.176ex;" alt="{\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.}"></span> For example, the value <span class="texhtml"><i>μ</i> + <i>zσ</i></span> for <span class="texhtml"><i>z</i> = −3</span> will never exceed <span class="texhtml"><i>Q</i>(<i>p</i> = 0.1)</span>, the first decile. </p> <div class="mw-heading mw-heading2"><h2 id="Estimating_quantiles_from_a_sample">Estimating quantiles from a sample</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=6" title="Edit section: Estimating quantiles from a sample"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size <span class="texhtml mvar" style="font-style:italic;">N</span>. </p><p>Modern statistical packages rely on a number of techniques to <a href="/wiki/Estimation_theory" title="Estimation theory">estimate</a> the quantiles. </p><p><a href="/wiki/Rob_J._Hyndman" title="Rob J. Hyndman">Hyndman</a> and Fan compiled a <a href="/wiki/Taxonomy_(general)" class="mw-redirect" title="Taxonomy (general)">taxonomy</a> of nine algorithms<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> used by various software packages. All methods compute <span class="texhtml mvar" style="font-style:italic;">Q<sub>p</sub></span>, the estimate for the <span class="texhtml mvar" style="font-style:italic;">p</span>-quantile (the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <span class="texhtml mvar" style="font-style:italic;">q</span>-quantile, where <span class="texhtml"><i>p</i> = <i>k</i>/<i>q</i></span>) from a sample of size <span class="texhtml mvar" style="font-style:italic;">N</span> by computing a real valued index <span class="texhtml mvar" style="font-style:italic;">h</span>. When <span class="texhtml mvar" style="font-style:italic;">h</span> is an integer, the <span class="texhtml mvar" style="font-style:italic;">h</span>-th smallest of the <span class="texhtml mvar" style="font-style:italic;">N</span> values, <span class="texhtml mvar" style="font-style:italic;">x<sub>h</sub></span>, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from <span class="texhtml mvar" style="font-style:italic;">h</span>, <span class="texhtml"><i>x</i><sub>⌊<i>h</i>⌋</sub></span>, and <span class="texhtml"><i>x</i><sub>⌈<i>h</i>⌉</sub></span>. (For notation, see <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor and ceiling functions</a>). </p><p>The first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between data points, and differ only in how the index <span class="texhtml mvar" style="font-style:italic;">h</span> used to choose the point along the piecewise linear interpolation curve, is chosen. </p><p><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</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> <a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a>,<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> <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a><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> and <a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a><sup id="cite_ref-Function_Reference:_quantile_–_Octave-Forge_–_SourceForge_6-0" class="reference"><a href="#cite_note-Function_Reference:_quantile_–_Octave-Forge_–_SourceForge-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> programming languages support all nine sample quantile methods. <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a> includes five sample quantile methods, <a href="/wiki/SciPy" title="SciPy">SciPy</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a><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> both include eight, <a href="/wiki/EViews" title="EViews">EViews</a><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> and <a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a><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> include the six piecewise linear functions, <a href="/wiki/Stata" title="Stata">Stata</a><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> includes two, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> includes two, and <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a> includes two. Mathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods. </p><p>The estimate types and interpolation schemes used include: </p> <table class="wikitable"> <tbody><tr> <th>Type </th> <th><span class="texhtml mvar" style="font-style:italic;">h</span> </th> <th><span class="texhtml mvar" style="font-style:italic;">Q<sub>p</sub></span> </th> <th>Notes </th></tr> <tr> <td>R‑1, SAS‑3, Maple‑1 </td> <td><span class="texhtml mvar" style="font-style:italic;">Np</span> </td> <td><span class="texhtml"><i>x</i><sub>⌈<i>h</i>⌉</sub></span> </td> <td>Inverse of <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a>. </td></tr> <tr> <td>R‑2, SAS‑5, Maple‑2, Stata </td> <td><span class="texhtml"><i>Np</i> + 1/2</span> </td> <td><span class="texhtml">(<i>x</i><sub>⌈<i>h</i> – 1/2⌉</sub> + <i>x</i><sub>⌊<i>h</i> + 1/2⌋</sub>) / 2</span> </td> <td>The same as R-1, but with averaging at discontinuities. </td></tr> <tr> <td>R‑3, SAS‑2 </td> <td><span class="texhtml"><i>Np</i> − 1/2</span> </td> <td><span class="texhtml"><i>x</i><sub>⌊<i>h</i>⌉</sub></span> </td> <td>The observation numbered closest to <span class="texhtml mvar" style="font-style:italic;">Np</span>. Here, <span class="texhtml">⌊<i>h</i>⌉</span> indicates rounding to the nearest integer, <a href="/wiki/Rounding#Round_half_to_even" title="Rounding">choosing the even integer in the case of a tie</a>. </td></tr> <tr> <td>R‑4, SAS‑1, SciPy‑(0,1), Julia‑(0,1), Maple‑3 </td> <td><span class="texhtml mvar" style="font-style:italic;">Np</span> </td> <td rowspan="6"><span class="texhtml"><i>x</i><sub>⌊<i>h</i>⌋</sub> + (<i>h</i> − ⌊<i>h</i>⌋) (<i>x</i><sub>⌈<i>h</i>⌉</sub> − <i>x</i><sub>⌊<i>h</i>⌋</sub>)</span> </td> <td>Linear interpolation of the inverse of the empirical distribution function. </td></tr> <tr> <td>R‑5, SciPy‑(1/2,1/2), Julia‑(1/2,1/2), Maple‑4 </td> <td><span class="texhtml"><i>Np</i> + 1/2</span> </td> <td>Piecewise linear function where the knots are the values midway through the steps of the empirical distribution function. </td></tr> <tr> <td>R‑6, Excel, Python, SAS‑4, SciPy‑(0,0), Julia-(0,0), Maple‑5, Stata‑altdef </td> <td><span class="texhtml">(<i>N</i> + 1)<i>p</i></span> </td> <td>Linear interpolation of the expectations for the order statistics for the uniform distribution on [0,1]. That is, it is the linear interpolation between points <span class="texhtml">(<i>p</i><sub><i>h</i></sub>, <i>x</i><sub><i>h</i></sub>)</span>, where <span class="texhtml"><i>p</i><sub><i>h</i></sub> = <i>h</i>/(<i>N</i>+1)</span> is the probability that the last of (<span class="texhtml"><i>N</i>+1</span>) randomly drawn values will not exceed the <span class="texhtml mvar" style="font-style:italic;">h</span>-th smallest of the first <span class="texhtml mvar" style="font-style:italic;">N</span> randomly drawn values. </td></tr> <tr> <td>R‑7, Excel, Python, SciPy‑(1,1), Julia-(1,1), Maple‑6, NumPy </td> <td><span class="texhtml">(<i>N</i> − 1)<i>p</i> + 1</span> </td> <td>Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1]. </td></tr> <tr> <td>R‑8, SciPy‑(1/3,1/3), Julia‑(1/3,1/3), Maple‑7 </td> <td><span class="texhtml">(<i>N</i> + 1/3)<i>p</i> + 1/3</span> </td> <td>Linear interpolation of the approximate medians for order statistics. </td></tr> <tr> <td>R‑9, SciPy‑(3/8,3/8), Julia‑(3/8,3/8), Maple‑8 </td> <td><span class="texhtml">(<i>N</i> + 1/4)<i>p</i> + 3/8</span> </td> <td>The resulting quantile estimates are approximately unbiased for the expected order statistics if <span class="texhtml mvar" style="font-style:italic;">x</span> is normally distributed. </td></tr></tbody></table> <p>Notes: </p> <ul><li>R‑1 through R‑3 are piecewise constant, with discontinuities.</li> <li>R‑4 and following are piecewise linear, without discontinuities, but differ in how <span class="texhtml mvar" style="font-style:italic;">h</span> is computed.</li> <li>R‑3 and R‑4 are not symmetric in that they do not give <span class="texhtml"><i>h</i> = (<i>N</i> + 1) / 2</span> when <span class="texhtml"><i>p</i> = 1/2</span>.</li> <li>Excel's PERCENTILE.EXC and Python's default "exclusive" method are equivalent to R‑6.</li> <li>Excel's PERCENTILE and PERCENTILE.INC and Python's optional "inclusive" method are equivalent to R‑7. This is R's and Julia's default method.</li> <li>Packages differ in how they estimate quantiles beyond the lowest and highest values in the sample, i.e. <span class="texhtml"><i>p</i> < 1/<i>N</i></span> and <span class="texhtml"><i>p</i> > (<i>N</i> − 1)/<i>N</i></span>. Choices include returning an error value, computing linear extrapolation, or assuming a constant value.</li></ul> <p>Of the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as the default.<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> </p><p>The <a href="/wiki/Standard_error_(statistics)" class="mw-redirect" title="Standard error (statistics)">standard error</a> of a quantile estimate can in general be estimated via the <a href="/wiki/Bootstrap_(statistics)" class="mw-redirect" title="Bootstrap (statistics)">bootstrap</a>. The Maritz–Jarrett method can also be used.<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> </p> <div class="mw-heading mw-heading2"><h2 id="The_asymptotic_distribution_of_the_sample_median">The asymptotic distribution of the sample median</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=7" title="Edit section: The asymptotic distribution of the sample median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic. Moreover, the sample median is a more robust estimator than the sample mean. </p><p>One peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution, </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{Sample median m}}\sim {\mathcal {N}}\left(\mu =m,\sigma ^{2}={\frac {1}{Nf(m)^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sample median m</mtext> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>=</mo> <mi>m</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>N</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Sample median m}}\sim {\mathcal {N}}\left(\mu =m,\sigma ^{2}={\frac {1}{Nf(m)^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ae3769b320388e36a03a70d4b92aaa5407a296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.144ex; height:6.343ex;" alt="{\displaystyle {\text{Sample median m}}\sim {\mathcal {N}}\left(\mu =m,\sigma ^{2}={\frac {1}{Nf(m)^{2}}}\right)}"></span></dd></dl> <p>This extends to the other quantiles, </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{Sample quantile p}}\sim {\mathcal {N}}\left(\mu =x_{p},\sigma ^{2}={\frac {p(1-p)}{Nf(x_{p})^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sample quantile p</mtext> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>N</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Sample quantile p}}\sim {\mathcal {N}}\left(\mu =x_{p},\sigma ^{2}={\frac {p(1-p)}{Nf(x_{p})^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f351d758326486370a7987d79e1880fdb48496e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.041ex; height:6.509ex;" alt="{\displaystyle {\text{Sample quantile p}}\sim {\mathcal {N}}\left(\mu =x_{p},\sigma ^{2}={\frac {p(1-p)}{Nf(x_{p})^{2}}}\right)}"></span></dd></dl> <p>where <span class="texhtml"><i>f</i>(<i>x<sub>p</sub></i>)</span> is the value of the distribution density at the <span class="texhtml mvar" style="font-style:italic;">p</span>-th population quantile (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}=F^{-1}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{p}=F^{-1}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c2ba91659ab23d9c5dc75f6a1d85283452c7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.614ex; height:3.343ex;" alt="{\displaystyle x_{p}=F^{-1}(p)}"></span>).<sup id="cite_ref-Stuart1994_15-0" class="reference"><a href="#cite_note-Stuart1994-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>But when the distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in <a rel="nofollow" class="external free" href="https://stats.stackexchange.com/a/86638/28746">https://stats.stackexchange.com/a/86638/28746</a>). </p><p>A solution to this problem is to use an alternative definition of sample quantiles through the concept of the "mid-distribution" function, which is defined as </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 F_{\text{mid}}(x)=P(X\leq x)-{\frac {1}{2}}P(X=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>mid</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{mid}}(x)=P(X\leq x)-{\frac {1}{2}}P(X=x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7281b107ae6902d540b4e8467423cc257ba2a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.469ex; height:5.176ex;" alt="{\displaystyle F_{\text{mid}}(x)=P(X\leq x)-{\frac {1}{2}}P(X=x)}"></span></dd></dl> <p>The definition of sample quantiles through the concept of mid-distribution function can be seen as a generalization that can cover as special cases the continuous distributions. For discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227–243. </p> <div class="mw-heading mw-heading2"><h2 id="Approximate_quantiles_from_a_stream">Approximate quantiles from a stream</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=8" title="Edit section: Approximate quantiles from a stream"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Computing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures. The most popular methods are t-digest<sup id="cite_ref-Dunning2019_16-0" class="reference"><a href="#cite_note-Dunning2019-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and KLL.<sup id="cite_ref-Karnin2016_17-0" class="reference"><a href="#cite_note-Karnin2016-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile. </p><p>Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If the stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by <i>k</i>-means clustering to group similar values. The KLL algorithm uses a more sophisticated "compactor" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to <span class="texhtml mvar" style="font-style:italic;">p</span>. </p><p>Both methods belong to the family of <i>data sketches</i> that are subsets of <a href="/wiki/Streaming_algorithm" title="Streaming algorithm">Streaming Algorithms</a> with useful properties: t-digest or KLL sketches can be combined. Computing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later. </p><p>The algorithms described so far directly approximate the empirical quantiles without any particular assumptions on the data, in essence the data are simply numbers or more generally, a set of items that can be ordered. These algorithms are computer science derived methods. Another class of algorithms exist which assume that the data are realizations of a random process. These are statistics derived methods, sequential nonparametric estimation algorithms in particular. There are a number of such algorithms such as those based on stochastic approximation<sup id="cite_ref-tierney1983_18-0" class="reference"><a href="#cite_note-tierney1983-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-chen2000_19-0" class="reference"><a href="#cite_note-chen2000-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> or Hermite series estimators.<sup id="cite_ref-stephanou2017_20-0" class="reference"><a href="#cite_note-stephanou2017-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>These statistics based algorithms typically have constant update time and space complexity, but have different error bound guarantees compared to computer science type methods and make more assumptions. The statistics based algorithms do present certain advantages however, particularly in the non-stationary streaming setting i.e. time-varying data. The algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed.<sup id="cite_ref-StephanouHermiter2022_21-0" class="reference"><a href="#cite_note-StephanouHermiter2022-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Discussion">Discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=9" title="Edit section: Discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the <i>interval</i> between (in this case) the 80th and the 81st scalar percentile.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This separate meaning of percentile is also used in peer-reviewed scientific research articles.<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> The meaning used can be derived from its context. </p><p>If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, any particular sample of this random variable will have roughly a 63% chance of being less than the mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers. </p><p>Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics. </p><p>Closely related is the subject of <a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">least absolute deviations</a>, a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error. The connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error. <a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a> shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of <a href="/wiki/Robust_regression" title="Robust regression">robust regression</a> are available. </p><p>The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if <span class="texhtml mvar" style="font-style:italic;">m</span> is the median of a random variable <span class="texhtml mvar" style="font-style:italic;">X</span>, then <span class="texhtml">2<sup><i>m</i></sup></span> is the median of <span class="texhtml">2<sup><i>X</i></sup></span>, unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only <a href="/wiki/Ordinal_scale" class="mw-redirect" title="Ordinal scale">ordinal</a> data are available. </p> <div class="mw-heading mw-heading2"><h2 id="Other_quantifications">Other quantifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=10" title="Edit section: Other quantifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Values that divide sorted data into equal subsets other than four have different names. </p> <ul><li>The only 2-quantile is called the <a href="/wiki/Median" title="Median">median</a></li> <li>The 3-quantiles are called <a href="https://en.wiktionary.org/wiki/tertile" class="extiw" title="wikt:tertile">tertiles</a> or <a href="https://en.wiktionary.org/wiki/tercile" class="extiw" title="wikt:tercile">terciles</a> → T</li> <li>The 4-quantiles are called <a href="/wiki/Quartile" title="Quartile">quartiles</a> → Q; the difference between upper and lower quartiles is also called the <a href="/wiki/Interquartile_range" title="Interquartile range">interquartile range</a>, <b>midspread</b> or <b>middle fifty</b> → IQR = <span class="texhtml"><i>Q</i><sub>3</sub> − <i>Q</i><sub>1</sub></span>.</li> <li>The 5-quantiles are called <a href="https://en.wiktionary.org/wiki/quintile" class="extiw" title="wikt:quintile">quintiles</a> or <a href="https://en.wiktionary.org/wiki/pentile" class="extiw" title="wikt:pentile">pentiles</a> → QU</li> <li>The 6-quantiles are called <a href="https://en.wiktionary.org/wiki/sextile" class="extiw" title="wikt:sextile">sextiles</a> → S</li> <li>The 7-quantiles are called <a href="https://en.wiktionary.org/wiki/septile" class="extiw" title="wikt:septile">septiles</a> → SP</li> <li>The 8-quantiles are called <a href="https://en.wiktionary.org/wiki/octile" class="extiw" title="wikt:octile">octiles</a> → O</li> <li>The 10-quantiles are called <a href="/wiki/Decile" title="Decile">deciles</a> → D</li> <li>The 12-quantiles are called duo-deciles or dodeciles → DD</li> <li>The 16-quantiles are called <a href="https://en.wiktionary.org/wiki/hexadecile" class="extiw" title="wikt:hexadecile">hexadeciles</a> → H</li> <li>The 20-quantiles are called <a href="https://en.wiktionary.org/wiki/ventile" class="extiw" title="wikt:ventile">ventiles</a>, <a href="https://en.wiktionary.org/wiki/vigintile" class="extiw" title="wikt:vigintile">vigintiles</a>, or demi-deciles → V</li> <li>The 100-quantiles are called <a href="/wiki/Percentile" title="Percentile">percentiles</a> or centiles → P</li> <li><span class="anchor" id="millile"></span><span class="anchor" id="permille"></span> The 1000-quantiles have been called permilles or milliles, but these are rare and largely obsolete<sup id="cite_ref-walker_24-0" class="reference"><a href="#cite_note-walker-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li></ul> <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=Quantile&action=edit&section=11" 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/Flashsort" title="Flashsort">Flashsort</a> – sort by first bucketing by quantile</li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></li> <li><a href="/wiki/Expectile" title="Expectile">Expectile</a> – related to expectations in a way analogous to that in which quantiles are related to medians</li> <li><a href="/wiki/Quartile" title="Quartile">Quartile</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Quantile_function" title="Quantile function">Quantile function</a></li> <li><a href="/wiki/Quantile_normalization" title="Quantile normalization">Quantile normalization</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile regression</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Summary_statistics" title="Summary statistics">Summary statistics</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a> ("<a href="/wiki/Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a> for the <i>p</i>th quantile"<sup id="cite_ref-vardeman_25-0" class="reference"><a href="#cite_note-vardeman-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>)</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=Quantile&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-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="CITEREFBaguiBhaumik2004" class="citation journal cs1">Bagui, S.; Bhaumik, D. 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Lambert, Diane; Pinheiro, Jose (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1145/347090.347195">"Incremental quantile estimation for massive tracking"</a>. <i>Proceedings of the sixth ACM SIGKDD international conference on Knowledge discovery and data mining</i>. p. 516-522. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F347090.347195">10.1145/347090.347195</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58113-233-6" title="Special:BookSources/1-58113-233-6"><bdi>1-58113-233-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Incremental+quantile+estimation+for+massive+tracking&rft.btitle=Proceedings+of+the+sixth+ACM+SIGKDD+international+conference+on+Knowledge+discovery+and+data+mining&rft.pages=516-522&rft.date=2000&rft_id=info%3Adoi%2F10.1145%2F347090.347195&rft.isbn=1-58113-233-6&rft.aulast=Chen&rft.aufirst=Fei&rft.au=Lambert%2C+Diane&rft.au=Pinheiro%2C+Jose&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%2F347090.347195&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> <li id="cite_note-stephanou2017-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-stephanou2017_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephanouVarugheseMacdonald2017" class="citation journal cs1">Stephanou, Michael; Varughese, Melvin; Macdonald, Iain (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.1214/17-EJS1245">"Sequential quantiles via Hermite series density estimation"</a>. <i>Electronic Journal of Statistics</i>. <b>11</b> (1): 570-607. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1507.05073">1507.05073</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/10.1214%2F17-EJS1245">10.1214/17-EJS1245</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+of+Statistics&rft.atitle=Sequential+quantiles+via+Hermite+series+density+estimation&rft.volume=11&rft.issue=1&rft.pages=570-607&rft.date=2017&rft_id=info%3Aarxiv%2F1507.05073&rft_id=info%3Adoi%2F10.1214%2F17-EJS1245&rft.aulast=Stephanou&rft.aufirst=Michael&rft.au=Varughese%2C+Melvin&rft.au=Macdonald%2C+Iain&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%2F17-EJS1245&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> <li id="cite_note-StephanouHermiter2022-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-StephanouHermiter2022_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephanou,_M._and_Varughese,_M2023" class="citation journal cs1">Stephanou, M. and Varughese, M (2023). "Hermiter: R package for sequential nonparametric estimation". <i>Computational Statistics</i>. <b>39</b> (3): 1127–1163. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2111.14091">2111.14091</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/10.1007%2Fs00180-023-01382-0">10.1007/s00180-023-01382-0</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:244715035">244715035</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computational+Statistics&rft.atitle=Hermiter%3A+R+package+for+sequential+nonparametric+estimation&rft.volume=39&rft.issue=3&rft.pages=1127-1163&rft.date=2023&rft_id=info%3Aarxiv%2F2111.14091&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A244715035%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00180-023-01382-0&rft.au=Stephanou%2C+M.+and+Varughese%2C+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://www.oxfordreference.com/view/10.1093/oi/authority.20110803100316401">"percentile"</a>. <i>Oxford Reference</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Oxford+Reference&rft.atitle=percentile&rft_id=https%3A%2F%2Fwww.oxfordreference.com%2Fview%2F10.1093%2Foi%2Fauthority.20110803100316401&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrugerDunning1999" class="citation journal cs1">Kruger, J.; Dunning, D. (December 1999). <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10626367/">"Unskilled and unaware of it: how difficulties in recognizing one's own incompetence lead to inflated self-assessments"</a>. <i>Journal of Personality and Social Psychology</i>. <b>77</b> (6): 1121–1134. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0022-3514.77.6.1121">10.1037/0022-3514.77.6.1121</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/0022-3514">0022-3514</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10626367">10626367</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:2109278">2109278</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Personality+and+Social+Psychology&rft.atitle=Unskilled+and+unaware+of+it%3A+how+difficulties+in+recognizing+one%27s+own+incompetence+lead+to+inflated+self-assessments&rft.volume=77&rft.issue=6&rft.pages=1121-1134&rft.date=1999-12&rft.issn=0022-3514&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2109278%23id-name%3DS2CID&rft_id=info%3Apmid%2F10626367&rft_id=info%3Adoi%2F10.1037%2F0022-3514.77.6.1121&rft.aulast=Kruger&rft.aufirst=J.&rft.au=Dunning%2C+D.&rft_id=https%3A%2F%2Fpubmed.ncbi.nlm.nih.gov%2F10626367%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> <li id="cite_note-walker-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-walker_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalkerLev1969" class="citation book cs1">Walker, Helen Mary; Lev, Joseph (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ogYnAQAAIAAJ&q=permille"><i>Elementary Statistical Methods</i></a>. Holt, Rinehart and Winston. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-081130-2" title="Special:BookSources/978-0-03-081130-2"><bdi>978-0-03-081130-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Statistical+Methods&rft.pub=Holt%2C+Rinehart+and+Winston&rft.date=1969&rft.isbn=978-0-03-081130-2&rft.aulast=Walker&rft.aufirst=Helen+Mary&rft.au=Lev%2C+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DogYnAQAAIAAJ%26q%3Dpermille&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> <li id="cite_note-vardeman-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-vardeman_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_B._Vardeman1992" class="citation journal cs1">Stephen B. Vardeman (1992). "What about the Other Intervals?". <i>The American Statistician</i>. <b>46</b> (3): 193–197. <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%2F2685212">10.2307/2685212</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/2685212">2685212</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=What+about+the+Other+Intervals%3F&rft.volume=46&rft.issue=3&rft.pages=193-197&rft.date=1992&rft_id=info%3Adoi%2F10.2307%2F2685212&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2685212%23id-name%3DJSTOR&rft.au=Stephen+B.+Vardeman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerfling1980" class="citation book cs1">Serfling, R. J. (1980). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/approximationthe0000serf"><i>Approximation Theorems of Mathematical Statistics</i></a></span>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-02403-1" title="Special:BookSources/0-471-02403-1"><bdi>0-471-02403-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Approximation+Theorems+of+Mathematical+Statistics&rft.pub=John+Wiley+%26+Sons&rft.date=1980&rft.isbn=0-471-02403-1&rft.aulast=Serfling&rft.aufirst=R.+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fapproximationthe0000serf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantile" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantile&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Quantiles" class="extiw" title="commons:Category:Quantiles">Quantiles</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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