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class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Empirical_testing" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Empirical_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Empirical testing</span> </div> </a> <ul id="toc-Empirical_testing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_explanations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statistical_explanations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Statistical explanations</span> </div> </a> <ul id="toc-Statistical_explanations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_laws" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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sizes</span> </div> </a> <ul id="toc-City_sizes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Word_frequencies_in_natural_languages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Word_frequencies_in_natural_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Word frequencies in natural languages</span> </div> </a> <ul id="toc-Word_frequencies_in_natural_languages-sublist" class="vector-toc-list"> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </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">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">9</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">10</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">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Zipf's law</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 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Available in 35 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-35" 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">35 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%B2%D9%8A%D9%81" title="قانون زيف – Arabic" lang="ar" hreflang="ar" data-title="قانون زيف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Zipf_qanunu" title="Zipf qanunu – Azerbaijani" lang="az" hreflang="az" data-title="Zipf qanunu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%A6%D1%96%D0%BF%D1%84%D0%B0" title="Закон Ціпфа – Belarusian" lang="be" hreflang="be" data-title="Закон Ціпфа" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Zipfov_zakon" title="Zipfov zakon – Bosnian" lang="bs" hreflang="bs" data-title="Zipfov zakon" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_Zipf" title="Llei de Zipf – Catalan" lang="ca" hreflang="ca" data-title="Llei de Zipf" 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/Zipf%C5%AFv_z%C3%A1kon" title="Zipfův zákon – Czech" lang="cs" hreflang="cs" data-title="Zipfův zákon" 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/Zipfs_lov" title="Zipfs lov – Danish" lang="da" hreflang="da" data-title="Zipfs lov" 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/Zipfsches_Gesetz" title="Zipfsches Gesetz – German" lang="de" hreflang="de" data-title="Zipfsches Gesetz" 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/Ley_de_Zipf" title="Ley de Zipf – Spanish" lang="es" hreflang="es" data-title="Ley de Zipf" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Le%C4%9Do_de_Zipf" title="Leĝo de Zipf – Esperanto" lang="eo" hreflang="eo" data-title="Leĝo de Zipf" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zipfen_legea" title="Zipfen legea – Basque" lang="eu" hreflang="eu" data-title="Zipfen legea" 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/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%B2%DB%8C%D9%81" title="قانون زیف – Persian" lang="fa" hreflang="fa" data-title="قانون زیف" 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/Loi_de_Zipf" title="Loi de Zipf – French" lang="fr" hreflang="fr" data-title="Loi de Zipf" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_Zipf" title="Dlí Zipf – Irish" lang="ga" hreflang="ga" data-title="Dlí Zipf" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Lei_de_Zipf" title="Lei de Zipf – Galician" lang="gl" hreflang="gl" data-title="Lei de Zipf" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%80%ED%94%84%EC%9D%98_%EB%B2%95%EC%B9%99" title="지프의 법칙 – Korean" lang="ko" hreflang="ko" data-title="지프의 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_di_Zipf" title="Legge di Zipf – Italian" lang="it" hreflang="it" data-title="Legge di Zipf" 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%95%D7%A7_%D7%96%D7%99%D7%A3" title="חוק זיף – Hebrew" lang="he" hreflang="he" data-title="חוק זיף" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ltg mw-list-item"><a href="https://ltg.wikipedia.org/wiki/Zipfa_lykums" title="Zipfa lykums – Latgalian" lang="ltg" hreflang="ltg" data-title="Zipfa lykums" data-language-autonym="Latgaļu" data-language-local-name="Latgalian" class="interlanguage-link-target"><span>Latgaļu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Zipf-eloszl%C3%A1s" title="Zipf-eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Zipf-eloszlás" 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/Wet_van_Zipf" title="Wet van Zipf – Dutch" lang="nl" hreflang="nl" data-title="Wet van Zipf" 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/%E3%82%B8%E3%83%83%E3%83%97%E3%81%AE%E6%B3%95%E5%89%87" title="ジップの法則 – Japanese" lang="ja" hreflang="ja" data-title="ジップの法則" 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-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B2%D9%BE%D9%81_%D8%AF%D8%A7_%D9%82%D9%86%D9%88%D9%86" title="زپف دا قنون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="زپف دا قنون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawo_Zipfa" title="Prawo Zipfa – Polish" lang="pl" hreflang="pl" data-title="Prawo Zipfa" 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/Lei_de_Zipf" title="Lei de Zipf – Portuguese" lang="pt" hreflang="pt" data-title="Lei de Zipf" 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%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%A6%D0%B8%D0%BF%D1%84%D0%B0" title="Закон Ципфа – Russian" lang="ru" hreflang="ru" data-title="Закон Ципфа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Zipf%27s_law" title="Zipf's law – Simple English" lang="en-simple" hreflang="en-simple" data-title="Zipf's law" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zipfov_zakon" title="Zipfov zakon – Slovenian" lang="sl" hreflang="sl" data-title="Zipfov zakon" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Zipfin_laki" title="Zipfin laki – Finnish" lang="fi" hreflang="fi" data-title="Zipfin laki" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Zipfs_lag" title="Zipfs lag – Swedish" lang="sv" hreflang="sv" data-title="Zipfs lag" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%83%E0%AE%AA%E0%AF%81%E0%AE%B5%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="சிஃபுவின் விதி – Tamil" lang="ta" hreflang="ta" data-title="சிஃபுவின் விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Zipf_yasas%C4%B1" title="Zipf yasası – Turkish" lang="tr" hreflang="tr" data-title="Zipf yasası" 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%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%A6%D0%B8%D0%BF%D1%84%D0%B0" title="Закон Ципфа – Ukrainian" lang="uk" hreflang="uk" data-title="Закон Ципфа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B0%D9%81_%DA%A9%D8%A7_%D9%82%D8%A7%D9%86%D9%88%D9%86" title="ذف کا قانون – Urdu" lang="ur" hreflang="ur" data-title="ذف کا قانون" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%BD%8A%E5%A4%AB%E5%AE%9A%E5%BE%8B" title="齊夫定律 – Chinese" lang="zh" hreflang="zh" data-title="齊夫定律" 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/Q205472#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs 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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">For the linguistics law on word length, see <a href="/wiki/Zipf%27s_law_of_abbreviation" class="mw-redirect" title="Zipf's law of abbreviation">Zipf's law of abbreviation</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zipf%27s_law_on_War_and_Peace.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Zipf%27s_law_on_War_and_Peace.png/220px-Zipf%27s_law_on_War_and_Peace.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Zipf%27s_law_on_War_and_Peace.png/330px-Zipf%27s_law_on_War_and_Peace.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Zipf%27s_law_on_War_and_Peace.png/440px-Zipf%27s_law_on_War_and_Peace.png 2x" data-file-width="2011" data-file-height="2411" /></a><figcaption>Zipf's Law on <a href="/wiki/War_and_Peace" title="War and Peace">War and Peace</a>.<sup id="cite_ref-piant2014_1-0" class="reference"><a href="#cite_note-piant2014-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The lower plot shows the remainder when the Zipf law is divided away. It shows that there remains significant pattern not fitted by Zipf law.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg/220px-Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg.png" decoding="async" width="220" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg/330px-Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg/440px-Zipf-engl-0_English_-_Culpeper_herbal_and_War_of_the_Worlds.svg.png 2x" data-file-width="512" data-file-height="504" /></a><figcaption>A plot of the frequency of each word as a function of its frequency rank for two English language texts: <a href="/wiki/Nicholas_Culpeper" title="Nicholas Culpeper">Culpeper</a>'s <a href="/wiki/List_of_plants_in_The_English_Physitian" title="List of plants in The English Physitian"><i>Complete Herbal</i></a> (1652) and <a href="/wiki/H._G._Wells" title="H. G. Wells">H. G. Wells</a>'s <i><a href="/wiki/The_War_of_the_Worlds" title="The War of the Worlds">The War of the Worlds</a></i> (1898) in a <a href="/wiki/Log-log" class="mw-redirect" title="Log-log">log-log</a> scale. The dotted line is the ideal law <span class="texhtml mvar" style="font-style:italic;">y</span> <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)"><span class="texhtml">∝</span></a> <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> <i>x</i> </span></span>⁠</span></span> </figcaption></figure> <p><b>Zipf's law</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'z' in 'zoom'">z</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="'f' in 'find'">f</span></span>/</a></span></span>, <style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">German:</span> <span class="IPA nowrap" lang="de-Latn-fonipa"><a href="/wiki/Help:IPA/Standard_German" title="Help:IPA/Standard German">[t͡sɪpf]</a></span>) is an <a href="/wiki/Empirical_law" class="mw-redirect" title="Empirical law">empirical law</a> stating that when a list of measured values is sorted in decreasing order, the value of the <span class="texhtml mvar" style="font-style:italic;"> n </span>th entry is often approximately <a href="/wiki/Inversely_proportional" class="mw-redirect" title="Inversely proportional">inversely proportional</a> to <span class="texhtml mvar" style="font-style:italic;"> n </span>. </p><p>The best known instance of Zipf's law applies to the <a href="/wiki/Frequency_table" class="mw-redirect" title="Frequency table">frequency table</a> of words in a text or <a href="/wiki/Text_corpus" title="Text corpus">corpus</a> of <a href="/wiki/Natural_language" title="Natural language">natural language</a>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">w</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">d</mi> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">q</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">y</mi> </mrow> </mrow> <mtext> </mtext> <mo>∝</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">w</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">d</mi> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">k</mi> </mrow> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6b59f31fc385d1c961795d53495ac1cdd1ff44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.351ex; height:5.343ex;" alt="{\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.}"></span> It is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the <a href="/wiki/Brown_Corpus" title="Brown Corpus">Brown Corpus</a> of American English text, the word "<i><a href="/wiki/English_articles#Definite_article" title="English articles">the</a></i>" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's law, the second-place word "<i>of</i>" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "<i>and</i>" (28,852).<sup id="cite_ref-fagan2010_2-0" class="reference"><a href="#cite_note-fagan2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is often used in the following form, called <a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf-Mandelbrot law</a>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">q</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">y</mi> </mrow> </mrow> <mtext> </mtext> <mo>∝</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">k</mi> </mrow> </mrow> <mo>+</mo> <mi>b</mi> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2633a123b8b5a4f273ce5e05c77bac7568578a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.959ex; height:6.009ex;" alt="{\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124de742ae987fe73be9ca9d3d4ba8586e28b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }"></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 \ b\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>b</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ b\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978008481f65e59642b3c5cf1ad23b84284fff31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.159ex; height:2.176ex;" alt="{\displaystyle \ b\ }"></span> are fitted parameters, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\approx 1\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mo>≈</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\approx 1\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001051f805ad55c34a47481eb435338ca4f86884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.299ex; height:2.509ex;" alt="{\displaystyle \ a\approx 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 \ b\approx 2.7~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>b</mi> <mo>≈</mo> <mn>2.7</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ b\approx 2.7~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a35ece5593a9acf4237f758aca05cbca08fddcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.876ex; height:2.176ex;" alt="{\displaystyle \ b\approx 2.7~.}"></span><sup id="cite_ref-piant2014_1-1" class="reference"><a href="#cite_note-piant2014-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>This law is named after the American <a href="/wiki/Linguistics" title="Linguistics">linguist</a> <a href="/wiki/George_Kingsley_Zipf" title="George Kingsley Zipf">George Kingsley Zipf</a>,<sup id="cite_ref-Powers1998_3-0" class="reference"><a href="#cite_note-Powers1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-zipf1935_4-0" class="reference"><a href="#cite_note-zipf1935-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-zipf1949_5-0" class="reference"><a href="#cite_note-zipf1949-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and is still an important concept in <a href="/wiki/Quantitative_linguistics" title="Quantitative linguistics">quantitative linguistics</a>. It has been found to apply to many other types of data studied in the <a href="/wiki/Physical_science" class="mw-redirect" title="Physical science">physical</a> and <a href="/wiki/Social_science" title="Social science">social</a> sciences. </p><p>In <a href="/wiki/Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>, the concept has been formalized as the <b>Zipfian distribution</b>: A family of related discrete <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> whose <a href="/wiki/Rank-frequency_distribution" class="mw-redirect" title="Rank-frequency distribution">rank-frequency distribution</a> is an inverse <a href="/wiki/Power_law" title="Power law">power law</a> relation. They are related to <a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> and the <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>. </p><p>Some sets of time-dependent empirical data deviate somewhat from Zipf's law. Such empirical distributions are said to be <b>quasi-Zipfian</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1913, the German physicist <a href="/wiki/Felix_Auerbach" title="Felix Auerbach">Felix Auerbach</a> observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.<sup id="cite_ref-Auerbach1913_6-0" class="reference"><a href="#cite_note-Auerbach1913-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Zipf's law had been discovered before Zipf,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> first by the French stenographer <a href="/wiki/Jean-Baptiste_Estoup" title="Jean-Baptiste Estoup">Jean-Baptiste Estoup</a> in 1916,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mann1999_8-1" class="reference"><a href="#cite_note-mann1999-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and also by <a href="/wiki/Godfrey_Dewey" title="Godfrey Dewey">G. Dewey</a> in 1923,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and by <a href="/wiki/Edward_Condon" title="Edward Condon">E. Condon</a> in 1928.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932,<sup id="cite_ref-zipf1935_4-1" class="reference"><a href="#cite_note-zipf1935-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> but he never claimed to have originated it. In fact, Zipf did not like mathematics. In his 1932 publication,<sup id="cite_ref-zipf1932_12-0" class="reference"><a href="#cite_note-zipf1932-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> the author speaks with disdain about mathematical involvement in linguistics, <i>a.o. ibidem</i>, p. 21: </p> <dl><dd><i>... let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\sqrt {-i\;}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>i</mi> <mspace width="thickmathspace" /> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\sqrt {-i\;}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ff7b200beb0c5b75da0c85ca80ce5caa284382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:3.009ex;" alt="{\displaystyle \ {\sqrt {-i\;}}~.}"></span></dd></dl> <p>The only mathematical expression Zipf used looks like <span class="nowrap"><span class="texhtml"><i>a</i>.<i>b</i><sup>2</sup> = </span>   constant,</span> which he "borrowed" from <a href="/wiki/Alfred_J._Lotka" title="Alfred J. Lotka">Alfred J. Lotka</a>'s 1926 publication.<sup id="cite_ref-king1942_13-0" class="reference"><a href="#cite_note-king1942-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The same relationship was found to occur in many other contexts, and for other variables besides frequency.<sup id="cite_ref-piant2014_1-2" class="reference"><a href="#cite_note-piant2014-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank.<sup id="cite_ref-axte2001_14-0" class="reference"><a href="#cite_note-axte2001-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The same relation is found for personal incomes (where it is called <a href="/wiki/Pareto_distribution#Occurrence_and_applications" title="Pareto distribution">Pareto principle</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>), number of people watching the same TV channel,<sup id="cite_ref-erik2014_16-0" class="reference"><a href="#cite_note-erik2014-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Musical_note" title="Musical note">notes</a> in music,<sup id="cite_ref-zann2004_17-0" class="reference"><a href="#cite_note-zann2004-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> cells <a href="/wiki/Transcriptomes" class="mw-redirect" title="Transcriptomes">transcriptomes</a>,<sup id="cite_ref-lazz2023_18-0" class="reference"><a href="#cite_note-lazz2023-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-chen2011_19-0" class="reference"><a href="#cite_note-chen2011-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and more. </p><p>In 1992 bioinformatician <a href="/wiki/Wentian_Li" title="Wentian Li">Wentian Li</a> published a short paper<sup id="cite_ref-liwe1992_20-0" class="reference"><a href="#cite_note-liwe1992-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> showing that Zipf's law emerges even in randomly generated texts. It included proof that the power law form of Zipf's law was a byproduct of ordering words by rank. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1247679731">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Zipf's law</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability mass function</div><span typeof="mw:File"><a href="/wiki/File:Zipf_distribution_PMF.png" class="mw-file-description" title="Plot of the Zipf PMF for N = 10 ."><img alt="Plot of the Zipf PMF for N = 10 ." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Zipf_distribution_PMF.png/325px-Zipf_distribution_PMF.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Zipf_distribution_PMF.png/488px-Zipf_distribution_PMF.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Zipf_distribution_PMF.png/650px-Zipf_distribution_PMF.png 2x" data-file-width="1300" data-file-height="975" /></a></span><br /><small>Zipf PMF for <span class="nowrap"><span class="texhtml"><i>N</i> = 10</span> </span> on a log–log scale. The horizontal axis is the index <span class="texhtml mvar" style="font-style:italic;">k</span> . (The function is only defined at integer values of <span class="texhtml mvar" style="font-style:italic;">k</span> . The connecting lines are only visual guides; they do not indicate continuity.)</small></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><span typeof="mw:File"><a href="/wiki/File:Zipf_distribution_CMF.png" class="mw-file-description" title="Plot of the Zipf CDF for N=10"><img alt="Plot of the Zipf CDF for N=10" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Zipf_distribution_CMF.png/325px-Zipf_distribution_CMF.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Zipf_distribution_CMF.png/488px-Zipf_distribution_CMF.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Zipf_distribution_CMF.png/650px-Zipf_distribution_CMF.png 2x" data-file-width="1300" data-file-height="975" /></a></span><br /><small>Zipf CDF for <span class="nowrap"><span class="texhtml"><i>N</i> = 10</span> .</span> The horizontal axis is the index <span class="texhtml mvar" style="font-style:italic;">k</span> . (The function is only defined at integer values of <span class="texhtml mvar" style="font-style:italic;">k</span> . The connecting lines do not indicate continuity.)</small></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\geq 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\geq 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df1e5329d3e605c6489ccc6bef559aa1b5919e25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.738ex; height:2.343ex;" alt="{\displaystyle s\geq 0\,}"></span> (<a href="/wiki/Real_number" title="Real number">real</a>)<br /><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\in \{1,2,3\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in \{1,2,3\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9695f2f531e8f62752cab72d2bf86196812eaa36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.895ex; height:2.843ex;" alt="{\displaystyle N\in \{1,2,3\ldots \}}"></span> (<a href="/wiki/Integer" title="Integer">integer</a>)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <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\in \{1,2,\ldots ,N\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \{1,2,\ldots ,N\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2901de737533733105354dba92ef12fa5dfd444d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.977ex; height:2.843ex;" alt="{\displaystyle k\in \{1,2,\ldots ,N\}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1/k^{s}}{H_{N,s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1/k^{s}}{H_{N,s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0beacf1c1fdd669bae512343234fc9c26951db92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.687ex; height:6.343ex;" alt="{\displaystyle {\frac {1/k^{s}}{H_{N,s}}}}"></span> where <i>H<sub>N,s</sub></i> is the <i>N</i>th generalized <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2e24a4ad0920ceeda7e797e05e33f6034dd75a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.687ex; height:6.176ex;" alt="{\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de75685e9d5b89b2fddf89a37a83ae40511e795e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.788ex; height:6.176ex;" alt="{\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f64ead5fcf58e79457f4e67cc56e48d6286d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.416ex; height:7.843ex;" alt="{\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17af26bab5f739c118963b1133c07e16f21e7d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.053ex; height:7.343ex;" alt="{\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa2db46adb6afa6c6488016a11cc754214a44ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.548ex; height:7.343ex;" alt="{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f754ce96a12606a847ec97180f1f57c7a387dfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.116ex; height:7.343ex;" alt="{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}"></span></td></tr></tbody></table> <p>Formally, the Zipf distribution on <span class="texhtml mvar" style="font-style:italic;">N</span> elements assigns to the element of rank <span class="texhtml mvar" style="font-style:italic;">k</span> (counting from 1) the probability </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(k;N)~=~{\begin{cases}{\frac {1}{\ H_{N}}}\ {\frac {1}{\ k\ }}\ ,&\ {\mbox{ if }}\ 1\leq k\leq N~,\\{}\\~~0~~,&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> if </mtext> </mstyle> </mrow> <mtext> </mtext> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>N</mi> <mtext> </mtext> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mo>,</mo> </mtd> <mtd> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> if </mtext> </mstyle> </mrow> <mtext> </mtext> <mi>k</mi> <mo><</mo> <mn>1</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> or </mtext> </mstyle> </mrow> <mtext> </mtext> <mi>N</mi> <mo><</mo> <mi>k</mi> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(k;N)~=~{\begin{cases}{\frac {1}{\ H_{N}}}\ {\frac {1}{\ k\ }}\ ,&\ {\mbox{ if }}\ 1\leq k\leq N~,\\{}\\~~0~~,&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845725c0910515603912601e376ac2cdcaebcbd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:47.321ex; height:9.509ex;" alt="{\displaystyle \ f(k;N)~=~{\begin{cases}{\frac {1}{\ H_{N}}}\ {\frac {1}{\ k\ }}\ ,&\ {\mbox{ if }}\ 1\leq k\leq N~,\\{}\\~~0~~,&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">H</span><sub><span class="texhtml mvar" style="font-style:italic;">N</span></sub> is a normalization constant: The <span class="texhtml mvar" style="font-style:italic;">N</span>th <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac {\ 1\ }{k}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mi>k</mi> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac {\ 1\ }{k}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11ab23f3539ae896eef36cbcd41983a04a562ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.851ex; height:7.343ex;" alt="{\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac {\ 1\ }{k}}~.}"></span></dd></dl> <p>The distribution is sometimes generalized to an inverse power law with exponent <span class="texhtml mvar" style="font-style:italic;">s</span> instead of <span class="texhtml"> 1 .</span><sup id="cite_ref-adam2000_21-0" class="reference"><a href="#cite_note-adam2000-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Namely, </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(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ee28604422f8ed810b02983309d37e8cb58519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.745ex; height:5.843ex;" alt="{\displaystyle f(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">H</span><sub><span class="texhtml mvar" style="font-style:italic;">N</span>,<span class="texhtml mvar" style="font-style:italic;">s</span></sub> is a <a href="/wiki/Generalized_harmonic_number" class="mw-redirect" title="Generalized harmonic number">generalized harmonic number</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16bb5f8db6a17f9dd58ef5f361045bf9f1db39c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.97ex; height:7.343ex;" alt="{\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}~.}"></span></dd></dl> <p>The generalized Zipf distribution can be extended to infinitely many items (<span class="texhtml mvar" style="font-style:italic;">N</span> = ∞) only if the exponent <span class="texhtml mvar" style="font-style:italic;">s</span> exceeds <span class="texhtml"> 1 .</span> In that case, the normalization constant <span class="texhtml mvar" style="font-style:italic;">H</span><sub><span class="texhtml mvar" style="font-style:italic;">N</span>,<span class="texhtml mvar" style="font-style:italic;">s</span></sub> becomes <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann's zeta function</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01afe5724ffcad6adcd7d0cad8bed2228157b5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.536ex; height:6.843ex;" alt="{\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty ~.}"></span></dd></dl> <p>The infinite item case is characterized by the <a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta distribution</a> and is called <a href="/wiki/Lotka%27s_law" title="Lotka's law">Lotka's law</a>. If the exponent <span class="texhtml mvar" style="font-style:italic;">s</span> is <span class="texhtml"> 1 </span> or less, the normalization constant <span class="texhtml mvar" style="font-style:italic;">H</span><sub><span class="texhtml mvar" style="font-style:italic;">N</span>,<span class="texhtml mvar" style="font-style:italic;">s</span></sub> diverges as <span class="texhtml mvar" style="font-style:italic;">N</span> tends to infinity. </p> <div class="mw-heading mw-heading2"><h2 id="Empirical_testing">Empirical testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=3" title="Edit section: Empirical testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Empirically, a data set can be tested to see whether Zipf's law applies by checking the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a> of an empirical distribution to the hypothesized power law distribution with a <a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a>, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.<sup id="cite_ref-Clausetetal2009_22-0" class="reference"><a href="#cite_note-Clausetetal2009-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Zipf's law can be visuallized by <a href="/wiki/Graph_of_a_function" title="Graph of a function">plotting</a> the item frequency data on a <a href="/wiki/Log-log" class="mw-redirect" title="Log-log">log-log</a> graph, with the axes being the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent <span class="texhtml mvar" style="font-style:italic;">s</span> to the extent that the plot approximates a <a href="/wiki/Linear_equation" title="Linear equation">linear</a> (more precisely, <a href="/wiki/Affine_function" class="mw-redirect" title="Affine function">affine</a>) function with slope <span class="texhtml mvar" style="font-style:italic;">−s</span>. For exponent <span class="nowrap"><span class="texhtml"> <i>s</i> = 1 </span> ,</span> one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope <span class="texhtml"> 1 .</span><sup id="cite_ref-Powers1998_3-1" class="reference"><a href="#cite_note-Powers1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_explanations">Statistical explanations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=4" title="Edit section: Statistical explanations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Zipf's Law holds for most natural languages, and even some non-natural ones like <a href="/wiki/Esperanto" title="Esperanto">Esperanto</a><sup id="cite_ref-mana2006_23-0" class="reference"><a href="#cite_note-mana2006-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Toki_Pona" title="Toki Pona">Toki Pona</a>,<sup id="cite_ref-skot2020_24-0" class="reference"><a href="#cite_note-skot2020-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> the reason is still not well understood.<sup id="cite_ref-bril1959_25-0" class="reference"><a href="#cite_note-bril1959-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Recent reviews of generative processes for Zipf's law include <a href="/wiki/Michael_Mitzenmacher" title="Michael Mitzenmacher">Mitzenmacher</a>, "A Brief History of Generative Models for Power Law and Lognormal Distributions",<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and Simkin, "Re-inventing Willis".<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>However, it may be partly explained by statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" with different lengths follow the macro-trend of Zipf's law (the more probable words are the shortest and have equal probability).<sup id="cite_ref-liwe1992_20-1" class="reference"><a href="#cite_note-liwe1992-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In 1959, <a href="/wiki/Vitold_Belevitch" title="Vitold Belevitch">Vitold Belevitch</a> observed that if any of a large class of well-behaved <a href="/wiki/Statistical_distribution" class="mw-redirect" title="Statistical distribution">statistical distributions</a> (not only the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>) is expressed in terms of rank and expanded into a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in <a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Mandelbrot's law</a>.<sup id="cite_ref-bele1959_28-0" class="reference"><a href="#cite_note-bele1959-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-neum2011_29-0" class="reference"><a href="#cite_note-neum2011-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Principle_of_least_effort" title="Principle of least effort">principle of least effort</a> is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language wants to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.<sup id="cite_ref-zipf1949_5-2" class="reference"><a href="#cite_note-zipf1949-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ferr2003_30-0" class="reference"><a href="#cite_note-ferr2003-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>A minimal explanation assumes that words are generated by <a href="/wiki/Infinite_monkey_theorem" title="Infinite monkey theorem">monkeys typing randomly</a>. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>Another possible cause for the Zipf distribution is a <a href="/wiki/Preferential_attachment" title="Preferential attachment">preferential attachment</a> process, in which the value <span class="texhtml mvar" style="font-style:italic;">x</span> of an item tends to grow at a rate proportional to <span class="texhtml mvar" style="font-style:italic;">x</span> (intuitively, "<a href="/wiki/Matthew_effect" title="Matthew effect">the rich get richer</a>" or "success breeds success"). Such a growth process results in the <a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon distribution</a>, which has been shown to fit word frequency versus rank in language<sup id="cite_ref-linr2014_32-0" class="reference"><a href="#cite_note-linr2014-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and population versus city rank<sup id="cite_ref-vita2015_33-0" class="reference"><a href="#cite_note-vita2015-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon. </p><p>A similar explanation is based on <a href="/w/index.php?title=Atlas_model&action=edit&redlink=1" class="new" title="Atlas model (page does not exist)">atlas models</a>, systems of exchangeable positive-valued <a href="/wiki/Diffusion_process" title="Diffusion process">diffusion processes</a> with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.<sup id="cite_ref-fern2020_34-0" class="reference"><a href="#cite_note-fern2020-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-taot2012_35-0" class="reference"><a href="#cite_note-taot2012-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_laws">Related laws</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=5" title="Edit section: Related laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A generalization of Zipf's law is the <a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot law</a>, proposed by <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a>, whose frequencies are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;N,q,s)={\frac {1}{\ C\ }}\ {\frac {1}{\ \left(k+q\right)^{s}}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>N</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;N,q,s)={\frac {1}{\ C\ }}\ {\frac {1}{\ \left(k+q\right)^{s}}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfb5a80b9fdfd1f60673244fa4786ba423faee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.645ex; height:6.009ex;" alt="{\displaystyle f(k;N,q,s)={\frac {1}{\ C\ }}\ {\frac {1}{\ \left(k+q\right)^{s}}}~.}"></span><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="denominator missing from equation (September 2023)">clarification needed</span></a></i>]</sup></dd></dl> <p>The constant <span class="texhtml mvar" style="font-style:italic;">C</span> is the <a href="/wiki/Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a> evaluated at <span class="texhtml mvar" style="font-style:italic;">s</span>. </p><p>Zipfian distributions can be obtained from <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distributions</a> by an exchange of variables.<sup id="cite_ref-adam2000_21-1" class="reference"><a href="#cite_note-adam2000-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The Zipf distribution is sometimes called the <b>discrete Pareto distribution</b><sup id="cite_ref-john1992_36-0" class="reference"><a href="#cite_note-john1992-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> because it is analogous to the continuous <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a> in the same way that the <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">discrete uniform distribution</a> is analogous to the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">continuous uniform distribution</a>. </p><p>The tail frequencies of the <a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon distribution</a> are approximately </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(k;\rho )\approx {\frac {\ [{\mathsf {constant}}]\ }{k^{(\rho +1)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mtext> </mtext> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;\rho )\approx {\frac {\ [{\mathsf {constant}}]\ }{k^{(\rho +1)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1180bbcf7154a86bae8797758c7e7261afa7c447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.212ex; height:6.009ex;" alt="{\displaystyle f(k;\rho )\approx {\frac {\ [{\mathsf {constant}}]\ }{k^{(\rho +1)}}}}"></span></dd></dl> <p>for any choice of <span class="nowrap"> <span class="texhtml"> <i>ρ</i> > 0</span> .</span> </p><p>In the <a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">parabolic fractal distribution</a>, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.<sup id="cite_ref-Galien_37-0" class="reference"><a href="#cite_note-Galien-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.<sup id="cite_ref-efte2006_38-0" class="reference"><a href="#cite_note-efte2006-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>It has been argued that <a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> is a special bounded case of Zipf's law,<sup id="cite_ref-Galien_37-1" class="reference"><a href="#cite_note-Galien-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.<sup id="cite_ref-piet2001_39-0" class="reference"><a href="#cite_note-piet2001-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with <span class="nowrap"> <span class="texhtml">s = 1</span> ,</span> satisfy Benford's law. </p> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> </th> <th>Benford's law: <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(n)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26901bdfdbef7ae61cc7894029f331c665a68a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.403ex; height:2.843ex;" alt="{\displaystyle P(n)=}"></span><br /><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 \log _{10}(n+1)-\log _{10}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(n+1)-\log _{10}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a7ef47fd3dc7b403c5259569f638c7b042da69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.947ex; height:2.843ex;" alt="{\displaystyle \log _{10}(n+1)-\log _{10}(n)}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f5927dfd767857ef11954a0ca3fedc628f17fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.681ex; height:6.509ex;" alt="{\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}}"></span> </th></tr> <tr> <td>1 </td> <td>0.30103000 </td> <td> </td></tr> <tr> <td>2 </td> <td>0.17609126 </td> <td>−0.7735840 </td></tr> <tr> <td>3 </td> <td>0.12493874 </td> <td>−0.8463832 </td></tr> <tr> <td>4 </td> <td>0.09691001 </td> <td>−0.8830605 </td></tr> <tr> <td>5 </td> <td>0.07918125 </td> <td>−0.9054412 </td></tr> <tr> <td>6 </td> <td>0.06694679 </td> <td>−0.9205788 </td></tr> <tr> <td>7 </td> <td>0.05799195 </td> <td>−0.9315169 </td></tr> <tr> <td>8 </td> <td>0.05115252 </td> <td>−0.9397966 </td></tr> <tr> <td>9 </td> <td>0.04575749 </td> <td>−0.9462848 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Occurrences">Occurrences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=6" title="Edit section: Occurrences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="City_sizes">City sizes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=7" title="Edit section: City sizes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.<sup id="cite_ref-gaba1999_40-0" class="reference"><a href="#cite_note-gaba1999-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> However, more recent empirical<sup id="cite_ref-arsh2018_41-0" class="reference"><a href="#cite_note-arsh2018-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ganl2006_42-0" class="reference"><a href="#cite_note-ganl2006-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> and theoretical<sup id="cite_ref-verb2020_43-0" class="reference"><a href="#cite_note-verb2020-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> studies have challenged the relevance of Zipf's law for cities. </p> <div class="mw-heading mw-heading3"><h3 id="Word_frequencies_in_natural_languages">Word frequencies in natural languages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=8" title="Edit section: Word frequencies in natural languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zipf_30wiki_en_labels.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Zipf_30wiki_en_labels.png/220px-Zipf_30wiki_en_labels.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Zipf_30wiki_en_labels.png/330px-Zipf_30wiki_en_labels.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Zipf_30wiki_en_labels.png/440px-Zipf_30wiki_en_labels.png 2x" data-file-width="5600" data-file-height="4200" /></a><figcaption>Zipf's law plot for the first 10 million words in 30 Wikipedias (as of October 2015) in a <a href="/wiki/Log-log" class="mw-redirect" title="Log-log">log-log</a> scale</figcaption></figure> <p>In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent <span class="texhtml mvar" style="font-style:italic;">s</span> close to <span class="nowrap"> <span class="texhtml"> 1 </span> ;</span> that is, the most common word occurs about <span class="texhtml mvar" style="font-style:italic;">n</span> times the <span class="texhtml mvar" style="font-style:italic;">n</span>th most common one. </p><p>The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range. The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language, and on the spelling rules used.<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. (May 2023)">citation needed</span></a></i>]</sup> Some deviation is inevitable because of <a href="/wiki/Sampling_error" title="Sampling error">sampling error</a>. </p><p>At the low-frequency end, where the rank approaches <span class="texhtml mvar" style="font-style:italic;">N</span>, the plot takes a staircase shape, because each word can occur only an integer number of times. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Zipf's law plots for several languages</li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-euro-4_German,_Russian,_French,_Italian,_Medieval_English.svg" class="mw-file-description" title="Texts in German (1669), Russian (1972), French (1865), Italian (1840), and Medieval English (1460)"><img alt="Texts in German (1669), Russian (1972), French (1865), Italian (1840), and Medieval English (1460)" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg/280px-Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg/420px-Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg/560px-Zipf-euro-4_German%2C_Russian%2C_French%2C_Italian%2C_Medieval_English.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">Texts in <a href="/wiki/German_language" title="German language">German</a> (1669), <a href="/wiki/Russian_language" title="Russian language">Russian</a> (1972), <a href="/wiki/French_language" title="French language">French</a> (1865), <a href="/wiki/Italian_language" title="Italian language">Italian</a> (1840), and Medieval English (1460)</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-euro-3_Spanish_(Don_Quixote)_and_Portuguese_(Dom_Casmurro).svg" class="mw-file-description" title="Cervantes' Don Quixote Part I (Spanish, 1605) and Assis's Dom Casmurro (Portuguese, 1899)"><img alt="Cervantes' Don Quixote Part I (Spanish, 1605) and Assis's Dom Casmurro (Portuguese, 1899)" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg/280px-Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg/420px-Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg/560px-Zipf-euro-3_Spanish_%28Don_Quixote%29_and_Portuguese_%28Dom_Casmurro%29.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext"><a href="/wiki/Cervantes" class="mw-redirect" title="Cervantes">Cervantes</a>' <a href="/wiki/Don_Quixote" title="Don Quixote">Don Quixote</a> Part I (<a href="/wiki/Spanish_language" title="Spanish language">Spanish</a>, 1605) and <a href="/wiki/Machado_de_Assis" title="Machado de Assis">Assis</a>'s <a href="/wiki/Dom_Casmurro" title="Dom Casmurro">Dom Casmurro</a> (<a href="/wiki/Portuguese_language" title="Portuguese language">Portuguese</a>, 1899)</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-semi-1_Arabic,_Geez,_Hebraic.svg" class="mw-file-description" title="Ge'ez (14th century), Arabic (~650), Hebrew (500-800), all with vowels"><img alt="Ge'ez (14th century), Arabic (~650), Hebrew (500-800), all with vowels" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg/280px-Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg/420px-Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg/560px-Zipf-semi-1_Arabic%2C_Geez%2C_Hebraic.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext"><a href="/wiki/Ge%27ez_language" class="mw-redirect" title="Ge'ez language">Ge'ez</a> (14th century), <a href="/wiki/Arabic_language" class="mw-redirect" title="Arabic language">Arabic</a> (~650), <a href="/wiki/Hebrew_language" title="Hebrew language">Hebrew</a> (500-800), all with vowels</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-asia-1_Chinese,_Tibetan,_Vietnamese.svg" class="mw-file-description" title="Lhasa Tibetan, Chinese, Vietnamese, all with separated syllables"><img alt="Lhasa Tibetan, Chinese, Vietnamese, all with separated syllables" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg/280px-Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg/420px-Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg/560px-Zipf-asia-1_Chinese%2C_Tibetan%2C_Vietnamese.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext"><a href="/wiki/Lhasa_Tibetan" title="Lhasa Tibetan">Lhasa Tibetan</a>, <a href="/wiki/Chinese_language" title="Chinese language">Chinese</a>, <a href="/wiki/Vietnamese_language" title="Vietnamese language">Vietnamese</a>, all with separated syllables</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT,_Greek_NT,_Russian_OT.svg" class="mw-file-description" title="Biblical texts: Pentateuch from the Latin Vulgate and Russian Synodal Bible, the four Gospels from the Byzantine Greek Majority version"><img alt="Biblical texts: Pentateuch from the Latin Vulgate and Russian Synodal Bible, the four Gospels from the Byzantine Greek Majority version" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg/280px-Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg/420px-Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg/560px-Zipf-euro-2_Biblical_texts_-_Latin_Vulgate_OT%2C_Greek_NT%2C_Russian_OT.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">Biblical texts: <a href="/wiki/Pentateuch" class="mw-redirect" title="Pentateuch">Pentateuch</a> from the Latin <a href="/wiki/Vulgate" title="Vulgate">Vulgate</a> and <a href="/wiki/Russian_Synodal_Bible" title="Russian Synodal Bible">Russian Synodal Bible</a>, the four <a href="/wiki/Gospel" title="Gospel">Gospels</a> from the <a href="/wiki/Byzantine_Greek" class="mw-redirect" title="Byzantine Greek">Byzantine Greek</a> <a href="/wiki/Byzantine_text-type" title="Byzantine text-type">Majority version</a> </div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg" class="mw-file-description" title="Cervantes's Don Quixote, Part I (1605) and Part II (1615)"><img alt="Cervantes's Don Quixote, Part I (1605) and Part II (1615)" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg/280px-Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg/420px-Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg/560px-Zipf-span-1_Spanish_-_Don_Quixote_Parts_1_and_2.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">Cervantes's Don Quixote, Part I (1605) and Part II (1615)</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg" class="mw-file-description" title="First five books of the Old Testament (the Torah) in Hebrew, with vowels"><img alt="First five books of the Old Testament (the Torah) in Hebrew, with vowels" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg/280px-Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg/420px-Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg/560px-Zipf-heot-0_Hebrew_-_Books_of_the_Torah.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">First five books of the <a href="/wiki/Old_Testament" title="Old Testament">Old Testament</a> (the <a href="/wiki/Torah" title="Torah">Torah</a>) in Hebrew, with vowels</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-laot-0_Vulgate_Pentateuch_books.svg" class="mw-file-description" title="First five books of the Old Testament (the Pentateuch) in the Latin Vulgate version"><img alt="First five books of the Old Testament (the Pentateuch) in the Latin Vulgate version" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zipf-laot-0_Vulgate_Pentateuch_books.svg/280px-Zipf-laot-0_Vulgate_Pentateuch_books.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zipf-laot-0_Vulgate_Pentateuch_books.svg/420px-Zipf-laot-0_Vulgate_Pentateuch_books.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Zipf-laot-0_Vulgate_Pentateuch_books.svg/560px-Zipf-laot-0_Vulgate_Pentateuch_books.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">First five books of the <a href="/wiki/Old_Testament" title="Old Testament">Old Testament</a> (the <a href="/wiki/Pentateuch" class="mw-redirect" title="Pentateuch">Pentateuch</a>) in the Latin <a href="/wiki/Vulgate" title="Vulgate">Vulgate</a> version</div> </li> <li class="gallerybox" style="width: 315px"> <div class="thumb" style="width: 310px; height: 330px;"><span typeof="mw:File"><a href="/wiki/File:Zipf-lant-0_Vulgate_Gospels.svg" class="mw-file-description" title="First four books of the New Testament (the Gospels) in the Latin Vulgate version"><img alt="First four books of the New Testament (the Gospels) in the Latin Vulgate version" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zipf-lant-0_Vulgate_Gospels.svg/280px-Zipf-lant-0_Vulgate_Gospels.svg.png" decoding="async" width="280" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zipf-lant-0_Vulgate_Gospels.svg/420px-Zipf-lant-0_Vulgate_Gospels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zipf-lant-0_Vulgate_Gospels.svg/560px-Zipf-lant-0_Vulgate_Gospels.svg.png 2x" data-file-width="512" data-file-height="504" /></a></span></div> <div class="gallerytext">First four books of the <a href="/wiki/New_Testament" title="New Testament">New Testament</a> (the <a href="/wiki/Gospels" class="mw-redirect" title="Gospels">Gospels</a>) in the Latin <a href="/wiki/Vulgate" title="Vulgate">Vulgate</a> version</div> </li> </ul> <figure typeof="mw:File/Frame"><a href="/wiki/File:Wikipedia-n-zipf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b9/Wikipedia-n-zipf.png" decoding="async" width="640" height="480" class="mw-file-element" data-file-width="640" data-file-height="480" /></a><figcaption>A log-log plot of word frequency in the English Wikipedia (27 November 2006). 'Most popular words are "the", "of" and "and", as expected. Zipf's law corresponds to the middle linear portion of the curve, roughly following the <span style="color:green;"><b>green</b></span> <span class="nowrap">(<span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>x</i></span></span>⁠</span></span>) line,</span> while the early part is closer to the <span style="color:magenta;"><b>magenta</b></span> <span class="nowrap">(<span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>x</i></span></span></span></span>⁠</span></span>) line</span> while the later part is closer to the <span style="color:cyan;"><b>cyan</b></span> <span class="nowrap">(<span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> <i>x</i><sup>2</sup> </span></span>⁠</span></span>) line.</span> These lines correspond to three distinct parameterizations of the Zipf–Mandelbrot distribution, overall a <a href="/wiki/Broken_power_law" class="mw-redirect" title="Broken power law">broken power law</a> with three segments: a head, middle, and tail.<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. (November 2024)">citation needed</span></a></i>]</sup> Other descriptions highlight two segments or "regimes" instead.<sup id="cite_ref-cancho2001_44-0" class="reference"><a href="#cite_note-cancho2001-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dm2002_45-0" class="reference"><a href="#cite_note-dm2002-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></figcaption></figure><p>. </p><p>In some <a href="/wiki/Romance_languages" title="Romance languages">Romance languages</a>, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for <a href="/wiki/Grammatical_gender" title="Grammatical gender">grammatical gender</a> and <a href="/wiki/Grammatical_number" title="Grammatical number">number</a>.<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. (May 2023)">citation needed</span></a></i>]</sup> </p><p>In many <a href="/wiki/East_Asia" title="East Asia">East Asian</a> languages, such as <a href="/wiki/Chinese_language" title="Chinese language">Chinese</a>, <a href="/wiki/Lhasa_Tibetan" title="Lhasa Tibetan">Lhasa Tibetan</a>, and <a href="/wiki/Vietnamese_language" title="Vietnamese language">Vietnamese</a>, each "word" consists of a single <a href="/wiki/Syllable" title="Syllable">syllable</a>; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.<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. (May 2023)">citation needed</span></a></i>]</sup> </p><p>Even in English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in it have a good fit to Zipf's law. Slight changes in the definition of Zipf's law can increase this percentage up to close to 50%.<sup id="cite_ref-more2016_46-0" class="reference"><a href="#cite_note-more2016-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>In these cases, the observed frequency-rank relation can be modeled more accurately as by separate Zipf–Mandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the English Wikipedia. In particular, the frequencies of the closed class of <a href="/wiki/Function_word" title="Function word">function words</a> in English is better described with <span class="texhtml mvar" style="font-style:italic;">s</span> lower than <span class="texhtml"> 1 ,</span> while open-ended vocabulary growth with document size and corpus size require <span class="texhtml mvar" style="font-style:italic;">s</span> greater than 1 for convergence of the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Generalized Harmonic Series</a>.<sup id="cite_ref-Powers1998_3-2" class="reference"><a href="#cite_note-Powers1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Zipf-code-1_English_plain,_book-coded,_Vigenere_coded.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg/220px-Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg.png" decoding="async" width="220" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg/330px-Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg/440px-Zipf-code-1_English_plain%2C_book-coded%2C_Vigenere_coded.svg.png 2x" data-file-width="512" data-file-height="504" /></a><figcaption>Well's <i>War of the Worlds</i> in plain text, in a <a href="/wiki/Book_code" class="mw-redirect" title="Book code">book code</a>, and in a <a href="/wiki/Vigen%C3%A8re_cipher" title="Vigenère cipher">Vigenère cipher</a></figcaption></figure> <p>When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple <a href="/wiki/Substitution_cipher" title="Substitution cipher">substitution ciphers</a>, like the <a href="/wiki/Caesar_cipher" title="Caesar cipher">Caesar ciphers</a>, or simple <a href="/wiki/Codebook" title="Codebook">codebook</a> ciphers), the frequency-rank distribution is not affected. On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the <a href="/wiki/Vigen%C3%A8re_cipher" title="Vigenère cipher">Vigenère cipher</a>), the Zipf distribution will typically have a flat part at the high-frequency end.<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. (May 2023)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading4"><h4 id="Applications">Applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.<sup id="cite_ref-moha2016_47-0" class="reference"><a href="#cite_note-moha2016-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Laurance_Doyle" title="Laurance Doyle">Laurance Doyle</a> and others have suggested the application of Zipf's law for detection of <a href="/wiki/Alien_language" title="Alien language">alien language</a> in the <a href="/wiki/Search_for_extraterrestrial_intelligence" title="Search for extraterrestrial intelligence">search for extraterrestrial intelligence</a>.<sup id="cite_ref-doyle20162_48-0" class="reference"><a href="#cite_note-doyle20162-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kersh20212_49-0" class="reference"><a href="#cite_note-kersh20212-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.<sup id="cite_ref-droo2016_50-0" class="reference"><a href="#cite_note-droo2016-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-droo2019_51-0" class="reference"><a href="#cite_note-droo2019-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p>The word-like sign groups of the 15th-century codex <a href="/wiki/Voynich_manuscript" title="Voynich manuscript">Voynich Manuscript</a> have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.<sup id="cite_ref-boyle2022_52-0" class="reference"><a href="#cite_note-boyle2022-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mont2013_53-0" class="reference"><a href="#cite_note-mont2013-53"><span class="cite-bracket">[</span>52<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=Zipf%27s_law&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/1%25_rule_(Internet_culture)" class="mw-redirect" title="1% rule (Internet culture)">1% rule (Internet culture)</a> – Hypothesis that more people will lurk in a virtual community than will participate<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> – Observation that in many real-life datasets, the leading digit is likely to be small</li> <li><a href="/wiki/Bradford%27s_law" title="Bradford's law">Bradford's law</a> – Pattern of references in science journals</li> <li><a href="/wiki/Brevity_law" title="Brevity law">Brevity law</a> – Linguistics law</li> <li><a href="/wiki/Demographic_gravitation" title="Demographic gravitation">Demographic gravitation</a> – Social effect</li> <li><a href="/wiki/Frequency_list" class="mw-redirect" title="Frequency list">Frequency list</a> – Bare list of a language's words in corpus linguistics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a> – Economic principle</li> <li><a href="/wiki/Hapax_legomenon" title="Hapax legomenon">Hapax legomenon</a> – Word that only appears once in a given text or record</li> <li><a href="/wiki/Heaps%27_law" title="Heaps' law">Heaps' law</a> – Heuristic for distinct words in a document</li> <li><a href="/wiki/King_effect" title="King effect">King effect</a> – Phenomenon in statistics where highest-ranked data points are outliers</li> <li><a href="/wiki/Long_tail" title="Long tail">Long tail</a> – Feature of some statistical distributions</li> <li><a href="/wiki/Lorenz_curve" title="Lorenz curve">Lorenz curve</a> – Graphical representation of the distribution of income or of wealth</li> <li><a href="/wiki/Lotka%27s_law" title="Lotka's law">Lotka's law</a> – An application of Zipf's law describing the frequency of publication by authors in any given field</li> <li><a href="/wiki/Menzerath%27s_law" title="Menzerath's law">Menzerath's law</a> – Linguistic law</li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a> – Probability distribution</li> <li><a href="/wiki/Pareto_principle" title="Pareto principle">Pareto principle</a> – Statistical principle about ratio of effects to causes, a.k.a. the "80–20 rule"</li> <li><a href="/wiki/Price%27s_law" class="mw-redirect" title="Price's law">Price's law</a> – Physicist and science historian (1922–1983)<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Principle_of_least_effort" title="Principle of least effort">Principle of least effort</a> – Idea that agents prefer to do what's easiest</li> <li><a href="/wiki/Rank-size_distribution" class="mw-redirect" title="Rank-size distribution">Rank-size distribution</a> – distribution of size by rank<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Stigler%27s_law_of_eponymy" title="Stigler's law of eponymy">Stigler's law of eponymy</a> – Observation that no scientific discovery is named after its discoverer</li> <li><a href="/wiki/Letter_frequency" title="Letter frequency">Letter frequency</a></li> <li><a href="/wiki/Most_common_words_in_English" title="Most common words in English">Most common words in English</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&action=edit&section=11" title="Edit section: Notes"><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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">as Zipf acknowledged<sup id="cite_ref-zipf1949_5-1" class="reference"><a href="#cite_note-zipf1949-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 546">: 546 </span></sup></span> </li> </ol></div></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=Zipf%27s_law&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-piant2014-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-piant2014_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-piant2014_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-piant2014_1-2"><sup><i><b>c</b></i></sup></a></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="CITEREFPiantadosi2014" class="citation journal cs1">Piantadosi, Steven (25 March 2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4176592">"Zipf's word frequency law in natural language: A critical review and future directions"</a>. <i>Psychon Bull Rev</i>. <b>21</b> (5): 1112–1130. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3758%2Fs13423-014-0585-6">10.3758/s13423-014-0585-6</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4176592">4176592</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24664880">24664880</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychon+Bull+Rev&rft.atitle=Zipf%27s+word+frequency+law+in+natural+language%3A+A+critical+review+and+future+directions&rft.volume=21&rft.issue=5&rft.pages=1112-1130&rft.date=2014-03-25&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4176592%23id-name%3DPMC&rft_id=info%3Apmid%2F24664880&rft_id=info%3Adoi%2F10.3758%2Fs13423-014-0585-6&rft.aulast=Piantadosi&rft.aufirst=Steven&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4176592&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-fagan2010-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-fagan2010_2-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFaganGençay2010" class="citation book cs1">Fagan, Stephen; Gençay, Ramazan (2010). "An introduction to textual econometrics". In Ullah, Aman; Giles, David E.A. (eds.). <i>Handbook of Empirical Economics and Finance</i>. CRC Press. pp. 133–153, esp.&nbps, <a rel="nofollow" class="external text" href="https://books.google.com/books?hl=en&lr=&id=QAUv9R6bJzwC&oi=fnd&pg=PA139">139</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420070361" title="Special:BookSources/9781420070361"><bdi>9781420070361</bdi></a>. <q>For example, in the Brown Corpus, consisting of over one million words, half of the word volume consists of repeated uses of only 135 words.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+introduction+to+textual+econometrics&rft.btitle=Handbook+of+Empirical+Economics+and+Finance&rft.pages=133-153%2C+esp.%26nbps%2C+139&rft.pub=CRC+Press&rft.date=2010&rft.isbn=9781420070361&rft.aulast=Fagan&rft.aufirst=Stephen&rft.au=Gen%C3%A7ay%2C+Ramazan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Powers1998-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Powers1998_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Powers1998_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Powers1998_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPowers1998" class="citation conference cs1">Powers, David M.W. (1998). <a rel="nofollow" class="external text" href="http://aclweb.org/anthology/W98-1218"><i>Applications and explanations of Zipf's law</i></a>. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150910142650/http://aclweb.org/anthology/W98-1218">Archived</a> from the original on 10 September 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">2 February</span> 2015</span> – via aclweb.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Applications+and+explanations+of+Zipf%27s+law&rft.pages=151-160&rft.pub=Association+for+Computational+Linguistics&rft.date=1998&rft.aulast=Powers&rft.aufirst=David+M.W.&rft_id=http%3A%2F%2Faclweb.org%2Fanthology%2FW98-1218&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-zipf1935-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-zipf1935_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-zipf1935_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZipf1935" class="citation book cs1"><a href="/wiki/George_K._Zipf" class="mw-redirect" title="George K. Zipf">Zipf, G.K.</a> (1935). <i>The Psychobiology of Language</i>. New York, NY: Houghton-Mifflin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Psychobiology+of+Language&rft.place=New+York%2C+NY&rft.pub=Houghton-Mifflin&rft.date=1935&rft.aulast=Zipf&rft.aufirst=G.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-zipf1949-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-zipf1949_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-zipf1949_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-zipf1949_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZipf1949" class="citation book cs1"><a href="/wiki/George_K._Zipf" class="mw-redirect" title="George K. Zipf">Zipf, George K.</a> (1949). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.90211"><i>Human Behavior and the Principle of Least Effort</i></a>. Cambridge, MA: Addison-Wesley. p. 1 – via archive.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Human+Behavior+and+the+Principle+of+Least+Effort&rft.place=Cambridge%2C+MA&rft.pages=1&rft.pub=Addison-Wesley&rft.date=1949&rft.aulast=Zipf&rft.aufirst=George+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.90211&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Auerbach1913-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Auerbach1913_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuerbach1913" class="citation journal cs1 cs1-prop-foreign-lang-source">Auerbach, F. (1913). 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"The Unity of Nature, Least-Action, and Natural Social Science". <i>Sociometry</i>. <b>5</b> (1): 48–62. <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%2F2784953">10.2307/2784953</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/2784953">2784953</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sociometry&rft.atitle=The+Unity+of+Nature%2C+Least-Action%2C+and+Natural+Social+Science&rft.volume=5&rft.issue=1&rft.pages=48-62&rft.date=1942&rft_id=info%3Adoi%2F10.2307%2F2784953&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2784953%23id-name%3DJSTOR&rft.aulast=Zipf&rft.aufirst=George+Kingsley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-axte2001-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-axte2001_14-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxtell2001" class="citation journal cs1">Axtell, Robert L. 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Archived from <a rel="nofollow" class="external text" href="https://www.academia.edu/40029629">the original</a> on 30 September 2023 – via academia.edu.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=An+essential+rephrasing+of+the+Zipf-Mandelbrot+law+to+solve+authorship+attribution+applications+by+Gaussian+statistics&rft.date=2019&rft.aulast=van+Droogenbroeck&rft.aufirst=Frans+J.&rft_id=https%3A%2F%2Fwww.academia.edu%2F40029629&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-boyle2022-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-boyle2022_52-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyle" class="citation web cs1">Boyle, Rebecca. <a rel="nofollow" class="external text" href="https://www.newscientist.com/article/2106915-mystery-texts-language-like-patterns-may-be-an-elaborate-hoax/">"Mystery text's language-like patterns may be an elaborate hoax"</a>. <i>New Scientist</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220518100834/https://www.newscientist.com/article/2106915-mystery-texts-language-like-patterns-may-be-an-elaborate-hoax/">Archived</a> from the original on 18 May 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">25 February</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=New+Scientist&rft.atitle=Mystery+text%27s+language-like+patterns+may+be+an+elaborate+hoax&rft.aulast=Boyle&rft.aufirst=Rebecca&rft_id=https%3A%2F%2Fwww.newscientist.com%2Farticle%2F2106915-mystery-texts-language-like-patterns-may-be-an-elaborate-hoax%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-mont2013-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-mont2013_53-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontemurroZanette2013" class="citation journal cs1">Montemurro, Marcelo A.; Zanette, Damián H. (21 June 2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3689824">"Keywords and Co-Occurrence Patterns in the Voynich Manuscript: An Information-Theoretic Analysis"</a>. <i>PLOS ONE</i>. <b>8</b> (6): e66344. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0066344">10.1371/journal.pone.0066344</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3689824">3689824</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23805215">23805215</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Keywords+and+Co-Occurrence+Patterns+in+the+Voynich+Manuscript%3A+An+Information-Theoretic+Analysis&rft.volume=8&rft.issue=6&rft.pages=e66344&rft.date=2013-06-21&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3689824%23id-name%3DPMC&rft_id=info%3Apmid%2F23805215&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0066344&rft.aulast=Montemurro&rft.aufirst=Marcelo+A.&rft.au=Zanette%2C+Dami%C3%A1n+H.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3689824&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zipf%27s_law&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="CITEREFGelbukhSidorov2001" class="citation book cs1">Gelbukh, Alexander; Sidorov, Grigori (2001). "Zipf and Heaps Laws' Coefficients Depend on Language". <i>Computational Linguistics and Intelligent Text Processing</i>. Lecture Notes in Computer Science. Vol. 2004. pp. 332–335. <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%2F3-540-44686-9_33">10.1007/3-540-44686-9_33</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-41687-6" title="Special:BookSources/978-3-540-41687-6"><bdi>978-3-540-41687-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zipf+and+Heaps+Laws%27+Coefficients+Depend+on+Language&rft.btitle=Computational+Linguistics+and+Intelligent+Text+Processing&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=332-335&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F3-540-44686-9_33&rft.isbn=978-3-540-41687-6&rft.aulast=Gelbukh&rft.aufirst=Alexander&rft.au=Sidorov%2C+Grigori&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKali2003" class="citation journal cs1">Kali, Raja (15 September 2003). "The city as a giant component: a random graph approach to Zipf's law". <i>Applied Economics Letters</i>. <b>10</b> (11): 717–720. <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%2F1350485032000139006">10.1080/1350485032000139006</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Economics+Letters&rft.atitle=The+city+as+a+giant+component%3A+a+random+graph+approach+to+Zipf%27s+law&rft.volume=10&rft.issue=11&rft.pages=717-720&rft.date=2003-09-15&rft_id=info%3Adoi%2F10.1080%2F1350485032000139006&rft.aulast=Kali&rft.aufirst=Raja&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShyklo2017" class="citation report cs1">Shyklo, Alexandra Elizabeth (2017). Simple Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process (Report). <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2918642">2918642</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Simple+Explanation+of+Zipf%27s+Mystery+via+New+Rank-Share+Distribution%2C+Derived+from+Combinatorics+of+the+Ranking+Process&rft.date=2017&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D2918642%23id-name%3DSSRN&rft.aulast=Shyklo&rft.aufirst=Alexandra+Elizabeth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoskowitzFordChristiansen2024" class="citation journal cs1">Moskowitz, Clara; Ford, Ni-ka; Christiansen, Jen (January 2024). "Cells by Count and Size". <i>Scientific American</i>. <b>330</b> (1): 94. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0124-94">10.1038/scientificamerican0124-94</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/39017389">39017389</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Cells+by+Count+and+Size&rft.volume=330&rft.issue=1&rft.pages=94&rft.date=2024-01&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0124-94&rft_id=info%3Apmid%2F39017389&rft.aulast=Moskowitz&rft.aufirst=Clara&rft.au=Ford%2C+Ni-ka&rft.au=Christiansen%2C+Jen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" 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=Zipf%27s_law&action=edit&section=14" 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><div class="side-box metadata side-box-right"><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-abovebelow"> <a href="/wiki/Wikipedia:The_Wikipedia_Library" title="Wikipedia:The Wikipedia Library">Library resources</a> about <br /> <b>Zipf's law</b> <hr /></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Zipf%26%2339%3Bs+law">Resources in your library</a></li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Zipf%26%2339%3Bs+law&library=0CHOOSE0">Resources in other libraries</a></li> </ul></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/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:Zipf%27s_law" class="extiw" title="commons:Category:Zipf's law">Zipf's law</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrogatz2009" class="citation news cs1"><a href="/wiki/Steven_Strogatz" title="Steven Strogatz">Strogatz, Steven</a> (29 May 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150927204318/http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/">"Guest Column: Math and the City"</a>. <i>The New York Times</i>. Archived from <a rel="nofollow" class="external text" href="http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/">the original</a> on 27 September 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">29 May</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=Guest+Column%3A+Math+and+the+City&rft.date=2009-05-29&rft.aulast=Strogatz&rft.aufirst=Steven&rft_id=http%3A%2F%2Fjudson.blogs.nytimes.com%2F2009%2F05%2F19%2Fmath-and-the-city%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span>—An article on Zipf's law applied to city populations</li> <li><a rel="nofollow" class="external text" href="https://www.theatlantic.com/issues/2002/04/rauch.htm">Seeing Around Corners (Artificial societies turn up Zipf's law)</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20021018011011/http://planetmath.org/encyclopedia/ZipfsLaw.html">PlanetMath article on Zipf's law</a></li> <li><a rel="nofollow" class="external text" href="http://www.hubbertpeak.com/laherrere/fractal.htm">Distributions de type "fractal parabolique" dans la Nature (French, with English summary)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041024144850/http://www.hubbertpeak.com/laherrere/fractal.htm">Archived</a> 2004-10-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://www.newscientist.com/article.ns?id=mg18524904.300">An analysis of income distribution</a></li> <li><a rel="nofollow" class="external text" href="http://www.lexique.org/listes/liste_mots.txt">Zipf List of French words</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070623154627/http://www.lexique.org/listes/liste_mots.txt">Archived</a> 2007-06-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://1.1o1.in/en/webtools/semantic-depth">Zipf list for English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110408115104/http://1.1o1.in/en/webtools/semantic-depth">Archived</a> 2011-04-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9901035">Citations and the Zipf–Mandelbrot's law</a></li> <li><a rel="nofollow" class="external text" href="http://www.geoffkirby.co.uk/ZIPFSLAW.pdf">Zipf's Law examples and modelling (1985)</a></li> <li><a rel="nofollow" class="external text" href="http://www.nature.com/nature/journal/v474/n7350/full/474164a.html">Complex systems: Unzipping Zipf's law (2011)</a></li> <li><a rel="nofollow" class="external text" href="http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-the-pareto-distribution/">Benford's law, Zipf's law, and the Pareto distribution</a> by Terence Tao.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Zipf_law">"Zipf law"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zipf+law&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DZipf_law&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZipf%27s+law" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Probability_distributions_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite <br />support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a class="mw-selflink selflink">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's <i>z</i></a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral <i>t</i></a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <i>t</i></a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q205472#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td 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8.16% 129.222 1 Template:IPA"," 8.01% 126.862 1 Template:ProbDistributions"," 7.91% 125.357 4 Template:Navbox"," 5.09% 80.599 1 Template:Short_description"]},"scribunto":{"limitreport-timeusage":{"value":"0.869","limit":"10.000"},"limitreport-memusage":{"value":19503267,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAdamic2000\"] = 1,\n [\"CITEREFArshadHuAshraf2018\"] = 1,\n [\"CITEREFAuerbach1913\"] = 1,\n [\"CITEREFAxtell2001\"] = 1,\n [\"CITEREFBelevitch1959\"] = 1,\n [\"CITEREFBoyle\"] = 1,\n [\"CITEREFBrillouin2004\"] = 1,\n [\"CITEREFChennaGibson2011\"] = 1,\n [\"CITEREFClausetShaliziNewman2009\"] = 1,\n [\"CITEREFCondon1928\"] = 1,\n [\"CITEREFConradMitzenmacher2004\"] = 1,\n [\"CITEREFDewey1923\"] = 1,\n [\"CITEREFDorogovtsevMendes2001\"] = 1,\n [\"CITEREFDoyle2016\"] = 1,\n [\"CITEREFEftekhari2006\"] = 1,\n [\"CITEREFErikssonRahmanFraileSjostrom2013\"] = 1,\n [\"CITEREFEstoup1916\"] = 1,\n [\"CITEREFFaganGençay2010\"] = 1,\n [\"CITEREFFernholzFernholz2020\"] = 1,\n [\"CITEREFFerrer_CanchoSolé2001\"] = 1,\n [\"CITEREFFerrer_i_CanchoSole2003\"] = 1,\n [\"CITEREFGabaix1999\"] = 1,\n [\"CITEREFGanLiSong2006\"] = 1,\n [\"CITEREFGelbukhSidorov2001\"] = 1,\n [\"CITEREFJohnsonKotzKemp1992\"] = 1,\n [\"CITEREFKali2003\"] = 1,\n [\"CITEREFKershenbaum2021\"] = 1,\n [\"CITEREFLazzardiValleMazzoliniScialdone2023\"] = 1,\n [\"CITEREFLi1992\"] = 1,\n [\"CITEREFLinMaBian2014\"] = 1,\n [\"CITEREFManarisPellicoroPotheringHodges2006\"] = 1,\n [\"CITEREFManningSchütze1999\"] = 1,\n [\"CITEREFMitzenmacher2004\"] = 1,\n [\"CITEREFMohammadi2016\"] = 1,\n [\"CITEREFMontemurroZanette2013\"] = 1,\n [\"CITEREFMoreno-SánchezFont-ClosCorral2016\"] = 1,\n [\"CITEREFMoskowitzFordChristiansen2024\"] = 1,\n [\"CITEREFNeumannc._2022\"] = 1,\n [\"CITEREFPiantadosi2014\"] = 1,\n [\"CITEREFPietroneroTosattiTosattiVespignani2001\"] = 1,\n [\"CITEREFPowers1998\"] = 1,\n [\"CITEREFSandmo2015\"] = 1,\n [\"CITEREFShyklo2017\"] = 1,\n [\"CITEREFSimkinRoychowdhury2010\"] = 1,\n [\"CITEREFSkotarek2020\"] = 1,\n [\"CITEREFStrogatz2009\"] = 1,\n [\"CITEREFTao2012\"] = 1,\n [\"CITEREFVerbavatzBarthelemy2020\"] = 1,\n [\"CITEREFVitanovAusloos2015\"] = 1,\n [\"CITEREFZanette2004\"] = 1,\n [\"CITEREFZipf1932\"] = 1,\n [\"CITEREFZipf1935\"] = 1,\n [\"CITEREFZipf1942\"] = 1,\n [\"CITEREFZipf1949\"] = 1,\n [\"CITEREFvan_Droogenbroeck2016\"] = 1,\n [\"CITEREFvan_Droogenbroeck2019\"] = 1,\n [\"CITEREFvan_der_Galien2003\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 6,\n [\"Annotated link\"] = 20,\n [\"Authority control\"] = 1,\n [\"Citation needed\"] = 5,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 13,\n [\"Cite conference\"] = 5,\n [\"Cite journal\"] = 29,\n [\"Cite news\"] = 1,\n [\"Cite report\"] = 5,\n [\"Cite web\"] = 3,\n [\"Clarify\"] = 1,\n [\"Commons category\"] = 1,\n [\"Efn\"] = 1,\n [\"Fontcolor\"] = 3,\n [\"For\"] = 1,\n [\"Harvp\"] = 1,\n [\"IPA\"] = 1,\n [\"IPAc-en\"] = 1,\n [\"Library resources box\"] = 1,\n [\"Math\"] = 18,\n [\"Mvar\"] = 38,\n [\"Nobr\"] = 11,\n [\"Notelist\"] = 1,\n [\"ProbDistributions\"] = 1,\n [\"Probability distribution\"] = 1,\n [\"Reflist\"] = 1,\n [\"Refn\"] = 1,\n [\"Rp\"] = 1,\n [\"Sfrac\"] = 4,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Sqrt\"] = 1,\n [\"Sup\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 3,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-blxk6","timestamp":"20241124164507","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Zipf's law","url":"https:\/\/en.wikipedia.org\/wiki\/Zipf%27s_law","sameAs":"http:\/\/www.wikidata.org\/entity\/Q205472","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q205472","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-11-04T18:42:42Z","dateModified":"2024-11-19T15:38:45Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/96\/Zipf%27s_law_on_War_and_Peace.png","headline":"probability distribution"}</script> </body> </html>