Diversity index - Wikipedia

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sensitivity_of_the_diversity_value_to_rare_vs._abundant_species"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Sensitivity of the diversity value to rare vs. abundant species</span> </div> </a> <ul id="toc-Sensitivity_of_the_diversity_value_to_rare_vs._abundant_species-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Richness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Richness"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Richness</span> </div> </a> <ul id="toc-Richness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shannon_index" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shannon_index"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Shannon index</span> </div> </a> <button aria-controls="toc-Shannon_index-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Shannon index subsection</span> </button> <ul id="toc-Shannon_index-sublist" class="vector-toc-list"> <li id="toc-Rényi_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rényi_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rényi entropy</span> </div> </a> <ul id="toc-Rényi_entropy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Simpson_index" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simpson_index"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Simpson index</span> </div> </a> <button aria-controls="toc-Simpson_index-sublist" 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<input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 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-15" 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">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diversitetsindeks" title="Diversitetsindeks – Danish" lang="da" hreflang="da" data-title="Diversitetsindeks" 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/Diversit%C3%A4tsindex" title="Diversitätsindex – German" lang="de" hreflang="de" data-title="Diversitätsindex" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Liigierisus" title="Liigierisus – Estonian" lang="et" hreflang="et" data-title="Liigierisus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%8Dndice_de_diversidad" title="Índice de diversidad – Spanish" lang="es" hreflang="es" data-title="Índice de diversidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aniztasun_adierazle" title="Aniztasun adierazle – Basque" lang="eu" hreflang="eu" data-title="Aniztasun adierazle" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%A7%D8%AE%D8%B5_%D8%AA%D9%86%D9%88%D8%B9" title="شاخص تنوع – Persian" lang="fa" hreflang="fa" data-title="شاخص تنوع" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%96%91%EC%84%B1_%EC%A7%80%EC%88%98" title="다양성 지수 – Korean" lang="ko" hreflang="ko" data-title="다양성 지수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A5%D5%B6%D5%BD%D5%A1%D5%A2%D5%A1%D5%A6%D5%B4%D5%A1%D5%A6%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%B4%D5%A8" title="Կենսաբազմազանության չափումը – Armenian" lang="hy" hreflang="hy" data-title="Կենսաբազմազանության չափումը" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Indice_di_diversit%C3%A0" title="Indice di diversità – Italian" lang="it" hreflang="it" data-title="Indice di diversità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%8Dndice_de_diversidade" title="Índice de diversidade – Portuguese" lang="pt" hreflang="pt" data-title="Índice de diversidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Indice_de_diversitate" title="Indice de diversitate – Romanian" lang="ro" hreflang="ro" data-title="Indice de diversitate" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%80%D0%B0_%D1%80%D0%B0%D0%B7%D0%BD%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D1%8F" title="Мера разнообразия – Russian" lang="ru" hreflang="ru" data-title="Мера разнообразия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diversiteetti-indeksi" title="Diversiteetti-indeksi – Finnish" lang="fi" hreflang="fi" data-title="Diversiteetti-indeksi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D1%96%D1%80%D0%B0_%D1%80%D1%96%D0%B7%D0%BD%D0%BE%D0%BC%D0%B0%D0%BD%D1%96%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Міра різноманітності – Ukrainian" lang="uk" hreflang="uk" data-title="Міра різноманітності" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%9A%E6%A0%B7%E6%80%A7%E6%8C%87%E6%95%B0" title="多样性指数 – Chinese" lang="zh" hreflang="zh" data-title="多样性指数" 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.compact-ambox .hide-when-compact{display:none}</style><table class="box-Multiple_issues plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Diversity_index" title="Special:EditPage/Diversity index">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:Diversity_index" title="Talk:Diversity index">talk page</a></b>. <small><i>(<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove these messages</a>)</i></small> <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup_rewrite plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may need to be rewritten</b> to comply with Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>.<span class="hide-when-compact"> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Diversity_index&action=edit">You can help</a>. The <a href="/wiki/Talk:Diversity_index" title="Talk:Diversity index">talk page</a> may contain suggestions.</span> <span class="date-container"><i>(<span class="date">April 2020</span>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Diversity_index&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">April 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>diversity index</b> is a method of measuring how many different types (e.g. <a href="/wiki/Species" title="Species">species</a>) there are in a dataset (e.g. a community). Some more sophisticated indices also account for the <a href="/wiki/Phylogenetics" title="Phylogenetics">phylogenetic</a> relatedness among the types.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Diversity indices are statistical representations of different aspects of biodiversity (e.g. <a href="/wiki/Species_richness" title="Species richness">richness</a>, <a href="/wiki/Species_evenness" title="Species evenness">evenness</a>, and <a href="/wiki/Dominance_(ecology)" title="Dominance (ecology)">dominance</a>), which are useful simplifications for comparing different communities or sites. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Effective_number_of_species_or_Hill_numbers">Effective number of species or Hill numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=1" title="Edit section: Effective number of species or Hill numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When diversity indices are used in <a href="/wiki/Ecology" title="Ecology">ecology</a>, the types of interest are usually species, but they can also be other categories, such as <a href="/wiki/Genus" title="Genus">genera</a>, <a href="/wiki/Family_(biology)" title="Family (biology)">families</a>, <a href="/wiki/Plant_functional_type" title="Plant functional type">functional types</a>, or <a href="/wiki/Haplotype" title="Haplotype">haplotypes</a>. The entities of interest are usually individual organisms (e.g. plants or animals), and the measure of abundance can be, for example, number of individuals, biomass or coverage. In <a href="/wiki/Demography" title="Demography">demography</a>, the entities of interest can be people, and the types of interest various demographic groups. In <a href="/wiki/Information_science" title="Information science">information science</a>, the entities can be characters and the types of the different letters of the alphabet. The most commonly used diversity indices are simple transformations of the effective number of types (also known as 'true diversity'), but each diversity index can also be interpreted in its own right as a measure corresponding to some real phenomenon (but a different one for each diversity index).<sup id="cite_ref-Hill1973_2-0" class="reference"><a href="#cite_note-Hill1973-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jost2006_3-0" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010a_4-0" class="reference"><a href="#cite_note-Tuomisto2010a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010c_5-0" class="reference"><a href="#cite_note-Tuomisto2010c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Many indices only account for categorical diversity between subjects or entities. Such indices, however do not account for the total variation (diversity) that can be held between subjects or entities which occurs only when both categorical and qualitative diversity are calculated. </p><p>True diversity, or the effective number of types, refers to the number of equally abundant types needed for the average proportional abundance of the types to equal that observed in the dataset of interest (where all types may not be equally abundant). The true diversity in a dataset is calculated by first taking the weighted <a href="/wiki/Generalized_mean" title="Generalized mean">generalized mean</a> <span class="texhtml"><i>M</i><sub><i>q</i>−1</sub></span> of the proportional abundances of the types in the dataset, and then taking the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of this. The equation is:<sup id="cite_ref-Tuomisto2010a_4-1" class="reference"><a href="#cite_note-Tuomisto2010a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010c_5-1" class="reference"><a href="#cite_note-Tuomisto2010c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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 {}^{q}\!D={1 \over M_{q-1}}={1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </mroot> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{q}\!D={1 \over M_{q-1}}={1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf7e22f4730ed7b47ebb145d7a495dfd5ef9177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:49.434ex; height:9.509ex;" alt="{\displaystyle {}^{q}\!D={1 \over M_{q-1}}={1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}"></span></dd></dl> <p>The <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">denominator</a> <span class="texhtml"><i>M</i><sub><i>q</i>−1</sub></span> equals the average proportional abundance of the types in the dataset as calculated with the weighted <a href="/wiki/Generalized_mean" title="Generalized mean">generalized mean</a> with exponent <span class="texhtml"><i>q</i> − 1</span>. In the equation, <span class="texhtml"><i>R</i></span> is richness (the total number of types in the dataset), and the proportional abundance of the <span class="texhtml"><i>i</i></span>th type is <span class="texhtml"><i>p</i><sub><i>i</i></sub></span>. The proportional abundances themselves are used as the nominal weights. The numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{q}D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{q}D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a843767b9e4c56518b5442a543e5a4c4a749b8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.913ex; height:2.343ex;" alt="{\displaystyle ^{q}D}"></span> are called <b>Hill numbers of order</b> <i>q</i> or <b>effective number of species</b>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>When <span class="texhtml"><i>q</i> = 1</span>, the above equation is undefined. However, the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">mathematical limit</a> as <span class="texhtml"><i>q</i></span> approaches 1 is well defined and the corresponding diversity is calculated with the following equation: </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 {}^{1}\!D={1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}=\exp \left(-\sum _{i=1}^{R}p_{i}\ln(p_{i})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{1}\!D={1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}=\exp \left(-\sum _{i=1}^{R}p_{i}\ln(p_{i})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86b58b9bac6fa4e9f992a11974dcea8cef92be73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.204ex; height:7.676ex;" alt="{\displaystyle {}^{1}\!D={1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}=\exp \left(-\sum _{i=1}^{R}p_{i}\ln(p_{i})\right)}"></span></dd></dl> <p>which is the exponential of the <a href="/wiki/Shannon_entropy" class="mw-redirect" title="Shannon entropy">Shannon entropy</a> calculated with natural logarithms (see above). In other domains, this statistic is also known as the <i><a href="/wiki/Perplexity" title="Perplexity">perplexity</a></i>. </p><p>The general equation of diversity is often written in the form<sup id="cite_ref-Hill1973_2-1" class="reference"><a href="#cite_note-Hill1973-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jost2006_3-1" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 {}^{q}\!D=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{q}\!D=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416dfca97f0bcf9735a63d14068c8c6e510a747f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.218ex; height:8.009ex;" alt="{\displaystyle {}^{q}\!D=\left({\sum _{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}}"></span></dd></dl> <p>and the term inside the parentheses is called the basic sum. Some popular diversity indices correspond to the basic sum as calculated with different values of <span class="texhtml"><i>q</i></span>.<sup id="cite_ref-Jost2006_3-2" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Sensitivity_of_the_diversity_value_to_rare_vs._abundant_species">Sensitivity of the diversity value to rare vs. abundant species</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=2" title="Edit section: Sensitivity of the diversity value to rare vs. abundant species"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value of <span class="texhtml"><i>q</i></span> is often referred to as the order of the diversity. It defines the sensitivity of the true diversity to rare vs. abundant species by modifying how the weighted mean of the species' proportional abundances is calculated. With some values of the parameter <span class="texhtml"><i>q</i></span>, the value of the generalized mean <span class="texhtml"><i>M</i><sub><i>q</i>−1</sub></span> assumes familiar kinds of weighted means as special cases. In particular, </p> <ul><li><span class="texhtml"><i>q</i> = 0</span> corresponds to the weighted <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a>,</li> <li><span class="texhtml"><i>q</i> = 1</span> to the weighted <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>, and</li> <li><span class="texhtml"><i>q</i> = 2</span> to the weighted <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>.</li> <li>As <span class="texhtml"><i>q</i></span> approaches <a href="/wiki/Infinity" title="Infinity">infinity</a>, the weighted generalized mean with exponent <span class="texhtml"><i>q</i> − 1</span> approaches the maximum <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> value, which is the proportional abundance of the most abundant species in the dataset.</li></ul> <p>Generally, increasing the value of <span class="texhtml"><i>q</i></span> increases the effective weight given to the most abundant species. This leads to obtaining a larger <span class="texhtml"><i>M</i><sub><i>q</i>−1</sub></span> value and a smaller true diversity (<span class="texhtml"><i><sup>q</sup>D</i></span>) value with increasing <span class="texhtml"><i>q</i></span>. </p><p>When <span class="texhtml"><i>q</i> = 1</span>, the weighted geometric mean of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values is used, and each species is exactly weighted by its proportional abundance (in the weighted geometric mean, the weights are the exponents). When <span class="texhtml"><i>q</i> > 1</span>, the weight given to abundant species is exaggerated, and when <span class="texhtml"><i>q</i> < 1</span>, the weight given to rare species is. At <span class="texhtml"><i>q</i> = 0</span>, the species weights exactly cancel out the species proportional abundances, such that the weighted mean of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values equals <span class="texhtml">1 / <i>R</i></span> even when all species are not equally abundant. At <span class="texhtml"><i>q</i> = 0</span>, the effective number of species, <span class="texhtml"><sup>0</sup><i>D</i></span>, hence equals the actual number of species <span class="texhtml"><i>R</i></span>. In the context of diversity, <span class="texhtml"><i>q</i></span> is generally limited to non-negative values. This is because negative values of <span class="texhtml"><i>q</i></span> would give rare species so much more weight than abundant ones that <span class="texhtml"><sup><i>q</i></sup><i>D</i></span> would exceed <span class="texhtml"><i>R</i></span>.<sup id="cite_ref-Tuomisto2010a_4-2" class="reference"><a href="#cite_note-Tuomisto2010a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010c_5-2" class="reference"><a href="#cite_note-Tuomisto2010c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Richness">Richness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=3" title="Edit section: Richness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Species_richness" title="Species richness">Species richness</a></div> <p>Richness <span class="texhtml"><i>R</i></span> simply quantifies how many different types the dataset of interest contains. For example, species richness (usually noted <span class="texhtml"><i>S</i></span>) is simply the number of species, e.g. at a particular site. Richness is a simple measure, so it has been a popular diversity index in ecology, where abundance data are often not available.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> If true diversity is calculated with <span class="texhtml"><i>q</i> = 0</span>, the effective number of types (<span class="texhtml"><sup>0</sup><i>D</i></span>) equals the actual number of types, which is identical to Richness (<span class="texhtml"><i>R</i></span>).<sup id="cite_ref-Jost2006_3-3" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010c_5-3" class="reference"><a href="#cite_note-Tuomisto2010c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Shannon_index"><span class="anchor" id="Shannon_index"></span>Shannon index</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=4" title="Edit section: Shannon index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Shannon index</b> has been a popular diversity index in the ecological literature, where it is also known as <b>Shannon's diversity index</b>, <b>Shannon–<a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Wiener</a> index</b>, and (erroneously) <b>Shannon–<a href="/wiki/Warren_Weaver" title="Warren Weaver">Weaver</a> index</b>.<sup id="cite_ref-Spellerberg2003_8-0" class="reference"><a href="#cite_note-Spellerberg2003-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The measure was originally proposed by <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> in 1948 to quantify the <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a> (hence <i>Shannon entropy</i>, related to <a href="/wiki/Shannon_information_content" class="mw-redirect" title="Shannon information content">Shannon information content</a>) in strings of text.<sup id="cite_ref-Shannon1948_9-0" class="reference"><a href="#cite_note-Shannon1948-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The idea is that the more letters there are, and the closer their proportional abundances in the string of interest, the more difficult it is to correctly predict which letter will be the next one in the string. The Shannon entropy quantifies the uncertainty (entropy or degree of surprise) associated with this prediction. It is most often calculated as follows: </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'=-\sum _{i=1}^{R}p_{i}\ln p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff662d3135630b77bd4ca28c0ed99d1f5fb919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.476ex; height:7.343ex;" alt="{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}}"></span></dd></dl> <p>where <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> is the proportion of characters belonging to the <span class="texhtml"><i>i</i></span>th type of letter in the string of interest. In ecology, <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> is often the proportion of individuals belonging to the <span class="texhtml"><i>i</i></span>th species in the dataset of interest. Then the Shannon entropy quantifies the uncertainty in predicting the species identity of an individual that is taken at random from the dataset. </p><p>Although the equation is here written with natural logarithms, the base of the logarithm used when calculating the Shannon entropy can be chosen freely. Shannon himself discussed logarithm bases 2, 10 and <span class="texhtml"><i>e</i></span>, and these have since become the most popular bases in applications that use the Shannon entropy. Each log base corresponds to a different measurement unit, which has been called binary digits (bits), decimal digits (decits), and natural digits (nats) for the bases 2, 10 and <span class="texhtml"><i>e</i></span>, respectively. Comparing Shannon entropy values that were originally calculated with different log bases requires converting them to the same log base: change from the base <span class="texhtml"><i>a</i></span> to base <span class="texhtml"><i>b</i></span> is obtained with multiplication by <span class="texhtml">log<sub><i>b</i></sub><i>a</i></span>.<sup id="cite_ref-Shannon1948_9-1" class="reference"><a href="#cite_note-Shannon1948-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The Shannon index (<span class="texhtml"><i>H'</i></span>) is related to the <a href="/wiki/Weighted_geometric_mean" title="Weighted geometric mean">weighted geometric mean</a> of the proportional abundances of the types. Specifically, it equals the logarithm of true diversity as calculated with <span class="texhtml"><i>q</i> = 1</span>:<sup id="cite_ref-Tuomisto2010a_4-3" class="reference"><a href="#cite_note-Tuomisto2010a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </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'=-\sum _{i=1}^{R}p_{i}\ln p_{i}=-\sum _{i=1}^{R}\ln p_{i}^{p_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}=-\sum _{i=1}^{R}\ln p_{i}^{p_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271bd5b0f74d01654346ac6494d607c30a05000f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.691ex; height:7.343ex;" alt="{\displaystyle H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}=-\sum _{i=1}^{R}\ln p_{i}^{p_{i}}}"></span></dd></dl> <p>This can also be written </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'=-(\ln p_{1}^{p_{1}}+\ln p_{2}^{p_{2}}+\ln p_{3}^{p_{3}}+\cdots +\ln p_{R}^{p_{R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'=-(\ln p_{1}^{p_{1}}+\ln p_{2}^{p_{2}}+\ln p_{3}^{p_{3}}+\cdots +\ln p_{R}^{p_{R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095b5cd6f10ef541132cc667d4e714417ac833fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.481ex; height:3.176ex;" alt="{\displaystyle H'=-(\ln p_{1}^{p_{1}}+\ln p_{2}^{p_{2}}+\ln p_{3}^{p_{3}}+\cdots +\ln p_{R}^{p_{R}})}"></span></dd></dl> <p>which equals </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'=-\ln p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}=\ln \left({1 \over p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}}\right)=\ln \left({1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'=-\ln p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}=\ln \left({1 \over p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}}\right)=\ln \left({1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d0423ede079ae02b06f5ed8ac2ed4ec6ed8610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:70.02ex; height:7.676ex;" alt="{\displaystyle H'=-\ln p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}=\ln \left({1 \over p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}}\right)=\ln \left({1 \over {\prod _{i=1}^{R}p_{i}^{p_{i}}}}\right)}"></span></dd></dl> <p>Since the sum of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values equals 1 by definition, the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> equals the weighted geometric mean of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values, with the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values themselves being used as the weights (exponents in the equation). The term within the parentheses hence equals true diversity <span class="texhtml"><sup>1</sup><i>D</i></span>, and <span class="texhtml"><i>H'</i></span> equals <span class="texhtml">ln(<sup>1</sup><i>D</i>)</span>.<sup id="cite_ref-Hill1973_2-2" class="reference"><a href="#cite_note-Hill1973-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010a_4-4" class="reference"><a href="#cite_note-Tuomisto2010a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tuomisto2010c_5-4" class="reference"><a href="#cite_note-Tuomisto2010c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>When all types in the dataset of interest are equally common, all <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values equal <span class="texhtml">1 / <i>R</i></span>, and the Shannon index hence takes the value <span class="texhtml">ln(<i>R</i>)</span>. The more unequal the abundances of the types, the larger the weighted geometric mean of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values, and the smaller the corresponding Shannon entropy. If practically all abundance is concentrated to one type, and the other types are very rare (even if there are many of them), Shannon entropy approaches zero. When there is only one type in the dataset, Shannon entropy exactly equals zero (there is no uncertainty in predicting the type of the next randomly chosen entity). </p><p>In machine learning the Shannon index is also called as <a href="/wiki/Decision_tree_learning#Information_gain" title="Decision tree learning">Information gain</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Rényi_entropy"><span id="R.C3.A9nyi_entropy"></span>Rényi entropy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=5" title="Edit section: Rényi entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a> is a generalization of the Shannon entropy to other values of <span class="texhtml"><i>q</i></span> than 1. It can be expressed: </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 {}^{q}H={\frac {1}{1-q}}\;\ln \left(\sum _{i=1}^{R}p_{i}^{q}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{q}H={\frac {1}{1-q}}\;\ln \left(\sum _{i=1}^{R}p_{i}^{q}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c68459566e61a8b91ef5086e3437edb614607da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.612ex; height:7.509ex;" alt="{\displaystyle {}^{q}H={\frac {1}{1-q}}\;\ln \left(\sum _{i=1}^{R}p_{i}^{q}\right)}"></span></dd></dl> <p>which equals </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 {}^{q}H=\ln \left({1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}\right)=\ln({}^{q}\!D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mi>H</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </mroot> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{q}H=\ln \left({1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}\right)=\ln({}^{q}\!D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dadf05e280833af504d9f3339569ddd34e8b1b73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:37.37ex; height:10.509ex;" alt="{\displaystyle {}^{q}H=\ln \left({1 \over {\sqrt[{q-1}]{\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}\right)=\ln({}^{q}\!D)}"></span></dd></dl> <p>This means that taking the logarithm of true diversity based on any value of <span class="texhtml"><i>q</i></span> gives the Rényi entropy corresponding to the same value of <span class="texhtml"><i>q</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Simpson_index">Simpson index</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=6" title="Edit section: Simpson index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Simpson index was introduced in 1949 by <a href="/wiki/Edward_H._Simpson" title="Edward H. Simpson">Edward H. Simpson</a> to measure the degree of concentration when individuals are classified into types.<sup id="cite_ref-Simpson1949_10-0" class="reference"><a href="#cite_note-Simpson1949-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The same index was rediscovered by <a href="/wiki/Orris_C._Herfindahl" title="Orris C. Herfindahl">Orris C. Herfindahl</a> in 1950.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The square root of the index had already been introduced in 1945 by the economist <a href="/wiki/Albert_O._Hirschman" title="Albert O. Hirschman">Albert O. Hirschman</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> As a result, the same measure is usually known as the Simpson index in ecology, and as the <a href="/wiki/Herfindahl_index" class="mw-redirect" title="Herfindahl index">Herfindahl index</a> or the Herfindahl–Hirschman index (HHI) in economics. </p><p>The measure equals the probability that two entities taken at random from the dataset of interest represent the same type.<sup id="cite_ref-Simpson1949_10-1" class="reference"><a href="#cite_note-Simpson1949-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> It equals: </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 \lambda =\sum _{i=1}^{R}p_{i}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\sum _{i=1}^{R}p_{i}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bce2f20ba39f8b50aa3f9736b5fe7e4d65f5a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.066ex; height:7.343ex;" alt="{\displaystyle \lambda =\sum _{i=1}^{R}p_{i}^{2},}"></span></dd></dl> <p>where <span class="texhtml"><i>R</i></span> is richness (the total number of types in the dataset). This equation is also equal to the weighted arithmetic mean of the proportional abundances <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> of the types of interest, with the proportional abundances themselves being used as the weights.<sup id="cite_ref-Hill1973_2-3" class="reference"><a href="#cite_note-Hill1973-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Proportional abundances are by definition constrained to values between zero and one, but it is a weighted arithmetic mean, hence <span class="texhtml"><i>λ</i> ≥ 1/<i>R</i></span>, which is reached when all types are equally abundant. </p><p>By comparing the equation used to calculate λ with the equations used to calculate true diversity, it can be seen that <span class="texhtml">1/λ</span> equals <span class="texhtml"><sup>2</sup><i>D</i></span>, i.e., true diversity as calculated with <span class="texhtml"><i>q</i> = 2</span>. The original Simpson's index hence equals the corresponding basic sum.<sup id="cite_ref-Jost2006_3-4" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The interpretation of λ as the probability that two entities taken at random from the dataset of interest represent the same type assumes that the first entity is replaced to the dataset before taking the second entity. If the dataset is very large, sampling without replacement gives approximately the same result, but in small datasets, the difference can be substantial. If the dataset is small, and sampling without replacement is assumed, the probability of obtaining the same type with both random draws is: </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 \ell ={\frac {\sum _{i=1}^{R}n_{i}(n_{i}-1)}{N(N-1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ={\frac {\sum _{i=1}^{R}n_{i}(n_{i}-1)}{N(N-1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00276694949857dd6073fa3a17fa27907e39c691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.846ex; height:7.009ex;" alt="{\displaystyle \ell ={\frac {\sum _{i=1}^{R}n_{i}(n_{i}-1)}{N(N-1)}}}"></span></dd></dl> <p>where <span class="texhtml"><i>n</i><sub><i>i</i></sub></span> is the number of entities belonging to the <span class="texhtml"><i>i</i></span>th type and <span class="texhtml"><i>N</i></span> is the total number of entities in the dataset.<sup id="cite_ref-Simpson1949_10-2" class="reference"><a href="#cite_note-Simpson1949-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This form of the Simpson index is also known as the Hunter–Gaston index in microbiology.<sup id="cite_ref-Hunter1988_13-0" class="reference"><a href="#cite_note-Hunter1988-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Since the mean proportional abundance of the types increases with decreasing number of types and increasing abundance of the most abundant type, λ obtains small values in datasets of high diversity and large values in datasets of low diversity. This is counterintuitive behavior for a diversity index, so often, such transformations of λ that increase with increasing diversity have been used instead. The most popular of such indices have been the inverse Simpson index (1/λ) and the <a href="/wiki/Corrado_Gini" title="Corrado Gini">Gini</a>–Simpson index (1 − λ).<sup id="cite_ref-Hill1973_2-4" class="reference"><a href="#cite_note-Hill1973-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jost2006_3-5" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Both of these have also been called the Simpson index in the ecological literature, so care is needed to avoid accidentally comparing the different indices as if they were the same. </p> <div class="mw-heading mw-heading3"><h3 id="Inverse_Simpson_index">Inverse Simpson index</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=7" title="Edit section: Inverse Simpson index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inverse Simpson index equals: </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 {\frac {1}{\lambda }}={1 \over \sum _{i=1}^{R}p_{i}^{2}}={}^{2}D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={1 \over \sum _{i=1}^{R}p_{i}^{2}}={}^{2}D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc8eff396415812bcb1ae8a68d525cf43d83599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.168ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\lambda }}={1 \over \sum _{i=1}^{R}p_{i}^{2}}={}^{2}D}"></span></dd></dl> <p>This simply equals true diversity of order 2, i.e. the effective number of types that is obtained when the weighted arithmetic mean is used to quantify average proportional abundance of types in the dataset of interest. </p><p>The index is also used as a measure of the <a href="/wiki/Effective_number_of_parties" title="Effective number of parties">effective number of parties</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Gini–Simpson_index"><span id="Gini.E2.80.93Simpson_index"></span>Gini–Simpson index</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=8" title="Edit section: Gini–Simpson index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Gini-Simpson Index is also called <a href="/wiki/Decision_tree_learning#Gini_impurity" title="Decision tree learning">Gini impurity</a>, or <b>Gini's diversity index</b><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> in the field of <a href="/wiki/Machine_Learning" class="mw-redirect" title="Machine Learning">Machine Learning</a>. The original Simpson index λ equals the probability that two entities taken at random from the dataset of interest (with replacement) represent the same type. Its transformation 1 − λ, therefore, equals the probability that the two entities represent different types. This measure is also known in ecology as the probability of interspecific encounter (<i>PIE</i>)<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and the Gini–Simpson index.<sup id="cite_ref-Jost2006_3-6" class="reference"><a href="#cite_note-Jost2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It can be expressed as a transformation of the true diversity of order 2: </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 1-\lambda =1-\sum _{i=1}^{R}p_{i}^{2}=1-{\frac {1}{{}^{2}D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\lambda =1-\sum _{i=1}^{R}p_{i}^{2}=1-{\frac {1}{{}^{2}D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277d537e8802039177355e95c5a7ae02277b98d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.341ex; height:7.343ex;" alt="{\displaystyle 1-\lambda =1-\sum _{i=1}^{R}p_{i}^{2}=1-{\frac {1}{{}^{2}D}}}"></span></dd></dl> <p>The Gibbs–Martin index of sociology, psychology, and management studies,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> which is also known as the Blau index, is the same measure as the Gini–Simpson index. </p><p>The quantity is also known as the <a href="/wiki/Zygosity#Heterozygosity_in_population_genetics" title="Zygosity">expected heterozygosity</a> in population genetics. </p> <div class="mw-heading mw-heading2"><h2 id="Berger–Parker_index"><span id="Berger.E2.80.93Parker_index"></span>Berger–Parker index</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=9" title="Edit section: Berger–Parker index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Berger–Parker index, named after <a href="/wiki/Wolfgang_H._Berger" title="Wolfgang H. Berger">Wolfgang H. Berger</a> and <a href="/wiki/Frances_Lawrence_Parker" title="Frances Lawrence Parker">Frances Lawrence Parker</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> equals the maximum <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> value in the dataset, i.e., the proportional abundance of the most abundant type. This corresponds to the weighted <a href="/wiki/Generalized_mean" title="Generalized mean">generalized mean</a> of the <span class="texhtml"><i>p</i><sub><i>i</i></sub></span> values when <span class="texhtml"><i>q</i></span> approaches infinity, and hence equals the inverse of the true diversity of order infinity (<span class="texhtml">1/<sup>∞</sup><i>D</i></span>). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Alpha_diversity" title="Alpha diversity">Alpha diversity</a></li> <li><a href="/wiki/Beta_diversity" title="Beta diversity">Beta diversity</a></li> <li><a href="/wiki/Cultural_diversity" title="Cultural diversity">Cultural diversity</a></li> <li><a href="/wiki/Effective_number_of_parties" title="Effective number of parties">Effective number of parties</a>, a diversity index applied to political parties</li> <li><a href="/wiki/Gamma_diversity" title="Gamma diversity">Gamma diversity</a></li> <li><a href="/wiki/Generalized_entropy_index" title="Generalized entropy index">Generalized entropy index</a></li> <li><a href="/wiki/Gini_coefficient" title="Gini coefficient">Gini coefficient</a></li> <li><a href="/wiki/Isolation_index" title="Isolation index">Isolation index</a></li> <li><a href="/wiki/Measurement_of_biodiversity" title="Measurement of biodiversity">Measurement of biodiversity</a></li> <li><a href="/wiki/Qualitative_variation" title="Qualitative variation">Qualitative variation</a></li> <li><a href="/wiki/Relative_species_abundance" title="Relative species abundance">Relative abundance</a></li> <li><a href="/wiki/Species_diversity" title="Species diversity">Species diversity</a></li> <li><a href="/wiki/Species_richness" title="Species richness">Species richness</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversity_index&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 32em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTuckerCadotteCarvalhoDavies2017" class="citation journal cs1">Tucker, Caroline M.; Cadotte, Marc W.; Carvalho, Silvia B.; Davies, T. 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"Urbanization, technology and the division of labor". <i><a href="/wiki/American_Sociological_Review" title="American Sociological Review">American Sociological Review</a></i>. <b>27</b> (5): 667–677. <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%2F2089624">10.2307/2089624</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/2089624">2089624</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Sociological+Review&rft.atitle=Urbanization%2C+technology+and+the+division+of+labor&rft.volume=27&rft.issue=5&rft.pages=667-677&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F2089624&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2089624%23id-name%3DJSTOR&rft.au=Gibbs%2C+Jack+P.&rft.au=William+T.+Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversity+index" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBergerParker1970" class="citation journal cs1">Berger, Wolfgang H.; Parker, Frances L. (June 1970). "Diversity of Planktonic Foraminifera in Deep-Sea Sediments". <i><a href="/wiki/Science_(journal)" title="Science (journal)">Science</a></i>. <b>168</b> (3937): 1345–1347. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970Sci...168.1345B">1970Sci...168.1345B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.168.3937.1345">10.1126/science.168.3937.1345</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/17731043">17731043</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29553922">29553922</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Diversity+of+Planktonic+Foraminifera+in+Deep-Sea+Sediments&rft.volume=168&rft.issue=3937&rft.pages=1345-1347&rft.date=1970-06&rft_id=info%3Adoi%2F10.1126%2Fscience.168.3937.1345&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A29553922%23id-name%3DS2CID&rft_id=info%3Apmid%2F17731043&rft_id=info%3Abibcode%2F1970Sci...168.1345B&rft.aulast=Berger&rft.aufirst=Wolfgang+H.&rft.au=Parker%2C+Frances+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversity+index" 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=Diversity_index&action=edit&section=12" 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="CITEREFColinvaux,_Paul_A.1973" class="citation book cs1"><a href="/wiki/Paul_Colinvaux" title="Paul Colinvaux">Colinvaux, Paul A.</a> (1973). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec00coli"><i>Introduction to Ecology</i></a></span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-16498-4" title="Special:BookSources/0-471-16498-4"><bdi>0-471-16498-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Ecology&rft.pub=Wiley&rft.date=1973&rft.isbn=0-471-16498-4&rft.au=Colinvaux%2C+Paul+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoec00coli&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversity+index" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCover,_Thomas_M.Thomas,_Joy_A.1991" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (1991). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementsofinform0000cove"><i>Elements of Information Theory</i></a></span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-06259-6" title="Special:BookSources/0-471-06259-6"><bdi>0-471-06259-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Information+Theory&rft.pub=Wiley&rft.date=1991&rft.isbn=0-471-06259-6&rft.au=Cover%2C+Thomas+M.&rft.au=Thomas%2C+Joy+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsofinform0000cove&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversity+index" class="Z3988"></span> See chapter 5 for an elaboration of coding procedures described informally above.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaoShen2003" class="citation journal cs1"><a href="/wiki/Anne_Chao" title="Anne Chao">Chao, A.</a>; Shen, T-J. (2003). <a rel="nofollow" class="external text" href="http://chao.stat.nthu.edu.tw/paper/2003_EEST_10_P429.pdf">"Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample"</a> <span class="cs1-format">(PDF)</span>. <i>Environmental and Ecological Statistics</i>. <b>10</b> (4): 429–443. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1026096204727">10.1023/A:1026096204727</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:20389926">20389926</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Environmental+and+Ecological+Statistics&rft.atitle=Nonparametric+estimation+of+Shannon%27s+index+of+diversity+when+there+are+unseen+species+in+sample&rft.volume=10&rft.issue=4&rft.pages=429-443&rft.date=2003&rft_id=info%3Adoi%2F10.1023%2FA%3A1026096204727&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A20389926%23id-name%3DS2CID&rft.aulast=Chao&rft.aufirst=A.&rft.au=Shen%2C+T-J.&rft_id=http%3A%2F%2Fchao.stat.nthu.edu.tw%2Fpaper%2F2003_EEST_10_P429.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversity+index" 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=Diversity_index&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.countrysideinfo.co.uk/simpsons.htm">Simpson's Diversity index</a></li> <li><a rel="nofollow" class="external text" href="http://www.tiem.utk.edu/~gross/bioed/bealsmodules/simpsonDI.html">Diversity indices</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051219213715/http://www.tiem.utk.edu/~gross/bioed/bealsmodules/simpsonDI.html">Archived</a> 2005-12-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> gives some examples of estimates of Simpson's index for real ecosystems.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Diversity_index&oldid=1240968911">https://en.wikipedia.org/w/index.php?title=Diversity_index&oldid=1240968911</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Measurement_of_biodiversity" title="Category:Measurement of biodiversity">Measurement of biodiversity</a></li><li><a href="/wiki/Category:Index_numbers" title="Category:Index numbers">Index numbers</a></li><li><a href="/wiki/Category:Summary_statistics_for_categorical_data" title="Category:Summary statistics for categorical data">Summary statistics for categorical data</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_rewrite_from_April_2020" title="Category:Wikipedia articles needing rewrite from April 2020">Wikipedia articles needing rewrite from April 2020</a></li><li><a href="/wiki/Category:All_articles_needing_rewrite" title="Category:All articles needing rewrite">All articles needing rewrite</a></li><li><a href="/wiki/Category:Wikipedia_articles_that_are_too_technical_from_April_2020" title="Category:Wikipedia articles that are too technical from April 2020">Wikipedia articles that are too technical from April 2020</a></li><li><a href="/wiki/Category:All_articles_that_are_too_technical" title="Category:All articles that are too technical">All articles that are too technical</a></li><li><a href="/wiki/Category:Articles_with_multiple_maintenance_issues" title="Category:Articles with multiple maintenance issues">Articles with multiple maintenance issues</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 August 2024, at 15:34<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; 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Diversity index - Wikipedia