Anderson–Darling test

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<b>Anderson–Darling test</b> is a <a href="/wiki/Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">statistical test</a> of whether a given sample of data is drawn from a given <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>. In its basic form, the test assumes that there are no parameters to be estimated in the distribution being tested, in which case the test and its set of <a href="/wiki/Critical_value_(statistics)" class="mw-redirect" title="Critical value (statistics)">critical values</a> is distribution-free. However, the test is most often used in contexts where a family of distributions is being tested, in which case the parameters of that family need to be estimated and account must be taken of this in adjusting either the test-statistic or its critical values. When applied to testing whether a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> adequately describes a set of data, it is one of the most powerful statistical tools for detecting most departures from <a href="/wiki/Normal_distribution" title="Normal distribution">normality</a>.<sup id="cite_ref-Stephens74_1-0" class="reference"><a href="#cite_note-Stephens74-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stephens86_2-0" class="reference"><a href="#cite_note-Stephens86-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <b><i>K</i>-sample Anderson–Darling tests</b> are available for testing whether several collections of observations can be modelled as coming from a single population, where the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">distribution function</a> does not have to be specified. </p><p>In addition to its use as a test of fit for distributions, it can be used in parameter estimation as the basis for a form of <a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">minimum distance estimation</a> procedure. </p><p>The test is named after <a href="/wiki/Theodore_Wilbur_Anderson" title="Theodore Wilbur Anderson">Theodore Wilbur Anderson</a> (1918–2016) and <a href="/wiki/Donald_Allan_Darling" title="Donald Allan Darling">Donald A. Darling</a> (1915–2014), who invented it in 1952.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_single-sample_test">The single-sample test</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=1" title="Edit section: The single-sample test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Anderson–Darling and <a href="/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion" title="Cramér–von Mises criterion">Cramér–von Mises statistics</a> belong to the class of quadratic <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">EDF</a> statistics (tests based on the <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a>).<sup id="cite_ref-Stephens86_2-1" class="reference"><a href="#cite_note-Stephens86-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> If the hypothesized distribution is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, and empirical (sample) cumulative distribution function is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span>, then the quadratic EDF statistics measure the distance between <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></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 F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> by </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 n\int _{-\infty }^{\infty }(F_{n}(x)-F(x))^{2}\,w(x)\,dF(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\int _{-\infty }^{\infty }(F_{n}(x)-F(x))^{2}\,w(x)\,dF(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e70d6bb7b44c6332ef3643ede22ad11693f0ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.984ex; height:6.009ex;" alt="{\displaystyle n\int _{-\infty }^{\infty }(F_{n}(x)-F(x))^{2}\,w(x)\,dF(x),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> is the number of elements in the sample, 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 w(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504b9ce86ec0b1d4267109ee950497126f04713c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.803ex; height:2.843ex;" alt="{\displaystyle w(x)}"></span> is a weighting function. When the weighting function is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(x)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c180b2dd805adc4a3afffa7d44bf3bbff94de42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.064ex; height:2.843ex;" alt="{\displaystyle w(x)=1}"></span>, the statistic is the <a href="/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion" title="Cramér–von Mises criterion">Cramér–von Mises statistic</a>. The Anderson–Darling (1954) test<sup id="cite_ref-AD54_4-0" class="reference"><a href="#cite_note-AD54-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> is based on the distance </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 A^{2}=n\int _{-\infty }^{\infty }{\frac {(F_{n}(x)-F(x))^{2}}{F(x)\;(1-F(x))}}\,dF(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>n</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=n\int _{-\infty }^{\infty }{\frac {(F_{n}(x)-F(x))^{2}}{F(x)\;(1-F(x))}}\,dF(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777bb687cd72016605447d7b987150a2372e4943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.913ex; height:6.676ex;" alt="{\displaystyle A^{2}=n\int _{-\infty }^{\infty }{\frac {(F_{n}(x)-F(x))^{2}}{F(x)\;(1-F(x))}}\,dF(x),}"></span></dd></dl> <p>which is obtained when the weight function is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(x)=[F(x)\;(1-F(x))]^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(x)=[F(x)\;(1-F(x))]^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73cadb973dbd6f188470533d018bd65316c2e7af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.745ex; height:3.176ex;" alt="{\displaystyle w(x)=[F(x)\;(1-F(x))]^{-1}}"></span>. Thus, compared with the <a href="/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion" title="Cramér–von Mises criterion">Cramér–von Mises distance</a>, the Anderson–Darling distance places more weight on observations in the tails of the distribution. </p> <div class="mw-heading mw-heading3"><h3 id="Basic_test_statistic">Basic test statistic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=2" title="Edit section: Basic test statistic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Anderson–Darling test assesses whether a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> comes from a specified distribution. It makes use of the fact that, when given a hypothesized underlying distribution and assuming the data does arise from this distribution, the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) of the data can be transformed to what should follow a <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a>. The data can be then tested for uniformity with a distance test (Shapiro 1980). The formula for the <a href="/wiki/Test_statistic" title="Test statistic">test statistic</a> <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"> <mi>A</mi> </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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> to assess if 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 \{Y_{1}<\cdots <Y_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{1}<\cdots <Y_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e51ab4a37fecf15a1a82d2ed01408653d40af8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.219ex; height:2.843ex;" alt="{\displaystyle \{Y_{1}<\cdots <Y_{n}\}}"></span> (note that the data must be put in order) comes from a CDF <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> 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 A^{2}=-n-S\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=-n-S\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cf14f1073792dddad38e02aa56c3fcebad1c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.472ex; height:3.009ex;" alt="{\displaystyle A^{2}=-n-S\,,}"></span></dd></dl> <p>where </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 S=\sum _{i=1}^{n}{\frac {2i-1}{n}}\left[\ln(F(Y_{i}))+\ln \left(1-F(Y_{n+1-i})\right)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</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>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{i=1}^{n}{\frac {2i-1}{n}}\left[\ln(F(Y_{i}))+\ln \left(1-F(Y_{n+1-i})\right)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ce0f444154cb2dfe1b762d60cd5eea0f7254b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.965ex; height:6.843ex;" alt="{\displaystyle S=\sum _{i=1}^{n}{\frac {2i-1}{n}}\left[\ln(F(Y_{i}))+\ln \left(1-F(Y_{n+1-i})\right)\right].}"></span></dd></dl> <p>The test statistic can then be compared against the critical values of the theoretical distribution. In this case, no parameters are estimated in relation to the cumulative distribution function <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Tests_for_families_of_distributions">Tests for families of distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=3" title="Edit section: Tests for families of distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Essentially the same test statistic can be used in the test of fit of a family of distributions, but then it must be compared against the critical values appropriate to that family of theoretical distributions and dependent also on the method used for parameter estimation. </p> <div class="mw-heading mw-heading3"><h3 id="Test_for_normality">Test for normality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=4" title="Edit section: Test for normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Empirical testing has found<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> that the Anderson–Darling test is not quite as good as the <a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Shapiro–Wilk test</a>, but is better than other tests. Stephens<sup id="cite_ref-Stephens74_1-1" class="reference"><a href="#cite_note-Stephens74-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> found <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{2}}"></span> to be one of the best <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> statistics for detecting most departures from normality. </p><p>The computation differs based on what is known about the distribution:<sup id="cite_ref-RBD86_6-0" class="reference"><a href="#cite_note-RBD86-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Case 0: The mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and the variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> are both known.</li> <li>Case 1: The variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> is known, but the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is unknown.</li> <li>Case 2: The mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is known, but the variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> is unknown.</li> <li>Case 3: Both the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and the variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> are unknown.</li></ul> <p>The <i>n</i> observations, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span>, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079ea2e196dd04a164f5dd42b4b4e0827471a35d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.602ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots n}"></span>, of the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> must be sorted such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\leq X_{2}\leq ...\leq X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>≤</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\leq X_{2}\leq ...\leq X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b809a9580620ffa45f7fe5b50812f3e850a64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.207ex; height:2.509ex;" alt="{\displaystyle X_{1}\leq X_{2}\leq ...\leq X_{n}}"></span> and the notation in the following assumes that <i>X<sub>i</sub></i> represent the ordered observations. Let </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 {\hat {\mu }}={\begin{cases}\mu ,&{\text{if the mean is known.}}\\{\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i},&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>μ</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if the mean is known.</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}={\begin{cases}\mu ,&{\text{if the mean is known.}}\\{\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i},&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f364d1c6c5e7f554ec2672773d514b194ea4879f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.914ex; height:6.176ex;" alt="{\displaystyle {\hat {\mu }}={\begin{cases}\mu ,&{\text{if the mean is known.}}\\{\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i},&{\text{otherwise.}}\end{cases}}}"></span></dd></dl> <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 {\hat {\sigma }}^{2}={\begin{cases}\sigma ^{2},&{\text{if the variance is known.}}\\{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},&{\text{if the variance is not known, but the mean is.}}\\{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2},&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if the variance is known.</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if the variance is not known, but the mean is.</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{2}={\begin{cases}\sigma ^{2},&{\text{if the variance is known.}}\\{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},&{\text{if the variance is not known, but the mean is.}}\\{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2},&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7392f3fd62eb7d3e77b395f3313ec8f357362ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:76.198ex; height:10.176ex;" alt="{\displaystyle {\hat {\sigma }}^{2}={\begin{cases}\sigma ^{2},&{\text{if the variance is known.}}\\{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},&{\text{if the variance is not known, but the mean is.}}\\{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2},&{\text{otherwise.}}\end{cases}}}"></span></dd></dl> <p>The values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> are standardized to create new values <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 Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}"></span>, given by </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 Y_{i}={\frac {X_{i}-{\hat {\mu }}}{\hat {\sigma }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}={\frac {X_{i}-{\hat {\mu }}}{\hat {\sigma }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7d3e3fd75270c428c8e2b4afb9a12a70e0f710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.698ex; height:5.676ex;" alt="{\displaystyle Y_{i}={\frac {X_{i}-{\hat {\mu }}}{\hat {\sigma }}}.}"></span></dd></dl> <p>With the standard normal CDF <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span>, <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{2}}"></span> is calculated using </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 A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}(2i-1)(\ln \Phi (Y_{i})+\ln(1-\Phi (Y_{n+1-i}))).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}(2i-1)(\ln \Phi (Y_{i})+\ln(1-\Phi (Y_{n+1-i}))).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf20fe76ba5d13018ea3a8724bcf15ad9d57b171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.704ex; height:6.843ex;" alt="{\displaystyle A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}(2i-1)(\ln \Phi (Y_{i})+\ln(1-\Phi (Y_{n+1-i}))).}"></span></dd></dl> <p>An alternative expression in which only a single observation is dealt with at each step of the summation 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 A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}\left[(2i-1)\ln \Phi (Y_{i})+(2(n-i)+1)\ln(1-\Phi (Y_{i}))\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}\left[(2i-1)\ln \Phi (Y_{i})+(2(n-i)+1)\ln(1-\Phi (Y_{i}))\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b32681af767ed36b1dae7635f74f4909c5e01a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.193ex; height:6.843ex;" alt="{\displaystyle A^{2}=-n-{\frac {1}{n}}\sum _{i=1}^{n}\left[(2i-1)\ln \Phi (Y_{i})+(2(n-i)+1)\ln(1-\Phi (Y_{i}))\right].}"></span></dd></dl> <p>A modified statistic can be calculated using </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 A^{*2}={\begin{cases}A^{2}\left(1+{\frac {4}{n}}-{\frac {25}{n^{2}}}\right),&{\text{if the variance and the mean are both unknown.}}\\A^{2},&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if the variance and the mean are both unknown.</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}={\begin{cases}A^{2}\left(1+{\frac {4}{n}}-{\frac {25}{n^{2}}}\right),&{\text{if the variance and the mean are both unknown.}}\\A^{2},&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c2efbf987d724d9e9fe6b35a9f9f394c5c90b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:77.724ex; height:7.509ex;" alt="{\displaystyle A^{*2}={\begin{cases}A^{2}\left(1+{\frac {4}{n}}-{\frac {25}{n^{2}}}\right),&{\text{if the variance and the mean are both unknown.}}\\A^{2},&{\text{otherwise.}}\end{cases}}}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{2}}"></span> or <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^{*2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d3541c028456c2c49f359f1e2d915b15aeaf8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.619ex; height:2.676ex;" alt="{\displaystyle A^{*2}}"></span> exceeds a given critical value, then the hypothesis of normality is rejected with some significance level. The critical values are given in the table below for values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d3541c028456c2c49f359f1e2d915b15aeaf8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.619ex; height:2.676ex;" alt="{\displaystyle A^{*2}}"></span>.<sup id="cite_ref-Stephens74_1-2" class="reference"><a href="#cite_note-Stephens74-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-Marsaglia04_7-0" class="reference"><a href="#cite_note-Marsaglia04-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Note 1: If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e130d3780e4fc508e843485b3cfbf8347ba288fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {\sigma }}}"></span> = 0 or any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (Y_{i})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (Y_{i})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7222cf38425fec5b10c60c4ace805c20af1b1cf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.091ex; height:2.843ex;" alt="{\displaystyle \Phi (Y_{i})=}"></span>(0 or 1) then <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{2}}"></span> cannot be calculated and is undefined. </p><p>Note 2: The above adjustment formula is taken from Shorack & Wellner (1986, p239). Care is required in comparisons across different sources as often the specific adjustment formula is not stated. </p><p>Note 3: Stephens<sup id="cite_ref-Stephens74_1-3" class="reference"><a href="#cite_note-Stephens74-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> notes that the test becomes better when the parameters are computed from the data, even if they are known. </p><p>Note 4: Marsaglia & Marsaglia<sup id="cite_ref-Marsaglia04_7-1" class="reference"><a href="#cite_note-Marsaglia04-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> provide a more accurate result for Case 0 at 85% and 99%. </p> <table class="wikitable"> <tbody><tr> <th>Case</th> <th><i>n</i></th> <th>15%</th> <th>10%</th> <th>5%</th> <th>2.5%</th> <th>1% </th></tr> <tr> <td>0</td> <td>≥ 5</td> <td>1.621</td> <td>1.933</td> <td>2.492</td> <td>3.070</td> <td>3.878 </td></tr> <tr> <td>1</td> <td></td> <td></td> <td>0.908</td> <td>1.105</td> <td>1.304</td> <td>1.573 </td></tr> <tr> <td>2</td> <td>≥ 5</td> <td></td> <td>1.760</td> <td>2.323</td> <td>2.904</td> <td>3.690 </td></tr> <tr> <td>3</td> <td>10</td> <td>0.514</td> <td>0.578</td> <td>0.683</td> <td>0.779</td> <td>0.926 </td></tr> <tr> <td></td> <td>20</td> <td>0.528</td> <td>0.591</td> <td>0.704</td> <td>0.815</td> <td>0.969 </td></tr> <tr> <td></td> <td>50</td> <td>0.546</td> <td>0.616</td> <td>0.735</td> <td>0.861</td> <td>1.021 </td></tr> <tr> <td></td> <td>100</td> <td>0.559</td> <td>0.631</td> <td>0.754</td> <td>0.884</td> <td>1.047 </td></tr> <tr> <td></td> <td><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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span></td> <td>0.576</td> <td>0.656</td> <td>0.787</td> <td>0.918</td> <td>1.092 </td></tr></tbody></table> <p>Alternatively, for case 3 above (both mean and variance unknown), D'Agostino (1986) <sup id="cite_ref-RBD86_6-1" class="reference"><a href="#cite_note-RBD86-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> in Table 4.7 on p. 123 and on pages 372–373 gives the adjusted statistic: </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 A^{*2}=A^{2}\left(1+{\frac {0.75}{n}}+{\frac {2.25}{n^{2}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.75</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.25</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}=A^{2}\left(1+{\frac {0.75}{n}}+{\frac {2.25}{n^{2}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3be508a82e3a0bfdf861c95702d9385a01b6a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.141ex; height:6.176ex;" alt="{\displaystyle A^{*2}=A^{2}\left(1+{\frac {0.75}{n}}+{\frac {2.25}{n^{2}}}\right).}"></span></dd></dl> <p>and normality is rejected if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d3541c028456c2c49f359f1e2d915b15aeaf8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.619ex; height:2.676ex;" alt="{\displaystyle A^{*2}}"></span> exceeds 0.631, 0.754, 0.884, 1.047, or 1.159 at 10%, 5%, 2.5%, 1%, and 0.5% significance levels, respectively; the procedure is valid for sample size at least n=8. The formulas for computing the <a href="/wiki/P-value" title="P-value"><i>p</i>-values</a> for other values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d3541c028456c2c49f359f1e2d915b15aeaf8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.619ex; height:2.676ex;" alt="{\displaystyle A^{*2}}"></span> are given in Table 4.9 on p. 127 in the same book. </p> <div class="mw-heading mw-heading3"><h3 id="Tests_for_other_distributions">Tests for other distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=5" title="Edit section: Tests for other distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Above, it was assumed that the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> was being tested for normal distribution. Any other family of distributions can be tested but the test for each family is implemented by using a different modification of the basic test statistic and this is referred to critical values specific to that family of distributions. The modifications of the statistic and tables of critical values are given by Stephens (1986)<sup id="cite_ref-Stephens86_2-2" class="reference"><a href="#cite_note-Stephens86-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a> can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications to the test statistic and for the critical values for the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> and the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a>, are also given by Shorack & Wellner (1986, p239). Details for the <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a> are given by Stephens (1979). A test for the (two parameter) <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a> can be obtained by making use of the fact that the logarithm of a Weibull variate has a <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Non-parametric_k-sample_tests">Non-parametric <i>k</i>-sample tests</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=6" title="Edit section: Non-parametric k-sample tests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fritz Scholz and Michael A. Stephens (1987) discuss a test, based on the Anderson–Darling measure of agreement between distributions, for whether a number of random samples with possibly different sample sizes may have arisen from the same distribution, where this distribution is unspecified.<sup id="cite_ref-ksample_8-0" class="reference"><a href="#cite_note-ksample-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/R_(programming_language)" title="R (programming language)">R package</a> kSamples and the <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> package <a href="/wiki/SciPy" title="SciPy">Scipy</a> implements this rank test for comparing k samples among several other such rank tests.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> samples the statistic can be computed as follows under the assumption that the distribution function <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle F_{i}}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th sample is continuous </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 A_{kN}^{2}={\frac {1}{N}}\sum _{i=1}^{k}{\frac {1}{n_{i}}}\sum _{j=1}^{N-1}{\frac {(NM_{ij}-jn_{i})^{2}}{j(N-j)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi>j</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>j</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{kN}^{2}={\frac {1}{N}}\sum _{i=1}^{k}{\frac {1}{n_{i}}}\sum _{j=1}^{N-1}{\frac {(NM_{ij}-jn_{i})^{2}}{j(N-j)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354aab0da7c22906519aeba13f1c3cb805bdb06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.27ex; height:7.676ex;" alt="{\displaystyle A_{kN}^{2}={\frac {1}{N}}\sum _{i=1}^{k}{\frac {1}{n_{i}}}\sum _{j=1}^{N-1}{\frac {(NM_{ij}-jn_{i})^{2}}{j(N-j)}}}"></span></dd></dl> <p>where </p> <ul><li><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span> is the number of observations in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th sample</li> <li><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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the total number of observations in all samples</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1}<\cdots <Z_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{1}<\cdots <Z_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8431075a387a4c1bdbc9e4c9acc39a997d8521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.841ex; height:2.509ex;" alt="{\displaystyle Z_{1}<\cdots <Z_{N}}"></span> is the pooled ordered sample</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6136d919b3f10cc58cc023e05a5c36b2f44e111f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.731ex; height:2.843ex;" alt="{\displaystyle M_{ij}}"></span> is the number of observations in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th sample that are not greater than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64717aa20b0ceb8c112f8a94cac1a3acd1ea58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.497ex; height:2.843ex;" alt="{\displaystyle Z_{j}}"></span>.<sup id="cite_ref-ksample_8-1" class="reference"><a href="#cite_note-ksample-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a></li> <li><a href="/wiki/Kuiper%27s_test" title="Kuiper's test">Kuiper's test</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Shapiro–Wilk test</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera test</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Stephens74-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stephens74_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stephens74_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Stephens74_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Stephens74_1-3"><sup><i><b>d</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="CITEREFStephens1974" class="citation journal cs1">Stephens, M. A. (1974). "EDF Statistics for Goodness of Fit and Some Comparisons". <i>Journal of the American Statistical Association</i>. <b>69</b> (347): 730–737. <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%2F2286009">10.2307/2286009</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/2286009">2286009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=EDF+Statistics+for+Goodness+of+Fit+and+Some+Comparisons&rft.volume=69&rft.issue=347&rft.pages=730-737&rft.date=1974&rft_id=info%3Adoi%2F10.2307%2F2286009&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2286009%23id-name%3DJSTOR&rft.aulast=Stephens&rft.aufirst=M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-Stephens86-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stephens86_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stephens86_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Stephens86_2-2"><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="CITEREFM._A._Stephens1986" class="citation book cs1">M. A. Stephens (1986). "Tests Based on EDF Statistics". In D'Agostino, R. B.; Stephens, M. A. (eds.). <i>Goodness-of-Fit Techniques</i>. New York: Marcel Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8247-7487-6" title="Special:BookSources/0-8247-7487-6"><bdi>0-8247-7487-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tests+Based+on+EDF+Statistics&rft.btitle=Goodness-of-Fit+Techniques&rft.place=New+York&rft.pub=Marcel+Dekker&rft.date=1986&rft.isbn=0-8247-7487-6&rft.au=M.+A.+Stephens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonDarling,_D._A.1952" class="citation journal cs1"><a href="/wiki/Theodore_W._Anderson" class="mw-redirect" title="Theodore W. Anderson">Anderson, T. W.</a>; <a href="/wiki/Donald_Allan_Darling" title="Donald Allan Darling">Darling, D. A.</a> (1952). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177729437">"Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes"</a>. <i>Annals of Mathematical Statistics</i>. <b>23</b> (2): 193–212. <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/10.1214%2Faoms%2F1177729437">10.1214/aoms/1177729437</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Asymptotic+theory+of+certain+%22goodness-of-fit%22+criteria+based+on+stochastic+processes&rft.volume=23&rft.issue=2&rft.pages=193-212&rft.date=1952&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177729437&rft.aulast=Anderson&rft.aufirst=T.+W.&rft.au=Darling%2C+D.+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177729437&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-AD54-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-AD54_4-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson,_T.W.Darling,_D.A.1954" class="citation journal cs1">Anderson, T.W.; Darling, D.A. (1954). "A Test of Goodness-of-Fit". <i>Journal of the American Statistical Association</i>. <b>49</b> (268): 765–769. <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%2F2281537">10.2307/2281537</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/2281537">2281537</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=A+Test+of+Goodness-of-Fit&rft.volume=49&rft.issue=268&rft.pages=765-769&rft.date=1954&rft_id=info%3Adoi%2F10.2307%2F2281537&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2281537%23id-name%3DJSTOR&rft.au=Anderson%2C+T.W.&rft.au=Darling%2C+D.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRazaliWah,_Yap_Bee2011" class="citation journal cs1">Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests". <i>Journal of Statistical Modeling and Analytics</i>. <b>2</b> (1): 21–33.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Modeling+and+Analytics&rft.atitle=Power+comparisons+of+Shapiro%E2%80%93Wilk%2C+Kolmogorov%E2%80%93Smirnov%2C+Lilliefors+and+Anderson%E2%80%93Darling+tests&rft.volume=2&rft.issue=1&rft.pages=21-33&rft.date=2011&rft.aulast=Razali&rft.aufirst=Nornadiah&rft.au=Wah%2C+Yap+Bee&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-RBD86-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-RBD86_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-RBD86_6-1"><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="CITEREFRalph_B._D'Agostino1986" class="citation book cs1">Ralph B. 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New York: Marcel Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8247-7487-6" title="Special:BookSources/0-8247-7487-6"><bdi>0-8247-7487-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tests+for+the+Normal+Distribution&rft.btitle=Goodness-of-Fit+Techniques&rft.place=New+York&rft.pub=Marcel+Dekker&rft.date=1986&rft.isbn=0-8247-7487-6&rft.au=Ralph+B.+D%27Agostino&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-Marsaglia04-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Marsaglia04_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Marsaglia04_7-1"><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="CITEREFMarsaglia2004" class="citation journal cs1">Marsaglia, G. (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v009.i02">"Evaluating the Anderson-Darling Distribution"</a>. <i>Journal of Statistical Software</i>. <b>9</b> (2): 730–737. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.686.1363">10.1.1.686.1363</a></span>. <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/10.18637%2Fjss.v009.i02">10.18637/jss.v009.i02</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Software&rft.atitle=Evaluating+the+Anderson-Darling+Distribution&rft.volume=9&rft.issue=2&rft.pages=730-737&rft.date=2004&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.686.1363%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.18637%2Fjss.v009.i02&rft.aulast=Marsaglia&rft.aufirst=G.&rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v009.i02&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-ksample-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-ksample_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ksample_8-1"><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="CITEREFScholzStephens1987" class="citation journal cs1">Scholz, F. W.; Stephens, M. A. (1987). "K-sample Anderson–Darling Tests". <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>. <b>82</b> (399): 918–924. <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%2F01621459.1987.10478517">10.1080/01621459.1987.10478517</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=K-sample+Anderson%E2%80%93Darling+Tests&rft.volume=82&rft.issue=399&rft.pages=918-924&rft.date=1987&rft_id=info%3Adoi%2F10.1080%2F01621459.1987.10478517&rft.aulast=Scholz&rft.aufirst=F.+W.&rft.au=Stephens%2C+M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/kSamples/index.html">"kSamples: K-Sample Rank Tests and their Combinations"</a>. <i>R Project</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=R+Project&rft.atitle=kSamples%3A+K-Sample+Rank+Tests+and+their+Combinations&rft_id=https%3A%2F%2Fcran.r-project.org%2Fweb%2Fpackages%2FkSamples%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.anderson_ksamp.html">"The Anderson-Darling test for k-samples. Scipy package"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Anderson-Darling+test+for+k-samples.+Scipy+package.&rft_id=https%3A%2F%2Fdocs.scipy.org%2Fdoc%2Fscipy%2Freference%2Fgenerated%2Fscipy.stats.anderson_ksamp.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnderson%E2%80%93Darling+test" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anderson%E2%80%93Darling_test&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Corder, G.W., Foreman, D.I. (2009).<i>Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach</i> Wiley, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-45461-9" title="Special:BookSources/978-0-470-45461-9">978-0-470-45461-9</a></li> <li>Mehta, S. (2014) <i>Statistics Topics</i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1499273533" title="Special:BookSources/978-1499273533">978-1499273533</a></li> <li>Pearson E.S., Hartley, H.O. (Editors) (1972) <i>Biometrika Tables for Statisticians</i>, Volume II. CUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-06937-8" title="Special:BookSources/0-521-06937-8">0-521-06937-8</a>.</li> <li>Shapiro, S.S. (1980) How to test normality and other distributional assumptions. In: The ASQC basic references in quality control: statistical techniques 3, pp. 1–78.</li> <li><a href="/wiki/Galen_Shorack" title="Galen Shorack">Shorack, G.R.</a>, <a href="/wiki/Jon_A._Wellner" title="Jon A. Wellner">Wellner, J.A.</a> (1986) <i>Empirical Processes with Applications to Statistics</i>, Wiley. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-86725-X" title="Special:BookSources/0-471-86725-X">0-471-86725-X</a>.</li> <li>Stephens, M.A. (1979) <i>Test of fit for the logistic distribution based on the empirical distribution function</i>, Biometrika, 66(3), 591–5.</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=Anderson%E2%80%93Darling_test&action=edit&section=10" 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.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm">US NIST Handbook of Statistics</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Anderson–Darling_test&oldid=1258925848">https://en.wikipedia.org/w/index.php?title=Anderson–Darling_test&oldid=1258925848</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:Nonparametric_statistics" title="Category:Nonparametric statistics">Nonparametric statistics</a></li><li><a href="/wiki/Category:Normality_tests" title="Category:Normality tests">Normality tests</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></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 22 November 2024, at 11:29<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>; additional terms may apply. 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Anderson–Darling test - Wikipedia