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Chi-squared distribution
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{{Short description|Probability distribution and special case of gamma distribution}} {{About|the mathematics of the chi-squared distribution|its uses in statistics|chi-squared test|the music group|Chi2 (band)}} {{Probability distribution | name = Chi-squared | type = density | pdf_image = [[File:Chi-square pdf.svg|321px]] | cdf_image = [[File:Chi-square cdf.svg|321px]] | notation = <math>\chi^2(k)\;</math> or <math>\chi^2_k\!</math> | parameters = <math>k \in \mathbb{N}^{*}~~</math> (known as "degrees of freedom") | support = <math>x \in (0, +\infty)\;</math> | pdf = <math>\frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\; </math> | cdf = <math>\frac{1}{\Gamma(k/2 )} \; \gamma\left(\frac{k}{2},\,\frac{x}{2}\right)\;</math> | mean = <math>k</math> | median = <math>\approx k\bigg(1-\frac{2}{9k}\bigg)^3\;</math> | mode = <math>\max(k-2,0)\;</math> | variance = <math>2k\;</math> | skewness = <math>\sqrt{8/k}\,</math> | kurtosis = <math>\frac{12}{k}</math> | entropy = <math>\begin{align}\frac{k}{2}&+\log\left(2\Gamma\Bigl(\frac{k}{2}\Bigr)\right) \\ &\!+\left(1-\frac{k}{2}\right)\psi\left(\frac{k}{2}\right)\end{align}</math> | mgf = <math>(1-2t)^{-k/2} \text{ for } t < \frac{1}{2}\;</math> | char = <math>(1-2it)^{-k/2}</math><ref>{{cite web | url=http://www.planetmathematics.com/CentralChiDistr.pdf | title=Characteristic function of the central chi-square distribution | author=M.A. Sanders | access-date=2009-03-06 | archive-url=https://web.archive.org/web/20110715091705/http://www.planetmathematics.com/CentralChiDistr.pdf | archive-date=2011-07-15 | url-status=dead }}</ref> |pgf=<math>(1-2\ln t)^{-k/2} \text{ for } 0<t<\sqrt{e}\;</math>}} In [[probability theory]] and [[statistics]], the '''<math>\chi^2</math>-distribution''' with <math>k</math> [[Degrees of freedom (statistics)|degrees of freedom]] is the distribution of a sum of the squares of <math>k</math> [[Independence (probability theory)|independent]] [[standard normal]] random variables.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Chi-Squared Distribution |url=https://mathworld.wolfram.com/Chi-SquaredDistribution.html |access-date=2024-10-11 |website=mathworld.wolfram.com |language=en}}</ref> The chi-squared distribution <math> \chi^2_k </math> is a special case of the [[gamma distribution]] and the univariate [[Wishart distribution]]. Specifically if <math> X \sim \chi^2_k </math> then <math> X \sim \text{Gamma}(\alpha=\frac{k}{2}, \theta=2) </math> (where <math>\alpha</math> is the shape parameter and <math>\theta</math> the scale parameter of the gamma distribution) and <math> X \sim \text{W}_1(1,k) </math>. The '''scaled chi-squared distribution''' <math>s^2 \chi^2_k </math> is a reparametrization of the [[gamma distribution]] and the univariate [[Wishart distribution]]. Specifically if <math> X \sim s^2 \chi^2_k </math> then <math> X \sim \text{Gamma}(\alpha=\frac{k}{2}, \theta=2 s^2) </math> and <math> X \sim \text{W}_1(s^2,k) </math>. The chi-squared distribution is one of the most widely used [[probability distribution]]s in [[inferential statistics]], notably in [[hypothesis testing]] and in construction of [[confidence interval]]s.<ref name="United States Department of Commerce, National Bureau of Standards; Dover Publications-1983">{{Abramowitz Stegun ref|26|940}}</ref><ref>NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Engineering Statistics Handbook β Chi-Squared Distribution]</ref><ref name="Johnson-1994">{{cite book | last1 = Johnson | first1 = N. L. | first2 = S. |last2=Kotz |first3=N. |last3=Balakrishnan | title = Continuous Univariate Distributions |edition=Second |volume=1 |chapter=Chi-Square Distributions including Chi and Rayleigh |pages=415β493 | publisher = John Wiley and Sons | year = 1994 | isbn = 978-0-471-58495-7 }}</ref><ref>{{cite book | last1 = Mood | first1 = Alexander | first2=Franklin A. |last2=Graybill |first3=Duane C. |last3=Boes | title = Introduction to the Theory of Statistics |edition=Third |pages=241β246 | publisher = McGraw-Hill | year = 1974 | isbn = 978-0-07-042864-5 }}</ref> This distribution is sometimes called the '''central chi-squared distribution''', a special case of the more general [[noncentral chi-squared distribution]].<ref>{{Cite web |title=The Chi-Squared Distribution |url=https://uregina.ca/~gingrich/appchi.pdf |website=University of Regina}}</ref> The chi-squared distribution is used in the common [[chi-squared test]]s for [[goodness of fit]] of an observed distribution to a theoretical one, the [[statistical independence|independence]] of two criteria of classification of [[data analysis|qualitative data]], and in finding the confidence interval for estimating the population [[standard deviation]] of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as [[Friedman test|Friedman's analysis of variance by ranks]].
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