Editing Chi-squared distribution

{{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 – 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=415–493
  | 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=241–246
  | 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]].

== Definitions ==
If {{math|''Z''<sub>1</sub>, ..., ''Z''<sub>''k''</sub>}} are [[independence (probability theory)|independent]], [[standard normal]] random variables, then the sum of their squares,
: <math>X\ = \sum_{i=1}^k Z_i^2,</math>
is distributed according to the chi-squared distribution with {{mvar|k}} degrees of freedom. This is usually denoted as
: <math> X\ \sim\ \chi^2(k)\ \ \text{or}\ \ X\ \sim\ \chi^2_k.</math>

The chi-squared distribution has one parameter: a positive integer {{mvar|k}} that specifies the number of [[degrees of freedom (statistics)|degrees of freedom]] (the number of random variables being summed, ''Z''<sub>''i''</sub> s).

=== Introduction ===

The chi-squared distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the [[normal distribution]] and the [[exponential distribution]], the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:
* [[Pearson's chi-squared test|Chi-squared test]] of independence in [[contingency tables]]
* [[Pearson's chi-squared test|Chi-squared test]] of goodness of fit of observed data to hypothetical distributions
* [[Likelihood-ratio test]] for nested models
* [[Log-rank test]] in survival analysis
* [[Cochran–Mantel–Haenszel test]] for stratified contingency tables
* [[Wald test]]
* [[Score test]]

It is also a component of the definition of the [[Student's t-distribution|''t''-distribution]] and the [[F-distribution|''F''-distribution]] used in ''t''-tests, analysis of variance, and regression analysis.

The primary reason for which the chi-squared distribution is extensively used in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the [[t-statistic|''t''-statistic]] in a ''t''-test. For these hypothesis tests, as the sample size, {{mvar|n}}, increases, the [[sampling distribution]] of the test statistic approaches the normal distribution ([[central limit theorem]]). Because the test statistic (such as {{mvar|t}}) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Suppose that <math>Z</math> is a random variable sampled from the standard normal distribution, where the mean is <math>0</math> and the variance is <math>1</math>: <math>Z \sim N(0,1)</math>. Now, consider the random variable <math>X = Z^2</math>. The distribution of the random variable <math>X</math> is an example of a chi-squared distribution: <math>\ X\ \sim\ \chi^2_1</math>. The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized [[Likelihood-ratio test|likelihood ratio tests]] (LRT).<ref name="Westfall-2013">{{cite book|last1=Westfall|first1=Peter H.|title=Understanding Advanced Statistical Methods|date=2013|publisher=CRC Press|location=Boca Raton, FL|isbn=978-1-4665-1210-8}}</ref> LRTs have several desirable properties; in particular, simple LRTs commonly provide the highest power to reject the null hypothesis ([[Neyman–Pearson lemma]]) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the ''t'' distribution rather than the normal approximation or the chi-squared approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for a small sample size, and it is preferable to use [[Fisher's exact test]]. Ramsey shows that the exact [[binomial test]] is always more powerful than the normal approximation.<ref name="Ramsey-1988">{{cite journal|last1=Ramsey|first1=PH|title=Evaluating the Normal Approximation to the Binomial Test|journal=Journal of Educational Statistics|date=1988|volume=13|issue=2|pages=173–82|doi=10.2307/1164752|jstor=1164752}}</ref>

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows.<ref name="Lancaster-1969">{{Citation
|last=Lancaster
|first=H.O.
|title=The Chi-squared Distribution
|year=1969
|publisher=Wiley
}}</ref> De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

:<math> \chi = {m - Np \over \sqrt{Npq}} </math>

where <math>m</math> is the observed number of successes in <math>N</math> trials, where the probability of success is <math>p</math>, and <math>q = 1 - p</math>.

Squaring both sides of the equation gives

: <math style="block"> \chi^2 = {(m - Np)^2\over Npq} </math>

Using <math>N = Np + N(1 - p)</math>, <math>N = m + (N - m)</math>, and <math>q = 1 - p</math>, this equation can be rewritten as

: <math style="block"> \chi^2 = {(m - Np)^2\over Np} + {(N - m - Nq)^2\over Nq} </math>

The expression on the right is of the form that [[Karl Pearson]] would generalize to the form

: <math style="block"> \chi^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i} </math>

where

<math style="block"> \chi^2</math> = Pearson's cumulative test statistic, which asymptotically approaches a <math>\chi^2</math> distribution;
<math style="block">O_i</math> = the number of observations of type <math>i</math>;
<math style="block">E_i = N p_i</math> = the expected (theoretical) frequency of type <math>i</math>, asserted by the null hypothesis that the fraction of type <math>i</math> in the population is <math> p_i</math>; and
<math style="block">n</math> = the number of cells in the table.{{cn|date=November 2023}}

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large <math>n</math>). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a degenerate multivariate normal approximation to the multinomial distribution (the numbers in each category add up to the total sample size, which is considered fixed). Pearson showed that the chi-squared distribution arose from such a multivariate normal approximation to the multinomial distribution, taking careful account of the statistical dependence (negative correlations) between numbers of observations in different categories.<ref name="Lancaster-1969" />

=== Probability density function ===
The [[probability density function]] (pdf) of the chi-squared distribution is
:<math>
f(x;\,k) =
\begin{cases}
  \dfrac{x^{k/2 -1} e^{-x/2}}{2^{k/2} \Gamma\left(\frac k 2 \right)}, & x > 0; \\ 0, & \text{otherwise}.
\end{cases}
</math>
where <math display="inline">\Gamma(k/2)</math> denotes the [[gamma function]], which has [[particular values of the gamma function|closed-form values for integer <math>k</math>]].

For derivations of the pdf in the cases of one, two and <math>k</math> degrees of freedom, see [[Proofs related to chi-squared distribution]].

=== Cumulative distribution function ===

[[File:Chernoff-bound.svg|thumb|400px|Chernoff bound for the [[Cumulative distribution function|CDF]] and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (<math>k = 10</math>)]]

Its [[cumulative distribution function]] is:
: <math>
    F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),
  </math>
where <math>\gamma(s,t)</math> is the [[lower incomplete gamma function]] and <math display="inline">P(s,t)</math> is the [[Incomplete gamma function#Regularized gamma functions and Poisson random variables|regularized gamma function]].

In a special case of <math>k = 2</math> this function has the simple form:
: <math>
    F(x;\,2) = 1 - e^{-x/2}
  </math>
which can be easily derived by integrating <math>f(x;\,2)=\frac{1}{2}e^{-x/2}</math> directly. The integer recurrence of the gamma function makes it easy to compute <math>F(x;\,k)</math> for other small, even <math>k</math>.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many [[spreadsheet]]s and all [[statistical packages]].

Letting <math>z \equiv x/k</math>, [[Chernoff bound#The first step in the proof of Chernoff bounds|Chernoff bounds]] on the lower and upper tails of the CDF may be obtained.<ref>{{cite journal |last1=Dasgupta |first1=Sanjoy D. A. |last2=Gupta |first2=Anupam K. |date=January 2003 |title=An Elementary Proof of a Theorem of Johnson and Lindenstrauss |journal=Random Structures and Algorithms |volume=22 |issue=1 |pages=60–65 |doi=10.1002/rsa.10073 |s2cid=10327785 |url=http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf |access-date=2012-05-01 }}</ref> For the cases when <math>0 < z < 1</math> (which include all of the cases when this CDF is less than half):
<math style="block"> F(z k;\,k) \leq (z e^{1-z})^{k/2}.</math>

The tail bound for the cases when <math>z > 1</math>, similarly, is
: <math>
    1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.
  </math>

For another [[approximation]] for the CDF modeled after the cube of a Gaussian, see under [[Noncentral chi-squared distribution#Approximation|Noncentral chi-squared distribution]].

== Properties ==

=== Cochran's theorem ===
{{Main|Cochran's theorem}}
The following is a special case of Cochran's theorem.

'''Theorem.''' If <math>Z_1,...,Z_n</math> are [[independence (probability theory)|independent]] identically distributed (i.i.d.), [[standard normal]] random variables, then
<math>\sum_{t=1}^n(Z_t - \bar Z)^2 \sim \chi^2_{n-1}</math>
where <math>\bar Z = \frac{1}{n} \sum_{t=1}^n Z_t.</math>

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}
'''Proof.''' Let <math>Z\sim\mathcal{N}(\bar 0,1\!\!1)</math> be a vector of <math>n</math> independent normally distributed random variables,
and <math>\bar Z</math> their average.
Then
<math>
  \sum_{t=1}^n(Z_t-\bar Z)^2 ~=~ \sum_{t=1}^n Z_t^2 -n\bar Z^2 ~=~ Z^\top[1\!\!1 -{\textstyle\frac1n}\bar 1\bar 1^\top]Z ~=:~ Z^\top\!M Z
</math>
where <math>1\!\!1</math> is the identity matrix and <math>\bar 1</math> the all ones vector.
<math>M</math> has one eigenvector <math>b_1:={\textstyle\frac{1}{\sqrt{n}}} \bar 1</math> with eigenvalue <math>0</math>,
and <math>n-1</math> eigenvectors <math>b_2,...,b_n</math> (all orthogonal to <math>b_1</math>) with eigenvalue <math>1</math>,
which can be chosen so that <math>Q:=(b_1,...,b_n)</math> is an orthogonal matrix.
Since also <math>X:=Q^\top\!Z\sim\mathcal{N}(\bar 0,Q^\top\!1\!\!1 Q) =\mathcal{N}(\bar 0,1\!\!1)</math>,
we have
<math>
  \sum_{t=1}^n(Z_t-\bar Z)^2 ~=~ Z^\top\!M Z ~=~ X^\top\!Q^\top\!M Q X ~=~ X_2^2+...+X_n^2 ~\sim~ \chi^2_{n-1},
</math>
which proves the claim.
{{hidden end}}

=== Additivity ===
It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if <math>X_i,i=\overline{1,n}</math> are independent chi-squared variables with <math>k_i</math>, <math>i=\overline{1,n} </math> degrees of freedom, respectively, then <math>Y = X_1 + \cdots + X_n</math> is chi-squared distributed with <math>k_1 + \cdots + k_n</math> degrees of freedom.

=== Sample mean ===
The sample mean of <math>n</math> [[i.i.d.]] chi-squared variables of degree <math>k</math> is distributed according to a gamma distribution with shape <math>\alpha</math> and scale <math>\theta</math> parameters:
:<math> \overline X = \frac{1}{n} \sum_{i=1}^n X_i \sim \operatorname{Gamma}\left(\alpha=n\, k /2, \theta= 2/n \right) \qquad \text{where } X_i \sim \chi^2(k)</math>

[[#Asymptotic properties|Asymptotically]], given that for a shape parameter <math> \alpha </math> going to infinity, a Gamma distribution converges towards a normal distribution with expectation <math> \mu = \alpha\cdot \theta </math> and variance <math> \sigma^2 = \alpha\, \theta^2 </math>, the sample mean converges towards:

<math style="block"> \overline X \xrightarrow{n \to \infty} N(\mu = k, \sigma^2 = 2\, k /n ) </math>

Note that we would have obtained the same result invoking instead the [[central limit theorem]], noting that for each chi-squared variable of degree <math>k</math> the expectation is <math> k </math> , and its variance <math> 2\,k </math> (and hence the variance of the sample mean <math> \overline{X}</math> being <math> \sigma^2 = \frac{2k}{n} </math>).

=== Entropy ===
The [[differential entropy]] is given by
: <math>
    h = \int_{0}^\infty f(x;\,k)\ln f(x;\,k) \, dx
      = \frac k 2 + \ln \left[2\,\Gamma \left(\frac k 2 \right)\right] + \left(1-\frac k 2 \right)\, \psi\!\left(\frac k 2 \right),
  </math>
where <math>\psi(x)</math> is the [[Digamma function]].

The chi-squared distribution is the [[maximum entropy probability distribution]] for a random variate <math>X</math> for which <math>\operatorname{E}(X)=k</math> and <math>\operatorname{E}(\ln(X))=\psi(k/2)+\ln(2)</math> are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the [[gamma distribution#Logarithmic expectation and variance|Expectation of the log moment of gamma]]. For derivation from more basic principles, see the derivation in [[exponential family#Moment-generating function of the sufficient statistic|moment-generating function of the sufficient statistic]].

=== Noncentral moments ===
The noncentral moments (raw moments) of a chi-squared distribution with <math>k</math> degrees of freedom are given by<ref>[http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-squared distribution], from [[MathWorld]], retrieved Feb. 11, 2009</ref><ref>M. K. Simon, ''Probability Distributions Involving Gaussian Random Variables'', New York: Springer, 2002, eq. (2.35), {{ISBN|978-0-387-34657-1}}</ref>
: <math>
    \operatorname{E}(X^m) = k (k+2) (k+4) \cdots (k+2m-2) = 2^m \frac{\Gamma\left(m+\frac{k}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}.
  </math>

=== Cumulants ===
The [[cumulant]]s are readily obtained by a [[power series]] expansion of the logarithm of the characteristic function:
: <math>\kappa_n = 2^{n-1}(n-1)!\,k</math>
with [[cumulant generating function]] <math>\ln E[e^{tX}] = - \frac k2 \ln(1-2t) </math>.

=== Concentration ===

The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart<ref>{{Cite journal |last1=Laurent |first1=B. |last2=Massart |first2=P. |date=2000-10-01 |title=Adaptive estimation of a quadratic functional by model selection |journal=The Annals of Statistics |volume=28 |issue=5 |doi=10.1214/aos/1015957395 |s2cid=116945590 |issn=0090-5364|doi-access=free }}</ref> bounds are:
: <math>\operatorname{P}(X - k \ge 2 \sqrt{k x} + 2x) \le \exp(-x)</math>
: <math>\operatorname{P}(k - X \ge 2 \sqrt{k x}) \le \exp(-x)</math>
One consequence is that, if <math>Z \sim N(0, 1)^k</math> is a gaussian random vector in <math>\R^k</math>, then as the dimension <math>k</math> grows, the squared length of the vector is concentrated tightly around <math>k</math> with a width <math>k^{1/2 + \alpha}</math>:<math display="block">Pr(\|Z\|^2 \in [k - 2k^{1/2+\alpha}, k + 2k^{1/2+\alpha} + 2k^{\alpha}]) \geq 1-e^{-k^\alpha}</math>where the exponent <math>\alpha</math> can be chosen as any value in <math>\R</math>.

Since the cumulant generating function for <math>\chi^2(k)</math> is <math>K(t) = -\frac k2 \ln(1-2t) </math>, and its [[Convex conjugate|convex dual]] is <math>K^*(q) = \frac 12 (q-k + k\ln\frac kq) </math>, the standard [[Chernoff bound]] yields<math display="block">\begin{aligned}
\ln Pr(X \geq (1 + \epsilon) k) &\leq -\frac k2 ( \epsilon - \ln(1+\epsilon)) \\
\ln Pr(X \leq (1 - \epsilon) k) &\leq -\frac k2 ( -\epsilon - \ln(1-\epsilon))
\end{aligned}</math>where <math>0< \epsilon < 1</math>. By the union bound,<math display="block">Pr(X \in (1\pm \epsilon ) k ) \geq 1 - 2e^{-\frac k2 (\frac 12 \epsilon^2 - \frac 13 \epsilon^3)} </math>This result is used in proving the [[Johnson–Lindenstrauss lemma]].<ref>[https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/f9261308512f6b90e284599f94055bb4_MIT18_S096F15_Ses15_16.pdf MIT 18.S096 (Fall 2015): Topics in Mathematics of Data Science, Lecture 5, Johnson-Lindenstrauss Lemma and Gordons Theorem]</ref>

=== Asymptotic properties ===
[[File:Chi-square median approx.png|thumb|upright=1.818|Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference ({{font color|blue|blue}}) and relative difference ({{font color|red|red}}) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.]]

By the [[central limit theorem]], because the chi-squared distribution is the sum of <math>k</math> independent random variables with finite mean and variance, it converges to a normal distribution for large <math>k</math>. For many practical purposes, for <math>k>50</math> the distribution is sufficiently close to a [[normal distribution]], so the difference is ignorable.<ref>{{cite book|title=Statistics for experimenters|author=Box, Hunter and Hunter|publisher=Wiley|year=1978|isbn=978-0-471-09315-2|page=[https://archive.org/details/statisticsforexp00geor/page/118 118]|url-access=registration|url=https://archive.org/details/statisticsforexp00geor/page/118}}</ref> Specifically, if <math>X \sim \chi^2(k)</math>, then as <math>k</math> tends to infinity, the distribution of <math>(X-k)/\sqrt{2k}</math> [[convergence of random variables#Convergence in distribution|tends]] to a standard normal distribution. However, convergence is slow as the [[skewness]] is <math>\sqrt{8/k}</math> and the [[excess kurtosis]] is <math>12/k</math>.

The sampling distribution of <math>\ln(\chi^2)</math> converges to normality much faster than the sampling distribution of <math>\chi^2</math>,<ref>{{cite journal |first1=M. S. |last1=Bartlett |first2=D. G. |last2=Kendall |title=The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation |journal=Supplement to the Journal of the Royal Statistical Society |volume=8 |issue=1 |year=1946 |pages=128–138 |jstor=2983618 |doi=10.2307/2983618 }}</ref> as the [[logarithmic transformation|logarithmic transform]] removes much of the asymmetry.<ref name="Pillai-2016">{{Cite journal|last=Pillai|first=Natesh S.|year=2016|title=An unexpected encounter with Cauchy and Lévy|journal=[[Annals of Statistics]]|volume=44|issue=5|pages=2089–2097|doi=10.1214/15-aos1407|arxiv=1505.01957|s2cid=31582370}}</ref>

Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
* If <math>X \sim \chi^2(k)</math> then <math>\sqrt{2X}</math> is approximately normally distributed with mean <math>\sqrt{2k-1}</math> and unit variance (1922, by [[R. A. Fisher]], see (18.23), p.&nbsp;426 of Johnson.<ref name="Johnson-1994" />
* If <math>X \sim \chi^2(k)</math> then <math>\sqrt[3]{X/k}</math> is approximately normally distributed with mean <math> 1-\frac{2}{9k}</math> and variance <math>\frac{2}{9k} .</math><ref>{{cite journal |last1=Wilson |first1=E. B. |last2=Hilferty |first2=M. M. |year=1931 |title=The distribution of chi-squared |journal=[[Proc. Natl. Acad. Sci. USA]] |volume=17 |issue=12 |pages=684–688 |bibcode=1931PNAS...17..684W |doi=10.1073/pnas.17.12.684 |pmid=16577411 |pmc=1076144 |doi-access=free }}</ref> This is known as the '''Wilson–Hilferty transformation''', see (18.24), p.&nbsp;426 of Johnson.<ref name="Johnson-1994" />
** This [[Data transformation (statistics)#Transforming to normality|normalizing transformation]] leads directly to the commonly used median approximation <math>k\bigg(1-\frac{2}{9k}\bigg)^3\;</math> by back-transforming from the mean, which is also the median, of the normal distribution.

== Related distributions ==
{{More citations needed section|date=September 2011}}
* As <math>k\to\infty</math>, <math> (\chi^2_k-k)/\sqrt{2k} ~ \xrightarrow{d}\ N(0,1) \,</math> ([[normal distribution]])
* <math> \chi_k^2 \sim {\chi'}^2_k(0)</math> ([[noncentral chi-squared distribution]] with non-centrality parameter <math> \lambda = 0 </math>)
* If <math>Y \sim \mathrm{F}(\nu_1, \nu_2)</math> then <math>X = \lim_{\nu_2 \to \infty} \nu_1 Y</math> has the chi-squared distribution <math>\chi^2_{\nu_{1}}</math>
:*As a special case, if <math>Y \sim \mathrm{F}(1, \nu_2)\,</math> then <math>X = \lim_{\nu_2 \to \infty} Y\,</math> has the chi-squared distribution <math>\chi^2_{1}</math>
* <math> \|\boldsymbol{N}_{i=1,\ldots,k} (0,1) \|^2 \sim \chi^2_k </math> (The squared [[Norm (mathematics)|norm]] of ''k'' standard normally distributed variables is a chi-squared distribution with ''k'' [[degrees of freedom (statistics)|degrees of freedom]])
* If <math>X \sim \chi^2_\nu\,</math> and <math>c>0 \,</math>, then <math>cX \sim \Gamma(k=\nu/2, \theta=2c)\,</math>. ([[gamma distribution]])
* If <math>X \sim \chi^2_k</math> then <math>\sqrt{X} \sim \chi_k</math> ([[chi distribution]])
* If <math>X \sim \chi^2_2</math>, then <math>X \sim \operatorname{exp}(1/2)</math> is an [[exponential distribution]]. (See [[gamma distribution]] for more.)
* If <math>X \sim \chi^2_{2k}</math>, then <math>X \sim \operatorname{Erlang}(k, 1/2)</math> is an [[Erlang distribution]].
* If <math> X \sim \operatorname{Erlang}(k,\lambda)</math>, then <math> 2\lambda X\sim \chi^2_{2k}</math>
* If <math>X \sim \operatorname{Rayleigh}(1)\,</math> ([[Rayleigh distribution]]) then <math>X^2 \sim \chi^2_2\,</math>
* If <math>X \sim \operatorname{Maxwell}(1)\,</math> ([[Maxwell distribution]]) then <math>X^2 \sim \chi^2_3\,</math>
* If <math>X \sim \chi^2_\nu</math> then <math>\tfrac{1}{X} \sim \operatorname{Inv-}\chi^2_\nu\, </math> ([[Inverse-chi-squared distribution]])
* The chi-squared distribution is a special case of type III [[Pearson distribution]]
* If <math>X \sim \chi^2_{\nu_1}\,</math> and <math>Y \sim \chi^2_{\nu_2}\,</math> are independent then <math>\tfrac{X}{X+Y} \sim \operatorname{Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2})\,</math> ([[beta distribution]])
* If <math> X \sim \operatorname{U}(0,1)\, </math> ([[Uniform distribution (continuous)|uniform distribution]]) then <math> -2\log(X) \sim \chi^2_2\,</math>
* If <math>X_i \sim \operatorname{Laplace}(\mu,\beta)\,</math> then <math>\sum_{i=1}^n \frac{2 |X_i-\mu|}{\beta} \sim \chi^2_{2n}\,</math>
* If <math>X_i</math> follows the [[generalized normal distribution]] (version 1) with parameters <math>\mu,\alpha,\beta</math> then <math>\sum_{i=1}^n \frac{2 |X_i-\mu|^\beta}{\alpha} \sim \chi^2_{2n/\beta}\,</math> <ref>{{cite journal |last= Bäckström |first= T. |author2=Fischer, J. |date=January 2018|title= Fast Randomization for Distributed Low-Bitrate Coding of Speech and Audio|journal= IEEE/ACM Transactions on Audio, Speech, and Language Processing |volume= 26|issue= 1|pages= 19–30|doi= 10.1109/TASLP.2017.2757601|s2cid= 19777585 |url= https://research.aalto.fi/files/27158975/ELEC_backstrom_et_al_Fast_randomization.pdf }}</ref>
* The chi-squared distribution is a transformation of [[Pareto distribution]]
* [[Student's t-distribution]] is a transformation of chi-squared distribution
* [[Student's t-distribution]] can be obtained from chi-squared distribution and [[normal distribution]]
* The [[noncentral beta distribution]] can be obtained as a transformation of chi-squared distribution and [[noncentral chi-squared distribution]]
* The [[noncentral t-distribution]] can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with <math>k</math> degrees of freedom is defined as the sum of the squares of <math>k</math> independent [[standard normal]] random variables.

If <math>Y</math> is a <math>k</math>-dimensional Gaussian random vector with mean vector <math>\mu</math> and rank <math>k</math> covariance matrix <math>C</math>, then <math>X = (Y-\mu )^{T}C^{-1}(Y-\mu)</math> is chi-squared distributed with <math>k</math> degrees of freedom.

The sum of squares of [[statistically independent]] unit-variance Gaussian variables which do ''not'' have mean zero yields a generalization of the chi-squared distribution called the [[noncentral chi-squared distribution]].

If <math>Y</math> is a vector of <math>k</math> [[i.i.d.]] standard normal random variables and <math>A</math> is a <math>k\times k</math> [[symmetric matrix|symmetric]], [[idempotent matrix]] with [[rank (linear algebra)|rank]] <math>k-n</math>, then the [[quadratic form]] <math>Y^TAY</math> is chi-square distributed with <math>k-n</math> degrees of freedom.

If <math>\Sigma</math> is a <math>p\times p</math> positive-semidefinite covariance matrix with strictly positive diagonal entries, then for <math>X\sim N(0,\Sigma)</math> and <math>w</math> a random <math>p</math>-vector independent of <math>X</math> such that <math>w_1+\cdots+w_p=1</math> and <math>w_i\geq 0, i=1,\ldots,p,</math> then

: <math>\frac{1}{\left(\frac{w_1}{X_1},\ldots,\frac{w_p}{X_p}\right)\Sigma\left(\frac{w_1}{X_1},\ldots,\frac{w_p}{X_p}\right)^\top} \sim \chi_1^2.</math><ref name="Pillai-2016" />

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,
* <math>Y</math> is [[F-distribution|F-distributed]], <math>Y \sim F(k_1, k_2)</math> if <math>Y = \frac{ {X_1}/{k_1} }{ {X_2}/{k_2} }</math>, where <math>X_1 \sim \chi^2_{k_1}</math> and <math>X_2 \sim \chi^2_{k_2}</math> are statistically independent.
* If <math>X_1 \sim \chi^2_{k_1}</math> and <math>X_2 \sim \chi^2_{k_2}</math> are statistically independent, then <math>X_1 + X_2\sim \chi^2_{k_1+k_2}</math>. If <math>X_1</math> and <math>X_2</math> are not independent, then <math>X_1+X_2</math> is not chi-square distributed.

=== Generalizations ===
The chi-squared distribution is obtained as the sum of the squares of {{mvar|k}} independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

=== Linear combination ===
If <math>X_1,\ldots,X_n</math> are chi square random variables and <math>a_1,\ldots,a_n\in\mathbb{R}_{>0}</math>, then the distribution of <math>X=\sum_{i=1}^n a_i X_i</math> is a special case of a [[Generalized chi-squared distribution|Generalized Chi-squared Distribution]].
A closed expression for this distribution is not known. It may be, however, approximated efficiently using the [[Characteristic function (probability theory)#Properties|property of characteristic functions]] of chi-square random variables.<ref>{{cite journal
|first=J.
|last=Bausch
|title=On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua
|journal=J. Phys. A: Math. Theor.
|volume=46
|issue=50
|year=2013
|pages=505202
|doi=10.1088/1751-8113/46/50/505202 |bibcode=2013JPhA...46X5202B
|arxiv=1208.2691
|s2cid=119721108
}}</ref>

=== Chi-squared distributions ===

==== Noncentral chi-squared distribution ====
{{Main|Noncentral chi-squared distribution}}
The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and ''nonzero'' means.

==== Generalized chi-squared distribution ====
{{Main|Generalized chi-squared distribution}}
The generalized chi-squared distribution is obtained from the quadratic form {{math|z'Az}} where {{mvar|z}} is a zero-mean Gaussian vector having an arbitrary covariance matrix, and {{mvar|A}} is an arbitrary matrix.

=== Gamma, exponential, and related distributions ===
The chi-squared distribution <math>X \sim \chi_k^2</math> is a special case of the [[gamma distribution]], in that <math>X \sim \Gamma \left(\frac{k}2,\frac{1}2\right)</math> using the rate parameterization of the gamma distribution (or
<math>X \sim \Gamma \left(\frac{k}2,2 \right)</math> using the scale parameterization of the gamma distribution)
where {{mvar|k}} is an integer.

Because the [[exponential distribution]] is also a special case of the gamma distribution, we also have that if <math>X \sim \chi_2^2</math>, then <math>X\sim \operatorname{exp}\left(\frac 1 2\right)</math> is an [[exponential distribution]].

The [[Erlang distribution]] is also a special case of the gamma distribution and thus we also have that if <math>X \sim\chi_k^2</math> with even <math>k</math>, then <math>X</math> is Erlang distributed with shape parameter <math>k/2</math> and scale parameter <math>1/2</math>.

== Occurrence and applications{{anchor|Applications}} ==
The chi-squared distribution has numerous applications in inferential [[statistics]], for instance in [[chi-squared test]]s and in estimating [[variance]]s. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a [[linear regression|regression]] line via its role in [[Student's t-distribution]]. It enters all [[analysis of variance]] problems via its role in the [[F-distribution]], which is the distribution of the ratio of two independent chi-squared [[random variable]]s, each divided by their respective degrees of freedom.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.
* if <math>X_1, ..., X_n</math> are [[i.i.d.]] <math>N(\mu, \sigma^2)</math> [[random variable]]s, then <math>\sum_{i=1}^n(X_i - \overline{X})^2 \sim \sigma^2 \chi^2_{n-1}</math> where <math>\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i</math>.
* The box below shows some [[statistics]] based on <math>X_i \sim N(\mu_i, \sigma^2_i), i= 1, \ldots, k</math> independent random variables that have probability distributions related to the chi-squared distribution:
{| class="wikitable"  style="margin:1em auto;" align="center"
|-
! Name !! Statistic
|-
| chi-squared distribution || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
|-
| [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| [[chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
The chi-squared distribution is also often encountered in [[magnetic resonance imaging]].<ref>den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", ''Physica Medica'', [https://dx.doi.org/10.1016/j.ejmp.2014.05.002]</ref>

== Computational methods ==
=== Table of {{math|''χ''<sup>2</sup>}} values vs {{math|''p''}}-values ===
The [[p-value|<math display="inline">p</math>-value]] is the probability of observing a test statistic ''at least'' as extreme in a chi-squared distribution. Accordingly, since the [[cumulative distribution function]] (CDF) for the appropriate degrees of freedom ''(df)'' gives the probability of having obtained a value ''less extreme'' than this point, subtracting the CDF value from 1 gives the ''p''-value. A low ''p''-value, below the chosen significance level, indicates [[statistical significance]], i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results.

The table below gives a number of ''p''-values matching to <math> \chi^2 </math> for the first 10 degrees of freedom.
{| class="wikitable"
! Degrees of freedom (df)
!colspan=11| <math> \chi^2 </math> value<ref>[http://www2.lv.psu.edu/jxm57/irp/chisquar.html Chi-Squared Test] {{Webarchive|url=https://web.archive.org/web/20131118011437/http://www2.lv.psu.edu/jxm57/irp/chisquar.html |date=2013-11-18 }} Table B.2. Dr. Jacqueline S. McLaughlin at The Pennsylvania State University. In turn citing: R. A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV. Two values have been corrected, 7.82 with 7.81 and 4.60 with 4.61</ref>
|-
| style="text-align:center;" | 1
| 0.004
| 0.02
| 0.06
| 0.15
| 0.46
| 1.07
| 1.64
| 2.71
| 3.84
| 6.63
| 10.83
|-
| style="text-align:center;" | 2
| 0.10
| 0.21
| 0.45
| 0.71
| 1.39
| 2.41
| 3.22
| 4.61
| 5.99
| 9.21
| 13.82
|-
| style="text-align:center;" | 3
| 0.35
| 0.58
| 1.01
| 1.42
| 2.37
| 3.66
| 4.64
| 6.25
| 7.81
| 11.34
| 16.27
|-
| style="text-align:center;" | 4
| 0.71
| 1.06
| 1.65
| 2.20
| 3.36
| 4.88
| 5.99
| 7.78
| 9.49
| 13.28
| 18.47
|-
| style="text-align:center;" | 5
| 1.14
| 1.61
| 2.34
| 3.00
| 4.35
| 6.06
| 7.29
| 9.24
| 11.07
| 15.09
| 20.52
|-
| style="text-align:center;" | 6
| 1.63
| 2.20
| 3.07
| 3.83
| 5.35
| 7.23
| 8.56
| 10.64
| 12.59
| 16.81
| 22.46
|-
| style="text-align:center;" | 7
| 2.17
| 2.83
| 3.82
| 4.67
| 6.35
| 8.38
| 9.80
| 12.02
| 14.07
| 18.48
| 24.32
|-
| style="text-align:center;" | 8
| 2.73
| 3.49
| 4.59
| 5.53
| 7.34
| 9.52
| 11.03
| 13.36
| 15.51
| 20.09
| 26.12
|-
| style="text-align:center;" | 9
| 3.32
| 4.17
| 5.38
| 6.39
| 8.34
| 10.66
| 12.24
| 14.68
| 16.92
| 21.67
| 27.88
|-
| style="text-align:center;" | 10
| 3.94
| 4.87
| 6.18
| 7.27
| 9.34
| 11.78
| 13.44
| 15.99
| 18.31
| 23.21
| 29.59
 13.25
|-
! scope="row" style="text-align:right;" | ''p''-value (probability)
| style="background: #ffa2aa" | 0.95
| style="background: #efaaaa" | 0.90
| style="background: #e8b2aa" | 0.80
| style="background: #dfbaaa" | 0.70
| style="background: #d8c2aa" | 0.50
| style="background: #cfcaaa" | 0.30
| style="background: #c8d2aa" | 0.20
| style="background: #bfdaaa" | 0.10
| style="background: #b8e2aa" | 0.05
| style="background: #afeaaa" | 0.01
| style="background: #a8faaa" | 0.001
|}

These values can be calculated evaluating the [[quantile function]] (also known as "inverse CDF" or "ICDF") of the chi-squared distribution;<ref>{{Cite web|url=https://www.r-tutor.com/elementary-statistics/probability-distributions/chi-squared-distribution|title=Chi-squared Distribution &#124; R Tutorial|website=www.r-tutor.com}}</ref> e. g., the {{math|χ<sup>2</sup>}} ICDF for {{math|1=''p'' = 0.05}} and {{math|1=df = 7}} yields {{math|2.1673 ≈ 2.17}} as in the table above, noticing that {{math|1 – ''p''}} is the [[p-value|''p''-value]] from the table.

== History ==
This distribution was first described by the German geodesist and statistician [[Friedrich Robert Helmert]] in papers of 1875–6,{{sfn|Hald|1998|pp=633–692|loc=27. Sampling Distributions under Normality}}<ref>[[F. R. Helmert]], "[http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN599415665_0021&DMDID=DMDLOG_0018 Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen]", ''Zeitschrift für Mathematik und Physik'' [http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN599415665_0021 21], 1876, pp. 192–219</ref> where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the ''Helmert'sche'' ("Helmertian") or "Helmert distribution".

The distribution was independently rediscovered by the English mathematician [[Karl Pearson]] in the context of [[goodness of fit]], for which he developed his [[Pearson's chi-squared test]], published in 1900, with computed table of values published in {{Harv|Elderton|1902}}, collected in {{Harv|Pearson|1914|pp=xxxi–xxxiii, 26–28|loc=Table XII}}.
The name "chi-square" ultimately derives from Pearson's shorthand for the exponent in a [[multivariate normal distribution]] with the Greek letter [[Chi (letter)|Chi]], writing {{mvar|−½χ<sup>2</sup>}} for what would appear in modern notation as {{math|−½'''x'''<sup>T</sup>Σ<sup>−1</sup>'''x'''}} (Σ being the [[covariance matrix]]).<ref>R. L. Plackett, ''Karl Pearson and the Chi-Squared Test'', International Statistical Review, 1983, [https://www.jstor.org/stable/1402731?seq=3 61f.]
See also Jeff Miller, [http://jeff560.tripod.com/c.html Earliest Known Uses of Some of the Words of Mathematics].</ref> The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.{{sfn|Hald|1998|pp=633–692|loc=27. Sampling Distributions under Normality}}

== See also ==
{{Portal|Mathematics}}
{{Colbegin}}
* [[Chi distribution]]
* [[Scaled inverse chi-squared distribution]]
* [[Gamma distribution]]
* [[Generalized chi-squared distribution]]
* [[Noncentral chi-squared distribution]]
* [[Pearson's chi-squared test]]
* [[Reduced chi-squared statistic]]
* [[Wilks's lambda distribution]]
* [[Modified half-normal distribution]]<ref name="Sun-2021">{{cite journal |last1=Sun |first1=Jingchao |last2=Kong |first2=Maiying |last3=Pal |first3=Subhadip |title=The Modified-Half-Normal distribution: Properties and an efficient sampling scheme |journal=Communications in Statistics - Theory and Methods |date=22 June 2021 |volume=52 |issue=5 |pages=1591–1613 |doi=10.1080/03610926.2021.1934700 |s2cid=237919587 |url=https://figshare.com/articles/journal_contribution/The_Modified-Half-Normal_distribution_Properties_and_an_efficient_sampling_scheme/14825266/1/files/28535884.pdf |issn=0361-0926}}</ref> with the pdf on <math>(0, \infty)</math> is given as <math> f(x)= \frac{2\beta^{\alpha/2} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}</math>, where <math>\Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right)</math> denotes the [[Fox–Wright Psi function]].
{{Colend}}

== References ==
{{Reflist|30em}}

==Sources==
* {{cite book |title=A history of mathematical statistics from 1750 to 1930 |last=Hald |first=Anders |author-link=Anders Hald |year=1998 |publisher=Wiley |location=New York |isbn=978-0-471-17912-2 }}
* {{Cite journal |last=Elderton |first=William Palin |author-link=William Palin Elderton |title=Tables for Testing the Goodness of Fit of Theory to Observation |doi=10.1093/biomet/1.2.155 |journal=Biometrika |volume=1 |issue=2 |pages=155–163 |year=1902 |url=https://zenodo.org/record/1431595}}
* {{cite journal |last=Pearson |first=Karl |title=On the probability that two independent distributions of frequency are really samples of the same population, with special reference to recent work on the identity of Trypanosome strains |date=1914 |journal=Biometrika |volume=10 |pages=85–154 |doi=10.1093/biomet/10.1.85}}

== Further reading ==
{{refbegin}}
* {{cite book |title=An Expert's Chi-Square Testing Guide (Probability and Statistics) |last=Clay |first=Henry |author-link=Henry Clay |year=1852 |publisher=Ashland |location=Washington, D.C. |isbn=978-0-060173-22-7}}
* {{springer|title=Chi-squared distribution|id=Chi-squared_distribution}}
{{refend}}

== External links ==
* [https://jeff560.tripod.com/c.html Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history]
* [http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm Course notes on Chi-Squared Goodness of Fit Testing] from Yale University Stats 101 class.
* [https://demonstrations.wolfram.com/StatisticsAssociatedWithNormalSamples/ ''Mathematica'' demonstration showing the chi-squared sampling distribution of various statistics, e. g. Σ''x''², for a normal population]
* [https://www.jstor.org/stable/2348373 Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator]
* [https://www.medcalc.org/manual/chi-square-table.php Values of the Chi-squared distribution]

{{ProbDistributions|continuous-semi-infinite}}
{{Authority control}}

[[Category:Normal distribution]]
[[Category:Infinitely divisible probability distributions]]