Chi-squared distribution

If {{math|Z₁, ..., Z_k}} are independent, standard normal random variables, then the sum of their squares,

: $X\backslash \; =\; \backslash sum\_\{i=1\}^k\; Z\_i^2,$

is distributed according to the chi-squared distribution with {{mvar|k}} degrees of freedom. This is usually denoted as

: $X\backslash \; \backslash sim\backslash \; \backslash chi^2(k)\backslash \; \backslash \; \backslash text\{or\}\backslash \; \backslash \; X\backslash \; \backslash sim\backslash \; \backslash chi^2\_k.$

The chi-squared distribution has one parameter: a positive integer {{mvar|k}} that specifies the number of degrees of freedom (the number of random variables being summed, Z_i s).

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:

• Chi-squared test of independence in contingency tables
• 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 t-distribution and the 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 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 $Z$ is a random variable sampled from the standard normal distribution, where the mean is $0$ and the variance is $1$: $Z\; \backslash sim\; N(0,1)$. Now, consider the random variable $X\; =\; Z^2$. The distribution of the random variable $X$ is an example of a chi-squared distribution: $\backslash \; X\backslash \; \backslash sim\backslash \; \backslash chi^2\_1$. 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 tests (LRT).{{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}} 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.{{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}}

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows.{{Citation

|last=Lancaster

|first=H.O.

|title=The Chi-squared Distribution

|year=1969

|publisher=Wiley

}} 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

:$\backslash chi\; =\; \{m\; -\; Np\; \backslash over\; \backslash sqrt\{Npq\}\}$

where $m$ is the observed number of successes in $N$ trials, where the probability of success is $p$, and $q\; =\; 1\; -\; p$.

Squaring both sides of the equation gives

: $\backslash chi^2\; =\; \{(m\; -\; Np)^2\backslash over\; Npq\}$

Using $N\; =\; Np\; +\; N(1\; -\; p)$, $N\; =\; m\; +\; (N\; -\; m)$, and $q\; =\; 1\; -\; p$, this equation can be rewritten as

: $\backslash chi^2\; =\; \{(m\; -\; Np)^2\backslash over\; Np\}\; +\; \{(N\; -\; m\; -\; Nq)^2\backslash over\; Nq\}$

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

: $\backslash chi^2\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{(O\_i\; -\; E\_i)^2\}\{E\_i\}$

where

$\backslash chi^2$ = Pearson's cumulative test statistic, which asymptotically approaches a $\backslash chi^2$ distribution;

$O\_i$ = the number of observations of type $i$;

$E\_i\; =\; N\; p\_i$ = the expected (theoretical) frequency of type $i$, asserted by the null hypothesis that the fraction of type $i$ in the population is $p\_i$; and

$n$ = 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 $n$). 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.

The probability density function (pdf) of the chi-squared distribution is

:$$

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}

where $\backslash Gamma(k/2)$ denotes the gamma function, which has closed-form values for integer $k$.

For derivations of the pdf in the cases of one, two and $k$ degrees of freedom, see Proofs related to chi-squared distribution.

File:Chernoff-bound.svg and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom ($k\; =\; 10$)]]

Its cumulative distribution function is:

: $$

F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),

where $\backslash gamma(s,t)$ is the lower incomplete gamma function and $P(s,t)$ is the regularized gamma function.

In a special case of $k\; =\; 2$ this function has the simple form:

: $$

F(x;\,2) = 1 - e^{-x/2}

which can be easily derived by integrating $f(x;\backslash ,2)=\backslash frac\{1\}\{2\}e^\{-x/2\}$ directly. The integer recurrence of the gamma function makes it easy to compute $F(x;\backslash ,k)$ for other small, even $k$.

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

Letting $z\; \backslash equiv\; x/k$, Chernoff bounds on the lower and upper tails of the CDF may be obtained.{{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 }} For the cases when (which include all of the cases when this CDF is less than half):

$F(z\; k;\backslash ,k)\; \backslash leq\; (z\; e^\{1-z\})^\{k/2\}.$

The tail bound for the cases when $z\; >\; 1$, similarly, is

: $$

1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.

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

{{Main|Cochran's theorem}}

The following is a special case of Cochran's theorem.

Theorem. If $Z\_1,...,Z\_n$ are independent identically distributed (i.i.d.), standard normal random variables, then

$\backslash sum\_\{t=1\}^n(Z\_t\; -\; \backslash bar\; Z)^2\; \backslash sim\; \backslash chi^2\_\{n-1\}$

where $\backslash bar\; Z\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{t=1\}^n\; Z\_t.$

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}

Proof. Let $Z\backslash sim\backslash mathcal\{N\}(\backslash bar\; 0,1\backslash !\backslash !1)$ be a vector of $n$ independent normally distributed random variables,

and $\backslash bar\; Z$ their average.

Then

\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

where $1\backslash !\backslash !1$ is the identity matrix and $\backslash bar\; 1$ the all ones vector.

$M$ has one eigenvector $b\_1:=\{\backslash textstyle\backslash frac\{1\}\{\backslash sqrt\{n\}\}\}\; \backslash bar\; 1$ with eigenvalue $0$,

and $n-1$ eigenvectors $b\_2,...,b\_n$ (all orthogonal to $b\_1$) with eigenvalue $1$,

which can be chosen so that $Q:=(b\_1,...,b\_n)$ is an orthogonal matrix.

Since also $X:=Q^\backslash top\backslash !Z\backslash sim\backslash mathcal\{N\}(\backslash bar\; 0,Q^\backslash top\backslash !1\backslash !\backslash !1\; Q)\; =\backslash mathcal\{N\}(\backslash bar\; 0,1\backslash !\backslash !1)$,

we have

\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},

which proves the claim.

{{hidden end}}

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 $X\_i,i=\backslash overline\{1,n\}$ are independent chi-squared variables with $k\_i$, $i=\backslash overline\{1,n\}$ degrees of freedom, respectively, then $Y\; =\; X\_1\; +\; \backslash cdots\; +\; X\_n$ is chi-squared distributed with $k\_1\; +\; \backslash cdots\; +\; k\_n$ degrees of freedom.

The sample mean of $n$ i.i.d. chi-squared variables of degree $k$ is distributed according to a gamma distribution with shape $\backslash alpha$ and scale $\backslash theta$ parameters:

:$\backslash overline\; X\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; X\_i\; \backslash sim\; \backslash operatorname\{Gamma\}\backslash left(\backslash alpha=n\backslash ,\; k\; /2,\; \backslash theta=\; 2/n\; \backslash right)\; \backslash qquad\; \backslash text\{where\; \}\; X\_i\; \backslash sim\; \backslash chi^2(k)$

Asymptotically, given that for a shape parameter $\backslash alpha$ going to infinity, a Gamma distribution converges towards a normal distribution with expectation $\backslash mu\; =\; \backslash alpha\backslash cdot\; \backslash theta$ and variance $\backslash sigma^2\; =\; \backslash alpha\backslash ,\; \backslash theta^2$, the sample mean converges towards:

$\backslash overline\; X\; \backslash xrightarrow\{n\; \backslash to\; \backslash infty\}\; N(\backslash mu\; =\; k,\; \backslash sigma^2\; =\; 2\backslash ,\; k\; /n\; )$

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

The differential entropy is given by

: $$

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),

where $\backslash psi(x)$ is the Digamma function.

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

The noncentral moments (raw moments) of a chi-squared distribution with $k$ degrees of freedom are given by[http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-squared distribution], from MathWorld, retrieved Feb. 11, 2009M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), {{ISBN|978-0-387-34657-1}}

: $$

\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)}.

The cumulants are readily obtained by a power series expansion of the logarithm of the characteristic function:

: $\backslash kappa\_n\; =\; 2^\{n-1\}(n-1)!\backslash ,k$

with cumulant generating function $\backslash ln\; E[e^\{tX\}]\; =\; -\; \backslash frac\; k2\; \backslash ln(1-2t)$.

The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart{{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 }} bounds are:

: $\backslash operatorname\{P\}(X\; -\; k\; \backslash ge\; 2\; \backslash sqrt\{k\; x\}\; +\; 2x)\; \backslash le\; \backslash exp(-x)$

: $\backslash operatorname\{P\}(k\; -\; X\; \backslash ge\; 2\; \backslash sqrt\{k\; x\})\; \backslash le\; \backslash exp(-x)$

One consequence is that, if $Z\; \backslash sim\; N(0,\; 1)^k$ is a gaussian random vector in $\backslash R^k$, then as the dimension $k$ grows, the squared length of the vector is concentrated tightly around $k$ with a width $k^\{1/2\; +\; \backslash alpha\}$:$$Pr(\backslash |Z\backslash |^2\; \backslash in\; [k\; -\; 2k^\{1/2+\backslash alpha\},\; k\; +\; 2k^\{1/2+\backslash alpha\}\; +\; 2k^\{\backslash alpha\}])\; \backslash geq\; 1-e^\{-k^\backslash alpha\}$$where the exponent $\backslash alpha$ can be chosen as any value in $\backslash R$.

Since the cumulant generating function for $\backslash chi^2(k)$ is $K(t)\; =\; -\backslash frac\; k2\; \backslash ln(1-2t)$, and its convex dual is $K^*(q)\; =\; \backslash frac\; 12\; (q-k\; +\; k\backslash ln\backslash frac\; kq)$, the standard Chernoff bound yields$$\backslash begin\{aligned\}$$

\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}where . By the union bound,$$Pr(X\; \backslash in\; (1\backslash pm\; \backslash epsilon\; )\; k\; )\; \backslash geq\; 1\; -\; 2e^\{-\backslash frac\; k2\; (\backslash frac\; 12\; \backslash epsilon^2\; -\; \backslash frac\; 13\; \backslash epsilon^3)\}$$This result is used in proving the Johnson–Lindenstrauss lemma.[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]

File:Chi-square median approx.png

By the central limit theorem, because the chi-squared distribution is the sum of $k$ independent random variables with finite mean and variance, it converges to a normal distribution for large $k$. For many practical purposes, for $k>50$ the distribution is sufficiently close to a normal distribution, so the difference is ignorable.{{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}} Specifically, if $X\; \backslash sim\; \backslash chi^2(k)$, then as $k$ tends to infinity, the distribution of $(X-k)/\backslash sqrt\{2k\}$ tends to a standard normal distribution. However, convergence is slow as the skewness is $\backslash sqrt\{8/k\}$ and the excess kurtosis is $12/k$.

The sampling distribution of $\backslash ln(\backslash chi^2)$ converges to normality much faster than the sampling distribution of $\backslash chi^2$,{{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 }} as the logarithmic transform removes much of the asymmetry.{{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}}

Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

• If $X\; \backslash sim\; \backslash chi^2(k)$ then $\backslash sqrt\{2X\}$ is approximately normally distributed with mean $\backslash sqrt\{2k-1\}$ and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of Johnson.
• If $X\; \backslash sim\; \backslash chi^2(k)$ then $\backslash sqrt[3]\{X/k\}$ is approximately normally distributed with mean $1-\backslash frac\{2\}\{9k\}$ and variance $\backslash frac\{2\}\{9k\}\; .${{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 }} This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of Johnson.
• This normalizing transformation leads directly to the commonly used median approximation $k\backslash bigg(1-\backslash frac\{2\}\{9k\}\backslash bigg)^3\backslash ;$ by back-transforming from the mean, which is also the median, of the normal distribution.

{{More citations needed section|date=September 2011}}

• As $k\backslash to\backslash infty$, $(\backslash chi^2\_k-k)/\backslash sqrt\{2k\}\; ~\; \backslash xrightarrow\{d\}\backslash \; N(0,1)\; \backslash ,$ (normal distribution)
• $\backslash chi\_k^2\; \backslash sim\; \{\backslash chi\text{'}\}^2\_k(0)$ (noncentral chi-squared distribution with non-centrality parameter $\backslash lambda\; =\; 0$)
• If $Y\; \backslash sim\; \backslash mathrm\{F\}(\backslash nu\_1,\; \backslash nu\_2)$ then $X\; =\; \backslash lim\_\{\backslash nu\_2\; \backslash to\; \backslash infty\}\; \backslash nu\_1\; Y$ has the chi-squared distribution $\backslash chi^2\_\{\backslash nu\_\{1\}\}$

:*As a special case, if $Y\; \backslash sim\; \backslash mathrm\{F\}(1,\; \backslash nu\_2)\backslash ,$ then $X\; =\; \backslash lim\_\{\backslash nu\_2\; \backslash to\; \backslash infty\}\; Y\backslash ,$ has the chi-squared distribution $\backslash chi^2\_\{1\}$

• $\backslash |\backslash boldsymbol\{N\}\_\{i=1,\backslash ldots,k\}\; (0,1)\; \backslash |^2\; \backslash sim\; \backslash chi^2\_k$ (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
• If $X\; \backslash sim\; \backslash chi^2\_\backslash nu\backslash ,$ and $c>0\; \backslash ,$, then $cX\; \backslash sim\; \backslash Gamma(k=\backslash nu/2,\; \backslash theta=2c)\backslash ,$. (gamma distribution)
• If $X\; \backslash sim\; \backslash chi^2\_k$ then $\backslash sqrt\{X\}\; \backslash sim\; \backslash chi\_k$ (chi distribution)
• If $X\; \backslash sim\; \backslash chi^2\_2$, then $X\; \backslash sim\; \backslash operatorname\{exp\}(1/2)$ is an exponential distribution. (See gamma distribution for more.)
• If $X\; \backslash sim\; \backslash chi^2\_\{2k\}$, then $X\; \backslash sim\; \backslash operatorname\{Erlang\}(k,\; 1/2)$ is an Erlang distribution.
• If $X\; \backslash sim\; \backslash operatorname\{Erlang\}(k,\backslash lambda)$, then $2\backslash lambda\; X\backslash sim\; \backslash chi^2\_\{2k\}$
• If $X\; \backslash sim\; \backslash operatorname\{Rayleigh\}(1)\backslash ,$ (Rayleigh distribution) then $X^2\; \backslash sim\; \backslash chi^2\_2\backslash ,$
• If $X\; \backslash sim\; \backslash operatorname\{Maxwell\}(1)\backslash ,$ (Maxwell distribution) then $X^2\; \backslash sim\; \backslash chi^2\_3\backslash ,$
• If $X\; \backslash sim\; \backslash chi^2\_\backslash nu$ then $\backslash tfrac\{1\}\{X\}\; \backslash sim\; \backslash operatorname\{Inv-\}\backslash chi^2\_\backslash nu\backslash ,$ (Inverse-chi-squared distribution)
• The chi-squared distribution is a special case of type III Pearson distribution
• If $X\; \backslash sim\; \backslash chi^2\_\{\backslash nu\_1\}\backslash ,$ and $Y\; \backslash sim\; \backslash chi^2\_\{\backslash nu\_2\}\backslash ,$ are independent then $\backslash tfrac\{X\}\{X+Y\}\; \backslash sim\; \backslash operatorname\{Beta\}(\backslash tfrac\{\backslash nu\_1\}\{2\},\; \backslash tfrac\{\backslash nu\_2\}\{2\})\backslash ,$ (beta distribution)
• If $X\; \backslash sim\; \backslash operatorname\{U\}(0,1)\backslash ,$ (uniform distribution) then $-2\backslash log(X)\; \backslash sim\; \backslash chi^2\_2\backslash ,$
• If $X\_i\; \backslash sim\; \backslash operatorname\{Laplace\}(\backslash mu,\backslash beta)\backslash ,$ then $\backslash sum\_\{i=1\}^n\; \backslash frac\{2\; |X\_i-\backslash mu|\}\{\backslash beta\}\; \backslash sim\; \backslash chi^2\_\{2n\}\backslash ,$
• If $X\_i$ follows the generalized normal distribution (version 1) with parameters $\backslash mu,\backslash alpha,\backslash beta$ then $\backslash sum\_\{i=1\}^n\; \backslash frac\{2\; |X\_i-\backslash mu|^\backslash beta\}\{\backslash alpha\}\; \backslash sim\; \backslash chi^2\_\{2n/\backslash beta\}\backslash ,$ {{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 }}
• 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 $k$ degrees of freedom is defined as the sum of the squares of $k$ independent standard normal random variables.

If $Y$ is a $k$-dimensional Gaussian random vector with mean vector $\backslash mu$ and rank $k$ covariance matrix $C$, then $X\; =\; (Y-\backslash mu\; )^\{T\}C^\{-1\}(Y-\backslash mu)$ is chi-squared distributed with $k$ 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 $Y$ is a vector of $k$ i.i.d. standard normal random variables and $A$ is a $k\backslash times\; k$ symmetric, idempotent matrix with rank $k-n$, then the quadratic form $Y^TAY$ is chi-square distributed with $k-n$ degrees of freedom.

If $\backslash Sigma$ is a $p\backslash times\; p$ positive-semidefinite covariance matrix with strictly positive diagonal entries, then for $X\backslash sim\; N(0,\backslash Sigma)$ and $w$ a random $p$-vector independent of $X$ such that $w\_1+\backslash cdots+w\_p=1$ and $w\_i\backslash geq\; 0,\; i=1,\backslash ldots,p,$ then

: $\backslash frac\{1\}\{\backslash left(\backslash frac\{w\_1\}\{X\_1\},\backslash ldots,\backslash frac\{w\_p\}\{X\_p\}\backslash right)\backslash Sigma\backslash left(\backslash frac\{w\_1\}\{X\_1\},\backslash ldots,\backslash frac\{w\_p\}\{X\_p\}\backslash right)^\backslash top\}\; \backslash sim\; \backslash chi\_1^2.$

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

• $Y$ is F-distributed, $Y\; \backslash sim\; F(k\_1,\; k\_2)$ if $Y\; =\; \backslash frac\{\; \{X\_1\}/\{k\_1\}\; \}\{\; \{X\_2\}/\{k\_2\}\; \}$, where $X\_1\; \backslash sim\; \backslash chi^2\_\{k\_1\}$ and $X\_2\; \backslash sim\; \backslash chi^2\_\{k\_2\}$ are statistically independent.
• If $X\_1\; \backslash sim\; \backslash chi^2\_\{k\_1\}$ and $X\_2\; \backslash sim\; \backslash chi^2\_\{k\_2\}$ are statistically independent, then $X\_1\; +\; X\_2\backslash sim\; \backslash chi^2\_\{k\_1+k\_2\}$. If $X\_1$ and $X\_2$ are not independent, then $X\_1+X\_2$ is not chi-square distributed.

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.

If $X\_1,\backslash ldots,X\_n$ are chi square random variables and $a\_1,\backslash ldots,a\_n\backslash in\backslash mathbb\{R\}\_\{>0\}$, then the distribution of $X=\backslash sum\_\{i=1\}^n\; a\_i\; X\_i$ is a special case of a Generalized Chi-squared Distribution.

A closed expression for this distribution is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-square random variables.{{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

}}

{{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.

{{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.

The chi-squared distribution $X\; \backslash sim\; \backslash chi\_k^2$ is a special case of the gamma distribution, in that $X\; \backslash sim\; \backslash Gamma\; \backslash left(\backslash frac\{k\}2,\backslash frac\{1\}2\backslash right)$ using the rate parameterization of the gamma distribution (or

$X\; \backslash sim\; \backslash Gamma\; \backslash left(\backslash frac\{k\}2,2\; \backslash right)$ 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 $X\; \backslash sim\; \backslash chi\_2^2$, then $X\backslash sim\; \backslash operatorname\{exp\}\backslash left(\backslash frac\; 1\; 2\backslash right)$ is an exponential distribution.

The Erlang distribution is also a special case of the gamma distribution and thus we also have that if $X\; \backslash sim\backslash chi\_k^2$ with even $k$, then $X$ is Erlang distributed with shape parameter $k/2$ and scale parameter $1/2$.

The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a 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 variables, 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 $X\_1,\; ...,\; X\_n$ are i.i.d. $N(\backslash mu,\; \backslash sigma^2)$ random variables, then $\backslash sum\_\{i=1\}^n(X\_i\; -\; \backslash overline\{X\})^2\; \backslash sim\; \backslash sigma^2\; \backslash chi^2\_\{n-1\}$ where $\backslash overline\{X\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; X\_i$.
• The box below shows some statistics based on $X\_i\; \backslash sim\; N(\backslash mu\_i,\; \backslash sigma^2\_i),\; i=\; 1,\; \backslash ldots,\; k$ independent random variables that have probability distributions related to the chi-squared distribution:

The chi-squared distribution is also often encountered in magnetic resonance imaging.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]

The $p$-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 $\backslash chi^2$ for the first 10 degrees of freedom.

These values can be calculated evaluating the quantile function (also known as "inverse CDF" or "ICDF") of the chi-squared distribution;{{Cite web|url=https://www.r-tutor.com/elementary-statistics/probability-distributions/chi-squared-distribution|title=Chi-squared Distribution | R Tutorial|website=www.r-tutor.com}} e. g., the {{math|χ²}} 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 from the table.

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}}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 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, writing {{mvar|−½χ²}} for what would appear in modern notation as {{math|−½x^TΣ⁻¹x}} (Σ being the covariance matrix).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]. 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}}

{{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{{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}} with the pdf on $(0,\; \backslash infty)$ is given as $f(x)=\; \backslash frac\{2\backslash beta^\{\backslash alpha/2\}\; x^\{\backslash alpha-1\}\; \backslash exp(-\backslash beta\; x^2+\; \backslash gamma\; x\; )\}\{\backslash Psi\{\backslash left(\backslash frac\{\backslash alpha\}\{2\},\; \backslash frac\{\; \backslash gamma\}\{\backslash sqrt\{\backslash beta\}\}\backslash right)\}\}$, where $\backslash Psi(\backslash alpha,z)=\{\}\_1\backslash Psi\_1\backslash left(\backslash begin\{matrix\}\backslash left(\backslash alpha,\backslash frac\{1\}\{2\}\backslash right)\backslash \backslash (1,0)\backslash end\{matrix\};z\; \backslash right)$ denotes the Fox–Wright Psi function.

{{Colend}}

{{Reflist|30em}}

• {{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}}

{{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}}

• [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. Σx², 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}}

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Category:Normal distribution

Category:Infinitely divisible probability distributions