Chi-squared distribution

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

Probability density function
Cumulative distribution function
notation:	$\chi^2(k)\!$ or $\chi^2_k\!$
parameters:	k ∈ N₁ — degrees of freedom
support:	x ∈ [0, +∞)
pdf:	$\frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\,$
cdf:	$\frac{1}{\Gamma(k/2)}\;\gamma(k/2,\,x/2)$
mean:	k
median:	$\approx k\bigg(1-\frac{2}{9k}\bigg)^3$
mode:	max{ k − 2, 0 }
variance:	2k
skewness:	$\scriptstyle\sqrt{8/k}\,$
ex.kurtosis:	12 / k
entropy:	$\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$
mgf:	(1 − 2 t)^−k/2 for t < ½
cf:	(1 − 2 i t)^−k/2 ^[1]

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It is one of the most widely used probability distributions in inferential statistics, e.g., in hypothesis testing or in construction of confidence intervals.^[2]^[3]^[4]^[5] When there is a need to contrast it with the noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, like Friedman's analysis of variance by ranks.

The chi-squared distribution is a special case of the gamma distribution.

Definition

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

$Q\ = \sum_{i=1}^k Z_i^2 ,$

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

$Q\ \sim\ \chi^2(k)\ \ \text{or}\ \ Q\ \sim\ \chi^2_k .$

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i.e. the number of Z_i’s)

Characteristics

Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article.

Probability density function

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

$f(x;\,k) = \begin{cases} \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{k/2 - 1} e^{-x/2}, & x \geq 0; \\ 0, & \text{otherwise}. \end{cases}$

where Γ(k/2) denotes the Gamma function, which has closed-form values at the half-integers.

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

Cumulative distribution function

Its cumulative distribution function is:

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

where γ(k,z) is the lower incomplete Gamma function and P(k,z) is the regularized Gamma function.

In a special case of k = 2 this function has a simple form:

$F(x;\,2) = 1 - e^{-\frac{x}{2}}.$

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages. For a closed form approximation for the CDF, see under Noncentral chi-squared distribution.

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 {X_i}_i=1ⁿ are independent chi-squared variables with {k_i}_i=1ⁿ degrees of freedom, respectively, then Y = X₁ + ⋯ + X_n is chi-squared distributed with k₁ + ⋯ + k_n degrees of freedom.

Information entropy

The information entropy is given by

$H = \int_{-\infty}^\infty f(x;\,k)\ln f(x;\,k) \, dx = \frac{k}{2} + \ln\big( 2\Gamma(k/2) \big) + \big(1 - k/2\big) \psi(k/2),$

where ψ(x) is the Digamma function.

The Chi-squared distribution is the maximum entropy probability distribution for a random variate X for which $E (X) = ν$ is fixed, and $E(\ln(X))=\psi\left(\frac{1}{2}\right)+\ln(2)$ is fixed. ^[6]

Noncentral moments

The moments about zero of a chi-squared distribution with k degrees of freedom are given by^[7]^[8]

$\operatorname{E}(X^m) = k (k+2) (k+4) \cdots (k+2m-2) = 2^m \frac{\Gamma(m+k/2)}{\Gamma(k/2)}.$

Cumulants

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

$\kappa_n = 2^{n-1}(n-1)!\,k$

Asymptotic properties

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 for the difference to be ignored.^[9] Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of $(X-k)/\sqrt{2k}$ tends to a standard normal distribution. However, convergence is slow as the skewness is $\sqrt{8/k}$ and the excess kurtosis is 12/k. Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

If X ~ χ²(k) then $\scriptstyle\sqrt{2X}$ is approximately normally distributed with mean $\scriptstyle\sqrt{2k-1}$ and unit variance (result credited to R. A. Fisher).
If X ~ χ²(k) then $\scriptstyle\sqrt[3]{X/k}$ is approximately normally distributed with mean $\scriptstyle 1-2/(9k)$ and variance $\scriptstyle 2/(9k) .$ ^[10] This is known as the Wilson-Hilferty transformation.

Related distributions

$\lim_{k \to \infty}\tfrac{\chi^2_k(x)-\mu_k}{\sigma_k} \xrightarrow{d}\ N(0,1) \,$ (normal distribution)
$\chi_k^2 \sim {\chi'}^2_k(0)$ (Noncentral chi-squared distribution with non-centrality parameter $λ = 0$ )
If $X \simF(ν 1,ν 2)$ then $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ has the chi-squared distribution $\chi^2_{\nu_{1}}$
As a special case, if $X \sim \mathrm{F}(1, \nu_2)\,$ then $Y = \lim_{\nu_2 \to \infty} X\,$ has the chi-squared distribution $\chi^2_{1}$
$\|\boldsymbol{N}_{i=1,...,k}{(0,1)}\|^2 \sim \chi^2_k(x)$ (The squared norm of n standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
If $X \sim {\chi}^2(\nu)\,$ and $c>0 \,$ , then $cX \sim {\Gamma}(k=\nu/2, \theta=2c)\,$ . (gamma distribution)
If $X \sim \chi^2_k(x)$ then $\sqrt{X} \sim \chi_k$ (chi distribution)
If $X \sim \mathrm{Rayleigh}(1)\,$ (Rayleigh distribution) then $X^2 \sim \chi^2(2)\,$
If $X \sim \mathrm{Maxwell}(1)\,$ (Maxwell distribution) then $X^2 \sim \chi^2(3)\,$
If $X \simχ 2 (ν)$ then $\tfrac{1}{X} \sim \mbox{Inv-}\chi^2(\nu)\,$ (Inverse-chi-squared distribution)
The chi-squared distribution is a special case of type 3 Pearson distribution
If $X \sim \chi^2(v_1)\,$ and $Y \sim \chi^2(v_2)\,$ then $\tfrac{X}{X+Y} \sim {\rm Beta}(\tfrac{v_1}{2}, \tfrac{v_2}{2})\,$ (beta distribution)
If $X \sim {\rm U}(0,1)\,$ (Uniform distribution (continuous)) then $-2\log{(U)} \sim \chi^2(2)\,$
$\chi^2(6)\,$ is a transformation of Laplace distribution
If $X_i \sim \mathrm{Laplace}(\mu,\beta)\,$ then $\sum_{i=1}^n{\frac{2}{\beta|X_i-\mu|}} \sim \chi^2(2n)\,$
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
Noncentral beta distribution can be obtained as a transformation of chi-squared distribution and Noncentral chi-squared distribution
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 μ and rank k covariance matrix C, then X = (Y−μ)^TC⁻¹(Y−μ) 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×k idempotent matrix with rank k−n then the quadratic form Y^TAY is chi-squared distributed with k−n degrees of freedom.

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

Y is F-distributed, Y ~ F(k₁,k₂) if $\scriptstyle Y = \frac{X_1 / k_1}{X_2 / k_2}$ where X₁ ~ χ²(k₁) and X₂ ~ χ²(k₂) are statistically independent.

If X is chi-squared distributed, then $\scriptstyle\sqrt{X}$ is chi distributed.

If X₁ ~ χ²_k₁ and X₂ ~ χ²_k₂ are statistically independent, then X₁ + X₂ ~ χ²_k₁+k₂. If X₁ and X₂ are not independent, then X₁ + X₂ is not chi-squared distributed.

Generalizations

The chi-squared distribution is obtained as the sum of the squares of 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.

Chi-squared distributions

Noncentral chi-squared distribution

Main article: 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 article: Generalized chi-squared distribution

The generalized chi-squared distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions

The chi-squared distribution X ~ χ²(k) is a special case of the gamma distribution, in that X ~ Γ(k/2, 2) (using the shape parameterization of the gamma distribution) where k/2 is an integer.

Because the exponential distribution is also a special case of the Gamma distribution, we also have that if X ~ χ²(2), then X ~ Exp(1/2) is an exponential distribution.

The Erlang distribution is also a special case of the Gamma distribution and thus we also have that if X ~ χ²(k) with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

Applications

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 X₁, ..., X_n are i.i.d. N(μ, σ²) random variables, then $\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2(n-1)$ where $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ .

The box below shows probability distributions with name starting with chi for some statistics based on X_i ∼ Normal(μ_i, σ²_i), i = 1, ⋯, k, independent random variables:

Name	Statistic
chi-squared distribution	$\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-squared distribution	$\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution	$\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution	$\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$

Table of χ² value vs p-value

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. The table below gives a number of p-values matching to χ² for the first 10 degrees of freedom.

A p-value of 0.05 or less is usually regarded as statistically significant, i.e. the observed deviation from the null hypothesis is significant.

Degrees of freedom (df)	χ² value ^[11]
1	0.004	0.02	0.06	0.15	0.46	1.07	1.64	2.71	3.84	6.64	10.83
2	0.10	0.21	0.45	0.71	1.39	2.41	3.22	4.60	5.99	9.21	13.82
3	0.35	0.58	1.01	1.42	2.37	3.66	4.64	6.25	7.82	11.34	16.27
4	0.71	1.06	1.65	2.20	3.36	4.88	5.99	7.78	9.49	13.28	18.47
5	1.14	1.61	2.34	3.00	4.35	6.06	7.29	9.24	11.07	15.09	20.52
6	1.63	2.20	3.07	3.83	5.35	7.23	8.56	10.64	12.59	16.81	22.46
7	2.17	2.83	3.82	4.67	6.35	8.38	9.80	12.02	14.07	18.48	24.32
8	2.73	3.49	4.59	5.53	7.34	9.52	11.03	13.36	15.51	20.09	26.12
9	3.32	4.17	5.38	6.39	8.34	10.66	12.24	14.68	16.92	21.67	27.88
10	3.94	4.86	6.18	7.27	9.34	11.78	13.44	15.99	18.31	23.21	29.59
P value (Probability)	0.95	0.90	0.80	0.70	0.50	0.30	0.20	0.10	0.05	0.01	0.001
	Nonsignificant								Significant

References

^ M.A. Sanders. "Characteristic function of the central chi-squared distribution". http://www.planetmathematics.com/CentralChiDistr.pdf. Retrieved 2009-03-06.
^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 940, ISBN 978-0486612720, MR 0167642, http://www.math.sfu.ca/~cbm/aands/page_940.htm .
^ NIST (2006). Engineering Statistics Handbook - Chi-Squared Distribution
^ Jonhson, N.L.; S. Kotz, , N. Balakrishnan (1994). Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18). John Willey and Sons. ISBN 0-471-58495-9.
^ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 241-246). McGraw-Hill. ISBN 0-07-042864-6.
^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf. Retrieved 2011-06-02.
^ Chi-squared distribution, from MathWorld, retrieved Feb. 11, 2009
^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1
^ Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 46.
^ Wilson, E.B.; Hilferty, M.M. (1931) "The distribution of chi-squared". Proceedings of the National Academy of Sciences, Washington, 17, 684–688.
^ Chi-Squared Test 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

External links

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · Beta-binomial · binomial · categorical · hypergeometric · Poisson binomial · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial · Boltzmann · Conway–Maxwell–Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine · ARGUS · Balding-Nichols · Bates · Beta · Noncentral beta · Irwin–Hall · Kumaraswamy · logit-normal · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini · Benktander 1st kind · Benktander 2nd kind · Beta prime · Bose–Einstein · Burr · chi-squared · chi · Coxian · Dagum · Davis · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-squared · hyper-exponential · hypoexponential · inverse chi-squared (scaled-inverse-chi-squared) · inverse Gaussian · inverse gamma · Kolmogorov · Lévy · log-Cauchy · log-Laplace · log-logistic · log-normal · Maxwell–Boltzmann · Maxwell speed · Mittag–Leffler · Nakagami · noncentral chi-squared · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · exponential power · Fisher's z · generalized normal · generalized hyperbolic · geometric stable · Gumbel · Holtsmark · hyperbolic secant · Landau · Laplace · Linnik · logistic · noncentral t · normal (Gaussian) · normal-inverse Gaussian · skew normal · slash · stable · Student's t · type-1 Gumbel · variance-gamma · Voigt

Continuous univariate with support whose type varies

generalized extreme value · generalized Pareto · Tukey lambda · q-Gaussian · q-exponential · shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Pólya · negative multinomial Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · Multivariate stable · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional

Univariate (circular) directional: Circular uniform · univariate von Mises · wrapped normal · wrapped Cauchy · wrapped exponential · wrapped Lévy Bivariate (spherical): Kent Bivariate (toroidal): bivariate von Mises Multivariate: von Mises–Fisher · Bingham

Degenerate and singular

Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

Circular · compound Poisson · elliptical · exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · Tweedie · wrapped

v · d · eSome common univariate probability distributions

Continuous	beta • Cauchy • chi-squared • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

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Chi-squared distribution

Contents

Definition