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	$k \in \mathbb{N}~~$ (known as "degrees of freedom")
Support	x ∈ [0, +∞)
pdf	$\frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\,$
CDF	$\frac{1}{\Gamma\left(\frac{k}{2}\right)}\;\gamma\left(\frac{k}{2},\,\frac{x}{2}\right)$
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

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. 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

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

where Γ(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.

Cumulative distribution function

Chernoff bound for the CDF and tail (1-CDF) of a chi-square 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 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:

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

For the cases when 0 < z < 1 (which include all of the cases when this CDF is less than half), the following Chernoff upper bound may be obtained:

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

The tail bound for the cases when z > 1 follows similarly

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

Tables of this cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages. For another approximation for the CDF modeled after the cube of a Gaussian, 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.

Entropy

The differential entropy is given by

$h = \int_{-\infty}^\infty f(x;\,k)\ln f(x;\,k) \, dx = \frac{k}{2} + \ln\!\left[2\,\Gamma\!\left(\frac{k}{2}\right)\right] + \left(1-\frac{k}{2}\right)\, \psi\!\left[\frac{k}{2}\right],$

where ψ(x) is the Digamma function.

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

Noncentral moments

The moments about zero of a chi-squared distribution with k degrees of freedom are given by

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

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. 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.

• The sampling distribution of ln(χ²) converges to normality much faster than the sampling distribution of χ², as the logarithm removes much of the asymmetry. 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 $\scriptstyle 2/(9k) .$ This is known as the Wilson-Hilferty transformation.

Relation to other distributions

Approximate formula for median compared with numerical quantile (top) as presented in SAS Software. Difference between numerical quantile and approximate formula (bottom).

• As $k\to\infty$, $(\chi^2_k-k)/\sqrt{2k} ~ \xrightarrow{d}\ N(0,1) \,$ (normal distribution)

• $\chi_k^2 \sim {\chi'}^2_k(0)$ (Noncentral chi-squared distribution with non-centrality parameter $\lambda = 0$)

• If $X \sim \mathrm{F}(\nu_1, \nu_2)$ 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 $\chi^2_{1}$

• $\|\boldsymbol{N}_{i=1,...,k}{(0,1)}\|^2 \sim \chi^2_k$ (The squared norm of k 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 $X \sim \chi^2_k$ then $\sqrt{X} \sim \chi_k$ (chi distribution)

• If $X \sim \chi^2 \left( 2 \right)$, then $X \sim \mathrm{Exp(1/2)}$ is an exponential distribution. (See Gamma distribution for more.)

• 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 $X \sim \chi^2(\nu)$ 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 $X \sim \chi^2(\nu_1)\,$ and $Y \sim \chi^2(\nu_2)\,$ are independent then $\tfrac{X}{X+Y} \sim {\rm Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2})\,$ (beta distribution)

• If $X \sim {\rm U}(0,1)\,$ (uniform distribution) 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 $\sum_{i=1}^n{\frac{2 |X_i-\mu|}{\beta}} \sim \chi^2(2n)\,$

• 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, 1/2) using the rate parameterization of the gamma distribution (or X ~ Γ(k/2, 2) using the scale parameterization of the gamma distribution) where k 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
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
	Non-significant								Significant

History

This distribution was first described by the German statistician Helmert in papers of 1875/1876, where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmertsche ("Helmertian") or "Helmert distribution". The distribution was independently rediscovered by Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-squared test, published in (Pearson 1900), with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII). Further developments and corrections to the early work were due to Fisher in the 1920s.

See also

• Cochran's theorem
• F-distribution
• Fisher's method for combining independent tests of significance
• Gamma distribution
• Generalized chi-squared distribution
• Hotelling's T-squared distribution
• Pearson's chi-squared test
• Student's t-distribution
• Wilks' lambda distribution
• Wishart distribution

References

• Hald, Anders (1998). A history of mathematical statistics from 1750 to 1930. New York: Wiley. ISBN 0-471-17912-4. 
• Elderton, William Palin (1902). "Tables for Testing the Goodness of Fit of Theory to Observation". Biometrika 1 (2): 155–163. doi:10.1093/biomet/1.2.155. 

External links

• Hazewinkel, Michiel, ed. (2001), "Chi-squared distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
• Course notes on Chi-Squared Goodness of Fit Testing from Yale University Stats 101 class.
• Mathematica demonstration showing the chi-squared sampling distribution of various statistics, e.g. Σx², for a normal population
• Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator