# Chi-squared distribution

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Chi-squared distribution
notation: Probability density function Cumulative distribution function $\chi^2(k)\!$ or $\chi^2_k\!$ k ∈ N1 — degrees of freedom x ∈ [0, +∞) $\frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\,$ $\frac{1}{\Gamma(k/2)}\;\gamma(k/2,\,x/2)$ k $\approx k\bigg(1-\frac{2}{9k}\bigg)^3$ max{ k − 2, 0 } 2k $\scriptstyle\sqrt{8/k}\,$ 12 / k $\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$ (1 − 2 t)−k/2   for  t  < ½ (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, ..., Zk 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 Zi’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

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

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.

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 {Xi}i=1n are independent chi-squared variables with {ki}i=1n degrees of freedom, respectively, then Y = X1 + ⋯ + Xn is chi-squared distributed with k1 + ⋯ + kn 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∼F(ν12) 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∼χ2(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 = (Yμ)TC−1(Yμ) 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 YTAY 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  ~  χ2k1 and X2  ~  χ2k2 are statistically independent, then X1 + X2  ~ χ2k1+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

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

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, ..., Xn 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 Xi ∼ Normal(μi, σ2i), 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

1. ^ M.A. Sanders. "Characteristic function of the central chi-squared distribution". Retrieved 2009-03-06.
2. ^ 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, MR0167642 .
3. ^ NIST (2006). Engineering Statistics Handbook - Chi-Squared Distribution
4. ^ 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.
5. ^ 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.
6. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. Retrieved 2011-06-02.
7. ^ Chi-squared distribution, from MathWorld, retrieved Feb. 11, 2009
8. ^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1
9. ^ Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 46.
10. ^ Wilson, E.B.; Hilferty, M.M. (1931) "The distribution of chi-squared". Proceedings of the National Academy of Sciences, Washington, 17, 684–688.
11. ^ 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

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