Chi-square distribution

Chi-square distribution

Encyclopedia

Tutorials

Encyclopedia

Dictionary

Directory

Email this to a friend Chi-square_distribution

Chi-square distribution

In probability theory and statistics, the chi-square distribution (also chi-squared or \chi^2 distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests.^[1]^[2]^[3]^[4] It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

The best-known situations in which the chi-square distribution are used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

Definition
Characteristics

Probability density function
Cumulative distribution function
Characteristic function
Expected value and variance
Median
Information entropy

Related distributions and properties
See also
External links
References

Definition

If X_i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable

Q = \sum_{i=1}^k X_i^2

is distributed according to the chi-square distribution with k degrees of freedom. This is usually written

Q\sim\chi^2_k.\,

The chi-square distribution has one parameter: k - a positive integer that specifies the number of degrees of freedom (i.e. the number of X_i)

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

Characteristics

Probability density function

A probability density function of the chi-square distribution is

f(x;k)= \begin{cases}\displaystyle \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{(k/2) - 1} e^{-x/2}&\text{for }x>0,\\ 0&\text{for }x\le0, \end{cases}

where \Gamma denotes the Gamma function, which has closed-form values at the half-integers.

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 \gamma(k,z) is the lower incomplete Gamma function and P(k, z) is the regularized Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.

Characteristic function

The characteristic function of the Chi-square distribution is

\chi(t;k)=(1-2it)^{-k/2}.\,

Expected value and variance

If X\sim\chi^2_k then

\mathrm{E}(X)=k

\mathrm{Var}(X)=2k

Median

The median of X\sim\chi^2_k is given approximately by

k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.

Information entropy

The information 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(k/2).

where \psi(x) is the Digamma function.

Related distributions and properties

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square 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 divided by their respective degrees of freedom.

If X\sim\chi^2_k, then as k tends to infinity, the distribution of X tends to normality (convergence is slow as the skewness is \sqrt{8/k} and the excess kurtosis is 12/k)
If X\sim\chi^2_k then \sqrt{2X} is approximately normally distributed with mean \sqrt{2k-1} and unit variance (result credited to R. A. Fisher).
If X\sim\chi^2_k then \sqrt[3]{X/k} is approximately normally distributed with mean 1-2/(9k) and variance 2/(9k) (Wilson and Hilferty,1931)
X \sim \mathrm{Exponential}(\lambda = \tfrac{1}{2}) is an exponential distribution if X \sim \chi_2^2 (with 2 degrees of freedom).
Y \sim \chi_{\nu}^2 is a chi-square distribution if Y = \sum_{m=1}^{\nu} X_m^2 for X_i \sim N(0,1) independent that are normally distributed.

If \boldsymbol{z}'=[Z_1,Z_2,\cdots,Z_n], where the Z_is are independent \rm{Normal}(0,\sigma^2) random variables or \boldsymbol{z}\sim N_p(\boldsymbol{0},\sigma^2 \mathrm{I}) and \boldsymbol{A} is an n\times n idempotent matrix with rank n-k then the quadratic form \frac{\boldsymbol{z}'\boldsymbol{A}\boldsymbol{z}}{\sigma^2}\sim \chi^2_{n-k}.

If the X_i\sim N(\mu_i,1) have nonzero means, then Y = \sum_{m=1}^k X_m^2 is drawn from a noncentral chi-square distribution.
The chi-square distribution X\sim\chi^2_\nu is a special case of the gamma distribution, in that X \sim {\Gamma}(\frac{\nu}{2}, \theta=2).
Y \sim \mathrm{F}(\nu_1, \nu_2) is an F-distribution if Y = \frac{X_1 / \nu_1}{X_2 / \nu_2} where X_1 \sim \chi_{\nu_1}^2 and X_2 \sim \chi_{\nu_2}^2 are independent with their respective degrees of freedom.
Y \sim \chi^2(\bar{\nu}) is a chi-square distribution if Y = \sum_{m=1}^N X_m where X_m \sim \chi^2(\nu_m) are independent and \bar{\nu} = \sum_{m=1}^N \nu_m.
if X is chi-square distributed, then \sqrt{X} is chi distributed.
in particular, if X \sim \chi_2^2 (chi-square with 2 degrees of freedom), then \sqrt{X} is Rayleigh distributed.
if X_1, \dots, X_n are i.i.d. N(\mu,\sigma^2) 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.
if X \sim \mathrm{SkewLogistic}(\tfrac{1}{2})\,, then \mathrm{log}(1 + e^{-X}) \sim \chi_2^2\,
The box below shows probability distributions with name starting with chi for some statistics based on X_i\sim \mathrm{Normal}(\mu_i,\sigma^2_i),i=1,\cdots,k, independent random variables:

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