Recent Articles

Chi-square

For any positive integer <math>k<math>, the chi-square distribution with k degrees of freedom is the probability distribution of the random variable

<math>X=Z_1^2 + \cdots + Z_k^2<math>

where Z₁, ..., Z_k are independent normal variables, each having expected value 0 and variance 1. This distribution is usually written

<math>

X\sim\chi^2_k. <math>

If <math>p<math> independent linear homogeneous constraints are imposed on these variables, the distribution of <math>X<math> conditional on these constriants is <math>\chi^2_{k-p}<math>, justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is

<math>

\phi(t)=(1-2it)^{k/2}.<math>

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 chi-squared random variables.

Its probability density function is

<math>

p_k(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2} \quad \mbox{ for }x > 0 <math> and p_k(x) = 0 for x≤0. Here Γ denotes the gamma function. Tables of this distribution - usually in its cumulative form - are widely available (see the External Links below for online versions), and the function is included in many spreadsheets (for example Microsoft Excel) and all statistical packages.

The normal approximation

If <math>X\sim\chi^2_k<math>, then as <math>k<math> tends to infinity, the distribution of <math>X<math> tends to normality. However, the tendency is slow (the skewness is <math>8/k<math> and the kurtosis is <math>12/k<math>) and two transformations are commonly considered, each of which approaches normality faster than <math>X<math> itself:

Fisher showed that <math>\sqrt{2X}<math> is approximately normally distributed with mean <math>\sqrt{2k-1}<math> and unit variance.

Wilson and Hilferty showed in 1931 that <math>\sqrt[3]{X/k}<math> is approximately normally distributed with mean <math>1-2/(9k)<math> and variance <math>2/(9k)<math>.

The expected value of a random variable having chi-square distribution with k degrees of freedom is k and the variance is 2k. The median is given approximately by

<math>

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

Note that 2 degrees of freedom leads to an exponential distribution.

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

See Cochran's theorem.