Chi-squared distribution

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The continuous probability distribution, concentrated on the positive semi-axis , with density

where is the gamma-function and the positive integral parameter is called the number of degrees of freedom. A "chi-squared" distribution is a special case of a gamma-distribution and has all the properties of the latter. The distribution function of a "chi-squared" distribution is an incomplete gamma-function, the characteristic function is expressed by the formula

and the mathematical expectation and variance are and , respectively. The family of "chi-squared" distributions is closed under the operation of convolution.

The "chi-squared" distribution with degrees of freedom can be derived as the distribution of the sum of the squares of independent random variables having identical normal distributions with mathematical expectation 0 and variance 1. This connection with a normal distribution determines the role that the "chi-squared" distribution plays in probability theory and in mathematical statistics.

Many distributions can be defined by means of the "chi-squared" distribution. For example, the distribution of the random variable — the length of the random vector with independent normally-distributed components — (sometimes called a "chi" -distribution, see also the special cases of a Maxwell distribution and a Rayleigh distribution), the Student distribution, and the Fisher -distribution. In mathematical statistics these distributions together with the "chi-squared" distribution describe sample distributions of various statistics of normally-distributed results of observations and are used to construct statistical interval estimators and statistical tests. A special reputation in connection with the "chi-squared" distribution has been gained by the "chi-squared" test, based on the so-called "chi-squared" statistic of E.S. Pearson.

There are detailed tables of the "chi-squared" distribution which are convenient for statistical calculations. For large one uses approximations by means of a normal distribution; for example, according to the central limit theorem, the distribution of the normalized variable converges to the standard normal distribution. More accurate is the approximation

where is the standard normal distribution function.

See also Non-central "chi-squared" distribution.


[1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2] M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 1 , Griffin (1969)
[3] H.O. Lancaster, "The chi-squared distribution" , Wiley (1969)
[4] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)


The name "chi-square" distribution is also used.

How to Cite This Entry:
Chi-squared distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article