Chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution. The most familiar examples are the Rayleigh distribution with chi distribution with 2 degrees of freedom, and the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If are k independent, normally distributed random variables with means and standard deviations , then the statistic

is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of n − 1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter: which specifies the number of degrees of freedom (i.e. the number of ).

Characterization

Chi-squared distribution

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 a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, e.g., in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general 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.