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chi dis·tri·bu·tion

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. It is thus related to the chi-squared distribution by describing the distribution of the positive square roots of a variable obeying a chi-squared distribution. If Z 1 , … , Z k {\displaystyle Z_{1},\ldots ,Z_{k}} are k {\displaystyle k} independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic Y = ∑ i = 1 k Z i 2 {\displaystyle Y={\sqrt {\sum _{i=1}^{k}Z_{i}^{2}}}} is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of n − 1 {\displaystyle n-1} ) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter, k {\displaystyle k} , which specifies the number of degrees of freedom (i.e. the number of Z i {\displaystyle Z_{i}} ). The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom).

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