Fisher z-distribution

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A continuous probability distribution on the real line with density

The parameters are called the degrees of freedom. The characteristic function has the form

The mathematical expectation and the variance are equal to and , respectively.

If the random variable has the Fisher -distribution with and degrees of freedom, then the quantity has the Fisher -distribution with and degrees of freedom. Along with the Fisher -distribution, known as the distribution of the dispersion proportion, the Fisher -distribution was originally introduced in the analysis of variance by R.A. Fisher (1924). His intention was that the -distribution should be the basic distribution for testing statistical hypotheses in the analysis of variance. The Fisher -distribution was tabulated at the same time, and the first research was concerned with the statistic , although in modern mathematical statistics one uses the simpler statistic .


[1] R.A. Fisher, "On a distribution yielding the error functions of several well-known statistics" , Proc. Internat. Congress mathematicians (Toronto 1924) , 2 , Univ. Toronto Press (1928) pp. 805–813


The dispersion proportion is also called the variance ratio.

How to Cite This Entry:
Fisher z-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