#
Fisher's *z*-distribution

**Fisher's z-distribution** is the statistical distribution of half the logarithm of an

*F*-distribution variate:

Probability density function | |||

Parameters | deg. of freedom | ||
---|---|---|---|

Support | |||

Mode |

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto.[1] Nowadays one usually uses the *F*-distribution instead.

The probability density function and cumulative distribution function can be found by using the *F*-distribution at the value of . However, the mean and variance do not follow the same transformation.

The probability density function is[2][3]

where *B* is the beta function.

When the degrees of freedom becomes large () the distribution approaches normality with mean[2]

and variance

## Related distribution

- If then (
*F*-distribution) - If then

## References

- Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF).
*Proceedings of the International Congress of Mathematics, Toronto*.**2**: 805–813. Archived from the original (PDF) on April 12, 2011. - Leo A. Aroian (December 1941). "A study of R. A. Fisher's
*z*distribution and the related F distribution".*The Annals of Mathematical Statistics*.**12**(4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955. - Charles Ernest Weatherburn.
*A first course in mathematical statistics*.

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