# 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 distributionEdit

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

## ReferencesEdit

**^**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.- ^
^{a}^{b}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 (1961).*A first course in mathematical statistics*.