# Fisher's z-distribution

Not to be confused with Fisher z-transformation.
Parameters $d_1>0,\ d_2>0$ deg. of freedom $x \in (-\infty; +\infty)\!$ $\frac{2d_1^{d_1/2}d_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}\!$ $0$

Fisher's z-distribution is the statistical distribution of half the logarithm of an F distribution variate:

$z = \frac{1}{2} \log F$

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled "On a distribution yielding the error functions of several well-known statistics" (Proceedings of the International Congress of Mathematics, Toronto, 2: 805-813 (1924). 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 $x' = e^{2x} \,$. However, the mean and variance do not follow the same transformation.

The probability density function is[1][2]

$f(x; d_1, d_2) = \frac{2d_1^{d_1/2} d_2^{d_2/2}}{B(d_1/2, d_2/2)} \frac{e^{d_1 x}}{\left(d_1 e^{2 x} + d_2\right)^{(d_1+d_2)/2}},$

where B is the beta function.

When the degrees of freedom becomes large ($d_1, d_2 \rightarrow \infty$) the distribution approach normality with mean[1]

$\bar{x} = (1/d_2 - 1/d_1)/2$

and variance

$\sigma^2_x = (1/d_1 + 1/d_2)/2.$

## Related Distribution

• If $X \sim \operatorname{FisherZ}(n,m)$ then $e^{2X} \sim \operatorname{F}(n,m) \,$ (F-distribution)
• If $X \sim \operatorname{F}(n,m)$ then $\tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m)$

## References

• Fisher, R.A. (1924) On a Distribution Yielding the Error Functions of Several Well Known Statistics Proceedings of the International Congress of Mathematics, Toronto, 2: 805-813 pdf copy
1. ^ Charles Ernest Weatherburn. A first course in mathematical statistics.