Fisher's z-distribution

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

where B is the beta function.

When the degrees of freedom becomes large ($$d_1, d_2 \rightarrow \infty$$), the distribution approaches normality with mean
 * $$\bar{x} = \frac 1 2 \left( \frac 1 {d_2} - \frac 1 {d_1} \right)$$

and variance
 * $$\sigma^2_x = \frac 1 2 \left( \frac 1 {d_1} + \frac 1 {d_2} \right).$$

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 $$\tfrac{\log X}{2} \sim \operatorname{FisherZ}(n,m)$$