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{{Cite journal | author-link = Leo A. Aroian | doi-access = free | author-link = Charles Ernest Weatherburn : 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).