Fisher's z-distribution

Short description: Statistical distribution

Fisher's z

Probability density function
Parameters	$d_1>0,d_2>0$ deg. of freedom
Support	$x\in (-\mathrm{\infty };+\mathrm{\infty })$
PDF	$\frac{2d_1^{d_1/2}{d}_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_{1}x}}{{(d_1{e}^{2x}+d_2)}^{(d_1+d_2)/2}}$
Mode	$0$

Ronald Fisher

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 $x^{\prime }=e^{2x}$. However, the mean and variance do not follow the same transformation.

The probability density function is

$f(x;d_1,d_2)=\frac{2d_1^{d_1/2}{d}_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_{1}x}}{{(d_1{e}^{2x}+d_2)}^{(d_1+d_2)/2}},$

where B is the beta function.

When the degrees of freedom becomes large ($d_1,d_2\to \mathrm{\infty }$), the distribution approaches normality with mean

$\overline{x}=\frac{1}{2}(\frac{1}{d_2}-\frac{1}{d_1})$

and variance

${\sigma }_x^2=\frac{1}{2}(\frac{1}{d_1}+\frac{1}{d_2}).$

Related distribution

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

External links

• MathWorld entry