Fisher's z-distribution

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

${\displaystyle z={\frac {1}{2}}\log F}${\displaystyle 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.^[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 ${\displaystyle x'=e^{2x},円}${\displaystyle x'=e^{2x},円}. However, the mean and variance do not follow the same transformation.

The probability density function is^{[2] [3]}

${\displaystyle f(x;d_{1},d_{2})={\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},}${\displaystyle 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^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},}

where B is the beta function.

When the degrees of freedom becomes large (${\displaystyle d_{1},d_{2}\rightarrow \infty }${\displaystyle d_{1},d_{2}\rightarrow \infty }), the distribution approaches normality with mean^[2]

${\displaystyle {\bar {x}}={\frac {1}{2}}\left({\frac {1}{d_{2}}}-{\frac {1}{d_{1}}}\right)}${\displaystyle {\bar {x}}={\frac {1}{2}}\left({\frac {1}{d_{2}}}-{\frac {1}{d_{1}}}\right)}

and variance

${\displaystyle \sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).}${\displaystyle \sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).}

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

• MathWorld entry

• Continuous distributions
• Ronald Fisher

• Articles with short description
• Short description matches Wikidata