Fisher's z-distribution

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

Fisher's z
Probability density function
Parameters deg. of freedom

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 . However, the mean and variance do not follow the same transformation.

The probability density function is[2][3]

where B is the beta function.

When the degrees of freedom becomes large () the distribution approaches normality with mean[2]

and variance

  • If then (F-distribution)
  • If then


  1. Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF). Proceedings of the International Congress of Mathematics, Toronto. 2: 805–813. Archived from the original (PDF) on April 12, 2011.
  2. Leo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution". The Annals of Mathematical Statistics. 12 (4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955.
  3. Charles Ernest Weatherburn. A first course in mathematical statistics.
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