Fisher's ''z''-distribution is the

statistical distribution In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step functio ...

of half the

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

of an ''F''-distribution variate: :

z = \frac 1 2 \log F

It was first described by

Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...

in a paper delivered at the International Mathematical Congress of 1924 in

Toronto Toronto ( , locally pronounced or ) is the List of the largest municipalities in Canada by population, most populous city in Canada. It is the capital city of the Provinces and territories of Canada, Canadian province of Ontario. With a p ...

. Nowadays one usually uses the ''F''-distribution instead. The

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

and

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

can be found by using the ''F''-distribution at the value of

x' = e^ \,

. However, the mean and variance do not follow the same transformation. The probability density function is :

f(x; d_1, d_2) = \frac \frac,

where ''B'' is the

beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

. When the

degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

becomes large (

d_1, d_2 \rightarrow \infty

), the distribution approaches normality with mean :

\bar = \frac 1 2 \left( \frac 1  - \frac 1  \right)

and variance :

\sigma^2_x = \frac 1 2 \left( \frac 1  + \frac 1  \right).

Related distribution

*If

X \sim \operatorname(n,m)

then

e^ \sim \operatorname(n,m) \,

( ''F''-distribution) *If

X \sim \operatorname(n,m)

then

\tfrac \sim \operatorname(n,m)

References

External links

MathWorld entry
{{ProbDistributions, continuous-infinite Continuous distributions Ronald Fisher