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

statistical distribution In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution functio ...

of half the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...

of an ''F''-distribution

variate In probability and statistics, a random variate or simply variate is a particular outcome of a ''random variable'': the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random ...

: :

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 ( ; or ) is the capital city of the Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the most populous city in Canada and the fourth most populous city in North America. The city is the ancho ...

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

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

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. Ev ...

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^(1 ...

. When the

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

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