In statistics and probability, the F-distribution is a continuous probability distribution. It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after Ronald Fisher and George W. Snedecor).

A random variate of the F-distribution arises as the ratio of two chi-squared variates:

where

• U₁ and U₂ have chi-square distributions with d₁ and d₂ degrees of freedom respectively, and

• U₁ and U₂ are independent (see Cochran's theorem for an application).

The F-distribution arises frequently as the null distribution of a test statistic, especially in likelihood-ratio tests, perhaps most notably in the analysis of variance; see F-test.

The probability density function of an F(d₁, d₂) distributed random variable is given by

for real x ≥ 0, where d₁ and d₂ are positive integers, and B is the beta function.

The cumulative distribution function is

where I is the regularized incomplete beta function.

An F(d₁, d₂) random variable has the following properties:

mode 

provided d₁ > 2

mean 

provided d₂ > 2

variance 

provided d₂ > 4

skewness 

provided d₂ > 6

Generalization

A generalization of the (central) F-distribution is the noncentral F-distribution.

External links

• Table of critical values of the F-distribution (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm)
• Online significance testing with the F-distribution (http://home.clara.net/sisa/signhlp.htm)
• Distribution Calculator (http://www.vias.org/simulations/simusoft_distcalc.html) Calculates probabilities and critical values for normal, t-, chi2- and F-distribution