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Encyclopedia > F distribution

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

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

  Results from FactBites:
 
Normal distribution - Wikipedia, the free encyclopedia (4114 words)
The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal.
The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n.
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant.
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