<< Click to Display Table of Contents >> Navigation: Probability distributions > Continuous univariate distributions > F distribution 
The F distribution, also known as Fisher's F or FisherSnedecor's distribution, is obtained as the ratio of two independent Chisquare distributions with n and m degrees of freedom. More specifically:
The probability density function (pdf) and higher moments are quite involved and details be found on the Mathworld and Wikipedia sites. The pdf can be written in various forms, but to keep the expression as simple as possible we will let a=n/2, b=m/2 and c=(m+n)/2. The pdf is then:
where B(a,b) is the Beta function. Because Chisquare distributions are themselves obtainable as the sum of squared Normal variates, the Fdistribution is particularly useful for comparing rations of variances of Normal variates, which is major feature of Analysis of Variance (ANOVA) and regression analysis. Plots of the Fdistribution for a range of values for n, and fixed m are provided in the diagram below.
Sample Fdistribution graphs, for n=2,3,4 and m=5
For any given n, the F distribution tends to a fixed form as m→∞. This form is the numerator of the ratio above, i.e. an adjusted Chisquare distribution. This is illustrated below, where n has been taken as fixed, at 4, and m increased to the limit.
Sample Fdistribution graphs, for n=4 and m=1,2,10 and ∞
Key measures for the F distribution are provided in the table below:
Item 
Value 

Mean 
m/(m2), m>2 
Variance 

Skewness 

Kurtosis 

MGF 
does not exist 
Distribution tables are provided in the Resources topic, Distribution tables section
References
[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, II. Houghton Mifflin/J Wiley & Sons, New York
Mathworld: Weisstein E W: Fdistribution: http://mathworld.wolfram.com/FDistribution.html
Wikipedia: F distribution: http://en.wikipedia.org/wiki/Fdistribution