= F Distribution = The '''F distribution''' is a continuous probability distribution function that represents the ratio of [[Statistics/Variance|variances]] between two [[Analysis/ChiSquaredDistribution|chi-squared distributed]] random variables. <> ---- == Description == The distribution is characterized by the degrees of freedom for each of the [[Analysis/ChiSquaredDistribution|chi-squared distributed]] random variables. They are ordered/referred to as '''numerator''' and '''denominator degrees of freedom'''/notated as ''d,,1,,'' and ''d,,2,,'' respectively. The square of a [[Analysis/StudentsTDistribution|t distributed]] random variable has an F distribution with 1 numerator degree of freedom and the same denominator degrees of freedom as the original t distributed variable. The [[Analysis/HotellingsTSquaredDistribution|T-squared distribution]] is a scaled version of the F distribution. The following are both valid formulations: {{attachment:ftotsquared1.svg}} {{attachment:ftotsquared2.svg}} ---- == Moments == The [[Statistics/Moments|first moment]] of the distribution is ''d,,2,,/(d,,2,, - 2)'' for ''d,,2,,>2''. ---- == Usage == === Probability Tests === The F distribution is almost exclusively used for [[Statistics/TestStatistic|test statistics]]: see the [[Statistics/FTest|F test]]. Furthermore, the test is usually formulated such that the denominator degrees of freedom are very large. As an example, for a very large denominator degrees of freedom and a significance level of 5%, the critical F statistics are: * 3.84 for 1 numerator degree of freedom * 3.00 for 2 * 2.6 for 3 * 2.37 for 4 * 2.21 for 5 * and so on * 1.00 for very large numerator degrees of freedom ---- CategoryRicottone