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| The '''F distribution''' is a continuous probability distribution function that represents the ratio of [[Statistics/Variance|variances]] between two [[Statistics/ChiSquaredDistribution|chi-squared distributed]] random variables. | 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. |
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| The distribution is characterized by the degrees of freedom for each of the [[Statistics/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 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. |
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| The square of a [[Statistics/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 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}} |
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| The [[Statistics/Moments|first moment]] of the distribution is ''d,,2,,/(d,,2,, - 2)'' for ''d,,2,,>2''. | The [[Analysis/ExpectedValue|expected value]] is given as ''d,,2,,/(d,,2,, - 2)'' for ''d,,2,,>2''. |
F Distribution
The F distribution is a continuous probability distribution function that represents the ratio of variances between two chi-squared distributed random variables.
Description
The distribution is characterized by the degrees of freedom for each of the chi-squared distributed random variables. They are ordered/referred to as numerator and denominator degrees of freedom/notated as d1 and d2 respectively.
The square of a 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 T-squared distribution is a scaled version of the F distribution. The following are both valid formulations:
Moments
The expected value is given as d2/(d2 - 2) for d2>2.
Usage
Probability Tests
The F distribution is almost exclusively used for test statistics: see the 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