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F-Distribution

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FDistribution

A continuous statistical distribution which arises in the testing of whether two observed samples have the same variance. Let chi_m^2 and chi_n^2 be independent variates distributed as chi-squared with m and n degrees of freedom.

Define a statistic F_(n,m) as the ratio of the dispersions of the two distributions

F_(n,m)=(chi_n^2/n)/(chi_m^2/m).
(1)

This statistic then has an F-distribution on domain [0,infty) with probability function f_(n,m)(x) and cumulative distribution function F_(n,m)(x) given by

f_(n,m)(x)=(Gamma((n+m)/2)n^(n/2)m^(m/2))/(Gamma(n/2)Gamma(m/2))(x^(n/2-1))/((m+nx)^((n+m)/2))
(2)
=(m^(m/2)n^(n/2)x^(n/2-1))/((m+nx)^((n+m)/2)B(1/2n,1/2m))
(3)
F_(n,m)(x)=I((nx)/(m+nx);1/2n,1/2m)
(4)
=2n^((n-2)/2)(x/m)^(n/2)×(_2F_1(1/2(m+n),1/2n;1+1/2n;-nx/m))/(B(1/2n,1/2m)),
(5)

where Gamma(z) is the gamma function, B(a,b) is the beta function, I(x;a,b) is the regularized beta function, and _2F_1(a,b;c;z) is a hypergeometric function.

The F-distribution is implemented in the Wolfram Language as FRatioDistribution[n, m].

The mean, variance, skewness and kurtosis excess are

mu=m/(m-2)
(6)
sigma^2=(2m^2(m+n-2))/(n(m-2)^2(m-4))
(7)
gamma_1=(2(m+2n-2))/(m-6)sqrt((2(m-4))/(n(m+n-2)))
(8)
gamma_2=(12(-16+20m-8m^2+m^3+44n-32mn+5m^2n-22n^2+5mn^2))/(n(m-6)(m-8)(n+m-2)).
(9)

The probability that F would be as large as it is if the first distribution has a smaller variance than the second is denoted Q(F_(n,m)).

See also

Beta Function, Gamma Function, Hotelling T2 Distribution, Noncentral F-Distribution, Regularized Beta Function, Snedecor's F-Distribution

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 946-949, 1972.David, F. N. "The Moments of the z and F Distributions." Biometrika 36, 394-403, 1949.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 117-118, 1992.

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F-Distribution

Cite this as:

Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/F-Distribution.html

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