On the exact calculation of cumulative F probabilities

Various approximate procedures have been employed in the tabulation of F distributions. These approximations generally perform poorly for small degrees of freedom and/or for extreme tail areas. Closed form expressions are presented in this paper for the exact calculation of cumulative probabilities of F distributions. For small degrees of freedom, the probabilities may be calculated with the aid of standard mathematical tables and a calculator. An APL computer program, also presented in this paper, may be used in calculating the cumulative probabilities according to the exact expressions.