An approximation of F distribution by binomial probabilities

It is well known that the F distribution function can be expressed as an incomplete beta function. In particular when both degrees of freedom are even, the F distribution function may be simply expressed as a sum of binomial probabilities. In this note we consider the F distribution with at least one odd degree of freedom. By relating the F distribution to a generalized logistic distribution we obtain an approximation of the F distribution in terms of binomial probabilities. Because it is very close to the exact at the tails, the approximation is particularly suitable for estimating P-values of F ratio tests. All the computations involved can be carried out easily with only a hand calculator.