Hello everyone

I'm trying to understand the F-distribution. I realize that the F-distribution is not symmetric in the degrees of freedom, and understand why. However I would think that it should be Fnm = 1/Fmn, but it isn't. Here is my reasoning:

When testing whether the variances of two sets are equal, one uses the F-statistic: F = S1^2/S2^2. This is then compared with the f-value at the desired significance level: Fnm (with degrees of freedom n and m respectively).

However the zero hypothesis is symmetric: H0: sigma1 = sigma2. So I should get the same result if I switch set1 and set2, and get the zero hypothesis H0': sigma2 = sigma1. The new F-statistic is

F' = S2^2/S1^2 = 1/F.

Now the degrees of freedom are also switched and so F' should be compared with the f-value Fmn. But if we want to reach the same conclusion for both hypotheses H0 and H0', we should have Fnm = 1/Fmn... I would think.

What is wrong with my reasoning? Because, in general, Fmn is not equal to 1/Fnm.

Thanks in advance

Lukas