I don't understand why in the F test we calculate the ratio between MSE between subject and MSE within subject. As far as I know, this is due because we want to use the F distribution, which is a rate between two $\chi^2$ distribution divided their degree of freedom.
My question is: Why is not used a simple version of this method? Couldn't we just take the ratio of the sum of squares between subjects with the sum of squares within subjects? Then we could compare this outcome with a distribution that is the result of the ratio between two $\chi^2$ distribution, instead of dividing the numerator and denominator for the degrees of freedom.
Does it makes sense? Why they use the other way?