The correcting factor n/(n-1) in R is what explains my paradox about the law of total variance Var(Y) = E(var(Y|X)) + Var(E(X|Y)), I was obtaining result that don't match this formula because I corrected all the variances with the factor 20/19 but the total variance should have the factor 40/39 just like you pointed.
Thanks for the comments and the correction.
I just added another comment that relates analysis of variance to this post to show that there is no real paradox here.
Finally, the formula for the total variance above is related to my intuition that having some information (having the data for each state) should make the means of the variances in each group smaller that the total variance, because variance is related to lack of information. But analysis of variance suggests (see other comment of mine) that the state factor is not representative because the high variance in each group (each state) and the low difference between the groups means and the total mean.
I just added another comment that relates analysis of variance to this post to show that there is no real paradox here.
Finally, the formula for the total variance above is related to my intuition that having some information (having the data for each state) should make the means of the variances in each group smaller that the total variance, because variance is related to lack of information. But analysis of variance suggests (see other comment of mine) that the state factor is not representative because the high variance in each group (each state) and the low difference between the groups means and the total mean.