“Any other such measure will indeed be isomorphic to variance when restricted to normal distributions.”
It’s true, but you should not restrict to normal distributions in this context. It is possible to find some distributions X1 and X2 with different variances but same value E(|x-mean|^p) for p≠2. Then X1 and X2 looks the same to this p-variance, but their normalized sample average will converge to different normal distributions. Hence variance is indeed the right and only measure of spreadout-ness to consider when applying the central limit theorem.
That’s exactly what I was trying to say, not a disagreement with it. The only step where I claimed all reasonable ways of measuring spreadout-ness agree was on the result you get after summing up a large number of iid random variables, not the random variables that were being summed up.
I don’t agree with the argument on the variance :
“Any other such measure will indeed be isomorphic to variance when restricted to normal distributions.”
It’s true, but you should not restrict to normal distributions in this context. It is possible to find some distributions X1 and X2 with different variances but same value E(|x-mean|^p) for p≠2. Then X1 and X2 looks the same to this p-variance, but their normalized sample average will converge to different normal distributions. Hence variance is indeed the right and only measure of spreadout-ness to consider when applying the central limit theorem.
That’s exactly what I was trying to say, not a disagreement with it. The only step where I claimed all reasonable ways of measuring spreadout-ness agree was on the result you get after summing up a large number of iid random variables, not the random variables that were being summed up.