> It's useful because if you have both and they are far apart you know there's a lot of variance
ummm. I would refrain from using nonparametric skew to make a comment about the magnitude of variance.
Essentially, the gap between mean and median will always be bounded by 1 sigma. The ratio abs(mean-median)/sigma is nonparametric skew. It is atmost 1, for any distribution( hence nonparametric, no distributional assumption required).
For unimodal distributions, especially symmetric unimodals, this ratio is 0. As the gap between the mean and median grows, the data gets more spread out, and the ratio captures that spread and consequent nonsymmetry. But you are using the value of this upper bound to make a comment about s^2. Which is very clever, but inaccurate. Say you standardize the rv and you have a nonsymmetric dist. Then mean 0, say median 100. Then stdev can be atmost 100, so variance can be atmost 10000. Which looks like “a lot of variance”. But is it really? Variance has a scaling problem, precisely why we take the square root, so the stdev remains in the scale of the mean. So at best one can say the stdev can be as big as the median. But that’s not very informative- because if the mean is -50 and median is +50, we are left with the same absolute gap of 100, so the same statement applies to the stdev even now.
I guess if I had to compare the variance of some sample X to another sample Y to make some claim that variance of X is much larger than Y, I would use a standard F test. Cooking up a test based on the gap between mean and median in a single sample seems somewhat shaky. It is very creative though, I grant you that.