I was wondering whether there is a condition, phrased in probability terms, for a median to be less or equal to the mean in a sample of positive numbers?
If $X$ is the positive Random variable corresponding to the values in the sample, can we give a condition for example on the variance of $X$ for this to be true?
This comes from my presumption that samples of weight, height , I.Q, and others should satisfy this property, and wondered whether one can justify it by showing the sample satisfies certain conditions. I have not dealt that much with quantiles, but I suspect this question was probably posed and answered long ago.
I would appreciate any insight on the matter.