There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, variance, and other central moments, maybe characteristic functions.
B. Everybody else would consider median and other percentiles, as well as max & min values for finite distributions.
Therefore a couple of questions:
How would one connect the above? For example, given a finite number of moments, say m1-m4, how one would estimate median and quartiles?
For finite distribution of size N, how one would estimate expected max & min values based on the 1st M moments, M << N?