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:


1. How would one connect the above? For example, given a finite number of moments, say m1-m4, how one would estimate median and quartiles?
2. For finite distribution of size N, how one would estimate expected max & min values based on the 1st M moments, M << N?



Relating percentiles to moments
Question 1. It is impossible to give any estimate without extra assumptions.
Take a random variable which takes only two values: $a>0$ and $b<0$, both with probability
$1/2$. The median is zero (see the remark below). 
All moments can be arbitrarily large or arbitrarily small: to make them large choose
$a$ very large, and $b$ moderate.


Remark. If you disagree that the median of this random variable is $0$, modify this distribution by making it continuous with density strictly positive on $[a,b]$,
but very small between $a$ and $b$. So that most of the density sits near $a$ and $b$.


With question 2, there are of course a trivial estimates: for example,
if the max is $a>0$ then the $m$-th
moment is at most $a^m$. Same with min. There is nothing else in general.


