Is it possible to figure out what is the nth percentile of a non-parametric distribution base on a few summary statistics I am working on a problem where I need to reliably measure the nth percentile of a non-parametric distribution without having to use all the data. Ideally, I was wondering if it is possible to do this like the case with a parametric distribution like Gaussian, where all you need for Gaussian is the mean and variance?
 A: Let us treat the larger set (whose quantile you want to estimate by a smaller sample) as the population.
You can estimate population quantiles by sample quantiles. This will generally work well except for extreme quantiles.
Draw a random sample from your population (the large set). Compute the quantile of your sample (e.g. a sample median for a population median). Use that as an estimate of the quantile you seek. This doesn't rely on any specific distributional assumptions.
You can even use other sample quantiles to obtain a non-parametric interval for the population quantile (via a binomial argument).
A: One way of doing this is by fitting some known distribution to the data, then calculating the quantiles on that distribution. In finance, for instance, the popular choices are the Gaussian and Student-t distribution. There's an area of financial risk management which has a similar problem to yours it's called value-at-risk (VaR). VaR is basically a quantile. 
So, in this application Johnson SU distribution is a popular choice. You fit it to the data by matching first three or four moments, then get the quantile estimation from the fitted distribution, see this paper for example.
UPDATE
Here's a link to a paper that describes how to fit Johnson SU distribution to the data, see p.10. The algorithm was first proposed in paper, which is behind the paywall, unfortunately: "An algorithm to determine the parameters of SU -curves in the johnson system of probabillity distributions by moment matching", Hans J.H. Tuenter
Another popular method for a similar problem is called Cornish-Fisher expansion, which is also popular in financial industry to calculate VaR. You may find it easier to use than Johnson SU. Here's a paper describing these and similar approaches for VaR.
