# Percentage Beyond a Given Value for Empirically Defined Distribution

It is my understanding that standard deviation does not work well as a measurement for distributions that are heavily skewed. If I have a heavily right-skewed distribution, should I simply use the CDF of an appropriate distribution to determine what percentage of values would lie beyond a certain point on the curve? What well-defined methods exist for determining the appropriate CDF given an empirically gathered distribution?

An example of the sort of 'rough' empirical distribution I'm talking about would be this: You could always use the the (complement of) the empirical CDF (ECDF) to estimate proportions beyond any particular value.

http://en.wikipedia.org/wiki/Empirical_distribution_function

You can even get nonparametric intervals for such quantities.

Asymptotically, you could use:

$$\sqrt{n}\big(\hat F_n(t) - F(t)\big)\ \ \xrightarrow{d}\ \ \mathcal{N}\Big( 0, F(t)\big(1-F(t)\big) \Big),$$

to get an interval for the estimate. See the linked article for a number of other results that might be of some use to you.

• Thanks. This is probably a silly question but what do you mean by nonparametric intervals here? – 114 Sep 30 '14 at 1:23
• @114 I mean intervals that are distribution-free, which is to say intervals that are of the same algebraic form irrespective of the distribution* from which they were drawn. $\quad$ * -- assuming we have continuous distributions, at least; discrete ones won't actually be distribution free. As an example, distribution-free intervals might be backed out of goodness of fit tests using the ECDF such as the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling. The asymptotic result above is distribution-free, as are bounds given by some of the inequalities at the link in the answer ...(ctd) – Glen_b -Reinstate Monica Sep 30 '14 at 2:13
• (ctd) ... there are also related distribution-free intervals for population quantiles - e.g. see here. There's also the bootstrap which can be used to generate intervals. ... One thing about your plot does concern me, though - it looks like your actual data may be discrete. – Glen_b -Reinstate Monica Sep 30 '14 at 2:21
• That's correct, there are discrete data points that I would like to build a continuous distribution around. I believe I see now what you mean by distribution-free, thanks! – 114 Sep 30 '14 at 3:30