the median and percentiles

The median is the middle of a distribution: a value with at most half the probability below it and at most half above it. If you lined up every possible outcome by size, the median sits at the halfway mark. Percentiles generalise this to any cutoff: the kth percentile is the value below which (about) k percent of the probability lies. The 25th, 50th, and 75th percentiles are the quartiles, and the median is just the 50th percentile.

These are simply named points read off the quantile function: the median is Q(0.5), the 90th percentile is Q(0.9), and so on. To find a median in practice, ask for the value where the cdf first reaches 0.5; for the 90th percentile, where it first reaches 0.9. For a symmetric distribution the median coincides with the mean, but in general they differ. A box plot is literally a picture of five of these numbers: the minimum, the three quartiles, and the maximum.

Percentiles matter because they are robust and easy to interpret. Telling a child she is at the 80th percentile in height means 80 percent of peers are shorter, no units or distribution shape needed. The median in particular shrugs off extreme values: one billionaire pulls the mean income way up but barely nudges the median, which is why typical income is reported as a median, not a mean. Honest caveat: for discrete or ties-heavy data the median can be slightly ambiguous (any value in a flat stretch of the cdf qualifies), and software conventions for percentiles differ, so a stated percentile can shift by a hair depending on the rule used.