Percentile
A percentile indicates the value below which a given percentage of data falls, placing an observation in context of the full dataset.
Definition
A percentile tells you what percentage of a group scored at or below a certain value: if you are in the 80th percentile, you scored higher than 80% of the group. The $k$-th percentile is the value below which $k\%$ of the data falls; quartiles are the $25$th, $50$th, and $75$th percentiles, and percentiles are computed from ordered data using interpolation, commonly used to report standardized test scores and medical measurements like height and weight. Formally, the $p$-th percentile is the $(100p)$-th quantile: the smallest $x$ such that $F(x) \ge p$, satisfying $F(x_p) = p$ for continuous distributions; percentile ranks are equivariant under monotone transformations but not under location-scale changes, making them scale-free measures of relative position.
Example
On a standardized test, scoring in the $90$th percentile means you scored higher than $90$ out of every $100$ test-takers; the $50$th percentile is the median. A child's height at the $65$th percentile means $65\%$ of children the same age are shorter and $35\%$ are taller. Percentile normalization (the probability integral transform) maps any continuous random variable $X$ to $U = F(X) \sim \text{Uniform}(0,1)$, the basis of copula models for multivariate distributions, and quantile regression (Koenker and Bassett, 1978) estimates conditional quantiles of the response rather than the conditional mean.
Key Insight
Percentiles are used on standardized tests, growth charts, and rankings to show where you stand compared to everyone else, conveying relative standing in a way raw scores cannot: a score of $85$ out of $100$ means less without knowing where most people scored. Quantile regression is more robust than OLS and gives a complete picture of how predictors affect the entire distribution, not just the average.