Percentile
Statistics & ProbabilityA 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.
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.
Key Insight
Percentiles are used on standardized tests, growth charts, and rankings to show where you stand compared to everyone else.
Definition
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. Percentiles are computed from ordered data using interpolation and are commonly used to report standardized test scores and medical measurements like height and weight.
Example
A child's height at the $65$th percentile means $65\%$ of children the same age are shorter and $35\%$ are taller. The median ($50$th percentile) is the boundary point.
Key Insight
Percentiles convey relative standing in a distribution, unlike raw scores which give no context. A score of $85$ out of $100$ means less without knowing where most people scored.
Definition
The $p$-th percentile is the $(100p)$-th quantile: the smallest $x$ such that $F(x) \ge p$, where $F$ is the empirical CDF. For continuous distributions, it satisfies $F(x_p) = p$. Percentile ranks are equivariant under monotone transformations but not under location-scale changes, making them scale-free measures of relative position.
Example
Converting a raw SAT score to a percentile rank involves looking up the empirical percentile from the score distribution. Percentile normalization (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.
Key Insight
Quantile regression (Koenker and Bassett, 1978) estimates conditional quantiles of the response, not the conditional mean. It is more robust than OLS and gives a complete picture of how predictors affect the entire distribution, not just the average.