Percentiles

A rigorous guide to percentile ranks, quantiles, interpolation conventions, and responsible interpretation.

A box plot divides ordered data into four quarter groups. A number line carries a box-and-whisker plot with minimum, quartiles, median, and maximum labelled. Unequal segment lengths reveal spread without implying unequal counts per quartile.

Precise definition

The $p$th percentile is a value at or below which approximately $p\%$ of observations fall, with the exact finite-sample definition depending on convention. More formally, a population $p$-quantile can be defined as $q_p=\inf\{x:F(x)\ge p\}$.

Notation and mathematical language

A percentile is a value; percentile rank describes a value's relative position. The median is the 50th percentile, and quartiles correspond roughly to 25%, 50%, and 75%. Software methods differ in index formulas and interpolation, especially for small samples.

Conceptual picture

Percentiles convert position in an ordered distribution into a common 0–100 scale. They are robust to extreme magnitudes because they use order, but they do not show how far apart neighbouring scores are.

Fully worked example

Interpretation and application

Percentiles report growth charts, test norms, response times, and incomes. They describe rank within a reference distribution; causal or ability claims require much more evidence, and norm changes can shift ranks without individual change.