pquantilen

pquantilen, or p-quantiles, are points in a distribution that separate the probability mass into p and 1−p. For a random variable X with distribution function F, the p-quantile q_p is defined as q_p = F^{-1}(p) when F is continuous. More generally, q_p is any value satisfying F(q_p−) ≤ p ≤ F(q_p). In continuous distributions this reduces to a unique inverse value.

Common special cases include the median (p = 0.5), and quartiles (p = 0.25 and 0.75). The term is sometimes written as p-th quantile or p-quantile.

From data, the population p-quantile is estimated by the sample p-quantile, obtained from the order statistics of a sample. There are several conventions for discrete data and small samples about how to interpolate; widely used is to take an order statistic near the p-th percentile, possibly with linear interpolation between adjacent values. Many statistical software implement multiple quantile types, and the result may depend on the chosen method.

Quantiles are used to summarize distributions, construct box plots, compare distributions, and set thresholds in risk assessment and quality control. The interquartile range, the distance between the 25th and 75th percentiles, is a common robustness measure.