pQuantil

pQuantil, or p-quantil, is a statistical concept that denotes the value in a distribution at which a portion p of the observations fall at or below that value. Formally, if X is a random variable with distribution function F, the p-quantil q_p satisfies F(q_p) = p for p in (0,1). When F is continuous, q_p is unique and equals the inverse function F^{-1}(p). If F has jumps, there may be an interval of values that satisfy the condition.

In continuous distributions, the p-quantil often corresponds to a percentile: the 50th percentile is the median, the 25th percentile is the first quartile, and the 75th percentile is the third quartile. Quantiles provide a concise summary of the distribution and are less sensitive to outliers than moments such as the mean.

Estimating p-quantiles from data typically uses order statistics. Given a sample x1,...,xn, sort them to x_(1) ≤ ... ≤ x_(n). A common estimator is x_(k) with k chosen according to a convention, such as k ≈ p(n+1). Other conventions use k = ⌈pn⌉ or interpolate between neighboring order statistics to obtain an intermediate value.

Applications include descriptive statistics, construction of robust measures like the interquartile range (Q3−Q1), and the estimation of conditional quantiles via quantile regression. Quantiles are widely used for setting thresholds, constructing confidence intervals in nonparametric contexts, and comparing distributions without assuming a specific parametric form.