percentileestimat

Percentile estimate, or percentile estimation, is a statistical method used to infer population percentiles from a finite sample. The p-th percentile is the value below which p percent of observations fall. Percentile estimates are widely used in descriptive statistics, growth charts, standardized testing, finance, and quality control. The p parameter is typically expressed as a number between 0 and 100 (or as a proportion between 0 and 1).

In a sample of size n with observations sorted in nondecreasing order x_(1) ≤ ... ≤ x_(n), several estimators exist. A common simple method is the nearest-rank estimator: the p-th percentile is x_(k) with k = ⌈p(n+1)⌉, with p in [0,1]. More refined approaches interpolate between order statistics: compute r = p(n+1) and use x_(⌊r⌋) and x_(⌈r⌉) to interpolate. Different definitions (Hyndman–Fan types) yield slightly different results, trading off bias, variance, and smoothness.

Implementation notes: percentile estimates depend on the chosen method, sample size, and data distribution. They can be affected by ties, missing values, and extreme observations. In skewed data, high percentiles may be less stable in small samples.

Applications and example: Percentile estimates are used to construct growth charts, benchmark performance, assess risk with value-at-risk concepts, and monitor process quality. Example: with the data set [1, 2, 3, 4], the 50th percentile (median) using linear interpolation yields 2.5, since r = 0.5×(4+1) = 2.5, interpolating between x_(2) = 2 and x_(3) = 3.