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In statistics, a percentile (or a centile) is a score below which a given percentage of scores in its frequency distribution falls (exclusive definition) or a score at or below which a given percentage falls (inclusive definition). For example, the 50th percentile (the median) is the score below which (exclusive) or at or below which (inclusive) 50% of the scores in the distribution may be found.
The percentile (or percentile score) and the percentile rank are related terms. The percentile rank of a score is the percentage of scores in its distribution that are less than it, an exclusive definition, and one that can be expressed with a single, simple formula. In contrast, there are many formulas or algorithms for a percentile score. Hyndman and Fan [1] identified nine and most statistical and spreadsheet software use one of the methods they describe.[2] Algorithms either return the value of a score that exists in the set of scores (nearest-rank methods) or interpolate between existing scores and are either exclusive or inclusive.
The figure shows a 10-score distribution, illustrates the percentile scores that result from these different algorithms, and serves as an introduction to the examples given subsequently. The simplest are nearest-rank methods that return a score from the distribution, although compared to interpolation methods, results can be a bit crude. The Nearest-Rank Methods table shows the computational steps for exclusive and inclusive methods.
Interpolation methods, as the name implies, can return a score that is between scores in the distribution. Algorithms used by statistical programs typically use interpolation methods, for example, the percentile.exl and percentile.inc function in Microsoft Excel. The Interpolated Methods table shows the computational steps.
The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests, but, as just noted, they are not the same. For percentile rank, a score is given and a percentage is computed. Percentile ranks are exclusive. If the percentile rank for a specified score is 90%, then 90% of the scores were lower. In contrast, for percentiles a percentage is given and a corresponding score is determined, which can be either exclusive or inclusive. The score for a specified percentage (e.g., 90th) indicates a score below which (exclusive definition) or at or below which (inclusive definition) other scores in the distribution fall.
The 25th percentile is also known as the first quartile (Q1), the 50th percentile as the median or second quartile (Q2), and the 75th percentile as the third quartile (Q3).
There is no standard definition of percentile,[1][5][6] however all definitions yield similar results when the number of observations is very large and the probability distribution is continuous.[7] In the limit, as the sample size approaches infinity, the 100pth percentile (0<p<1) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p, as p approximates the CDF. This can be seen as a consequence of the Glivenko–Cantelli theorem. Some methods for calculating the percentiles are given below.
One definition of percentile, often given in texts, is that the P-th percentile {\displaystyle (0<P\leq 100)} (0<P\leq 100) of a list of N ordered values (sorted from least to greatest) is the smallest value in the list such that no more than P percent of the data is strictly less than the value and at least P percent of the data is less than or equal to that value. This is obtained by first calculating the ordinal rank and then taking the value from the ordered list that corresponds to that rank. The ordinal rank n is calculated using this formula
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