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Percentiles

Jack Prins, Don McCormack, Di Michelson, Karen Horrell

edited by: Carroll Croarkin, Paul Tobias, James J. Filliben, Barry Hembree, William Guthrie, Paul Tobias, Ledi Trutna, Jack Prins



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Definitions of order statistics and ranks. For a series of measurements Y1, …, YN, denote the data ordered in increasing order of magnitude by Y〈1〉, …, Y〈N〉. These ordered data are called order statistics. If Y〈j〉 is the order statistic that corresponds to the measurement Yᵢ, then the rank for Yᵢ is j; i.e.,
Y〈j〉 ∼ Yᵢ, rᵢ=j.

Definition of percentiles. Order statistics provide a way of estimating proportions of the data that should fall above and below a given value, called a percentile. The pth percentile is a value, Y〈p〉, such that at most (100p) % of the measurements are less than this value and at most 100(1−p) % are greater. The 50th percentile is called the median.
Percentiles split a set of ordered data into hundredths. (Deciles split ordered data into tenths). For example, 70 % of the data should fall below the 70th percentile.

Estimation of percentiles. Percentiles can be estimated from N measurements as follows: for the pth percentile, set p(N+1) equal to k+d for k an integer, and d, a fraction greater than or equal to 0 and less than 1.
1 ▹ For 0 < k < N, Y(p) = Y〈k〉 + d(Y〈k+1〉 − Y〈k〉)
2 ▹ For k = 0, Y(p) = Y〈1〉
Note that any p ≤ 1/(N+1) will simply be set to the minimum value.
3 ▹ For k ≥ N, Y(p) = Y〈N〉
Note that any p ≥ N/(N+1) will simply be set to the maximum value.
[...]


In NIST/SEMATECH e-Handbook of Statistical Methods (2012), 7.2.6.2 
Key: INRMM:14376208

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http://www.itl.nist.gov/div898/handbook/p…, http://www.webcitation.org/6rC3p5wwW

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