Quantile

A distribution-free (non-parametric) quantile estimator based on the order statistics (the sorted values in the sample).

There are several definitions for the quantile estimator useful in defining reference limits.

Formula (N is the sample size and p is the quantile)	Description
Np+1/2	Piecewise linear function where the knots are the values midway through the steps of the empirical distribution function. Recommended by Linnet (2000) as having the lowest root-mean-squared-error .
(N+1)p	Linear interpolation of the expectations for the order statistics of the uniform distribution on [0,1]. Recommended by IFCC (1987) and CLSI (2010).
(N+1/3)p+1/3	Linear interpolation of the approximate medians for the order statistics. Has many excellent properties and is approximately median-unbiased regardless of the distribution.

A 95% reference interval (0.025 and 0.975 quantiles) requires a minimum sample size of 39. A 90% confidence interval for a 95% reference interval requires a minimum sample size of 119.

Related information

Linnet, K. (2000). Nonparametric estimation of reference intervals by simple and bootstrap-based procedures. Clinical Chemistry, 46(6), 867-869.

Solberg, H. E. (1987). Approved recommendation on the theory of reference values. Part 5. Statistical treatment of collected reference values. Determination of reference limits. Clinica Chimica Acta, 170(2), S13-S32.

CLSI. (2010). Defining, Establishing, and Verifying Reference Intervals in the Clinical Laboratory (EP28-A3C, formerly C28-A3C). Clinical and Laboratory Standards Institute.

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