Expressions for the First Two Moments of the Range of Normal Random Variables with Applications to the Range Control Chart

Don G. Wardell ()
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Don G. Wardell: David Eccles School of Business, University of Utah, Salt Lake City, UT 84112, USA

Mathematics, 2025, vol. 13, issue 9, 1-18

Abstract: A common and simple estimate of variability is the sample range, which is the difference between the maximum and minimum values in the sample. While other measures of variability are preferred in most instances, process owners and operators regularly use range (R) control charts to monitor process variability. The center line and limits of the R charts use constants that are based on the first two moments (mean and variance) of the distribution of the range of normal random variables. Historically, the computation of moments requires the use of tabulated constants approximated using numerical integration. We provide exact results for the moments for sample sizes 2 through 5. For sample sizes from 6 to 1000, we used the differential correction method to find Chebyshev minimax rational-function approximations of the moments. The rational function we recommend for the mean (R-chart constant d 2 ) has a polynomial of order two in the numerator and six in the denominator and achieves a maximum error of 4.4 × 10 −6 . The function for the standard deviation (R-chart constant d 3 ) has a polynomial of order two in the numerator and seven in the denominator and achieves a maximum error of 1.5 × 10 −5 . The exact and approximate expressions eliminate the need for table lookup in the control chart design phase.

Keywords: control charts; Six Sigma; process variation; order statistics; rational functions; minimax error approximation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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