Mathematics
Don Wardell
Department of Operations and Information Systems
Abstract
A common and simple estimate of variability is the sample range, which is the difference between the maximum and minimum values of 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 R charts use constants that are based on the mean and standard deviation of the distribution of the range of normal random variables. Historically, the computation of the moments requires the use of tabulated constants approximated using numerical integration. We provide exact results for the moments for sample sizes of 2 through 5. For sample sizes from 6 to 1000, we estimate the constants with rational functions. The rational function we recommend for the mean achieves a maximum error of 4.4×10-6 while that of the standard deviation achieves a maximum error of 1.5×10-5. The exact and approximate expressions remove the need for table lookup in the control chart design phase.
