Estimating the Standard Deviation in Quality-Control Applications

In estimating the standard deviation of a normally distributed random variable, a multiple of the sample range is often used instead of the sample standard deviation in view of the range's computational simplicity. Although it is well known that use of the sample standard deviation is more efficient if the sample size exceeds 2, many statistical quality-control textbooks argue that the loss in efficiency when using the sample range to estimate the process standard deviation is very small with relatively small sample sizes. In this paper, we show that this loss in efficiency can be relatively large even for very small sample sizes and thus strongly advise against using range-based methods. We found that some previously published tables of relative efficiencies were either mislabeled or inaccurate. We also make some recommendations when a number of samples have been taken over time.