Estimation of the standard deviation by order statistics: the range, the average range, and some quasi ranges

Abstract

Some rapid approximate methods of estimating the standard deviation from samples of moderate size (20 < n < 100) are presented. The emphasis is placed on solutions of problems commonly encountered in statistical quality control, especially in the electronics industry. Factors and efficiency values are given for the use of these estimators on normally distributed data. Statistical and practical engineering and administrative criteria are suggested for testing whether particular estimators are desirable in the usual industrial situation. The estimates discussed in this paper are all order statistics, i.e. statistics which are a function of only a small number of observations selected from the whole sample. These observations are selected because of the position they occupy among the other observations when all sample observations are arranged in order of magnitude. The first estimator discussed, for instance, is the range. The range of a sample is merely the numerical difference between the largest member of the sample and the smallest member of the sample. The standard deviation of the distribution from which the sample was drawn may be estimated by dividing the range by a suitable constant. The constant is a function of the sample size and of the shape of the distribution. Factors are given for sample sizes up to ten, for the normal distribution only. [TRUNCATED]