Sample Sizes to Set Tolerance Limits

Abstract

Statistical tolerance limits are used to obtain bounds from sample data to contain a specified proportion of a population with a high degree of confidence. Under the assumption of a normal distribution with sample mean and standard deviation, x̄ and s respectively, one-sided lower and upper tolerance limits are x̄ - ks and x̄ + ks, where k is a tabulated value. Two-sided statistical tolerance limits are of the form x̄ ± Ks where K is also a tabulated value.Ordered sample values can also be used as statistical tolerance limits. These limits are called distribution-free since they do not require a distributional assumption, such as normality for the measured variable. The one-sided statistical tolerance limit is the mth smallest sample value for a lower limit and the mth largest sample value for an upper limit in a sample of size n. A two-sided statistical tolerance interval is the interval from the rth smallest to the sth largest sample values where m = r + s.A recurring problem is to determine the random sample size to use for obtaining tolerance limits. Tables are provided in this paper for determining the sample size n such that the probability is β that p ± d of the population is within the computed statistical tolerance limits. Both the normal distribution and distribution free cases are considered for both one-sided and two-sided tolerance limits.