Tolerance Limits for a Normal Distribution

Abstract

The problem of constructing tolerance limits for a normal universe is considered. The tolerance limits are required to be such that the probability is equal to a preassigned value β that the tolerance limits include at least a given proportion γ of the population. A good approximation to such tolerance limits can be obtained as follows: Let x̄ denote the sample mean and s2 the sample estimate of the variance. Then the approximate tolerance limits are given by $\bar x - \sqrt\frac{n}{\chi^2_{n,\beta}} rs \text{and} \bar x + \sqrt\frac{n}{\chi^2_{n,\beta}} rs$ where n is one less than the number N of observations, χ2 n,β denotes the number for which the probability that χ2 with n degrees of freedom will exceed this number is β, and r is the root of the equation $\frac{1}{\sqrt{2\pi}} \int^{1/\sqrt{N} + r}_{1/\sqrt{N}-r} e^{-t^2/2} dt = \gamma.$ The number χ2 n,β can be obtained from a table of the χ2 distribution and r can be determined with the help of a table of the normal distribution.