Tolerance Limits for a Normal Distribution

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 x denote the sample mean and s² the sample estimate of the variance. Then the approximate tolerance limits are given by x - √nχ²n,β rs and x + √nχ²n,β rs where n is one less than the number N of observations, χ²n,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 1√2π ∫1/√N + r1/√N-r e-t²/2 dt = γ. The number χ²n,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.