The right bracket (]) shown in the figure is the calculated 90 percent normal tolerance bound for 95 percent of the population. To calculate this bound, we have to assume that the sample data are from a normal distribution. We make this assumption based on the 50 samples and any previous data that we have collected.

The right parenthesis) in the figure is the calculated 90 percent one-sided nonparametric tolerance bound for 95 percent of the population. Notice that the nonparametric tolerance bound is equal to the maximum observation.

In practice, one would calculate only one tolerance bound. If the tolerance bound is lower than the specification (as it is in this case), the test is “passed.” More formally, we are 90 percent confident that 95 percent of the population is below the specification.

As another example, suppose that we have 15 observations: 1.57, -0.57, -1.19, 0.08, 0.83, -1.55, 1.14, 0.63, -0.11, 1.64, 0.79, -0.44, 0.27, 1.18, -0.47. We want to calculate a 90 percent confidence bound for 95 percent of the population. We have x = 0.2533333 and s = 0.9725935.

Using the equation for the normal one-sided tolerance bound, we have g’ = 2.063 and Tp= 0.2533333 + 2.068(0.9725935) = 2.264657.

Using the nonparametric tolerance interval, we order the observations from smallest to largest: -1.55, -1.19, -0.57, -0.47, -0.44, -0.11, 0.08, 0.27, 0.63, 0.79, 0.83, 1.14, 1.18, 1.57, 1.64.

We find that k = 16, so our one-sided tolerance bound is 1.64. However, our confidence level is P(X image 14|5,0.95) = 0.5397, or 54 percent. We require at least n = log(0.1)/log(0.95) = 45 samples to achieve a 90 percent confidence level using the largest sample value as our upper tolerance bound.


Hahn, G.J., and W.Q. Meeker. 1991. Statistical Intervals: A Guide for Practitioners. New York, N.Y.: John Wiley & Sons.

Krishnamoorthy, K., and T. Mahew. 2009. Statistical Tolerance Regions: Theory, Applications, and Computation. New York, N.Y.: John Wiley & Sons.

Odeh, R.E., and D.B. Owen. 1980. Tables for Normal Tolerance Limits, Sampling Plans, and Screening. New York, N.Y.: Marcel Dekker, Inc.

Young, D. 2010. tolerance: An R package for estimating tolerance intervals. Journal of Statistical Software 36(5).

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