## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 17, Number 2 (1946), 208-215.

### Tolerance Limits for a Normal Distribution

A. Wald and J. Wolfowitz

#### 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 $\beta$ that the tolerance limits include at least a given proportion $\gamma$ of the population. A good approximation to such tolerance limits can be obtained as follows: Let $\bar x$ denote the sample mean and $s^2$ 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, $\chi^2_{n,\beta}$ denotes the number for which the probability that $\chi^2$ with $n$ degrees of freedom will exceed this number is $\beta$, 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 $\chi^2_{n,\beta}$ can be obtained from a table of the $\chi^2$ distribution and $r$ can be determined with the help of a table of the normal distribution.

#### Article information

**Source**

Ann. Math. Statist., Volume 17, Number 2 (1946), 208-215.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177730981

**Digital Object Identifier**

doi:10.1214/aoms/1177730981

**Mathematical Reviews number (MathSciNet)**

MR19901

**Zentralblatt MATH identifier**

0063.08130

**JSTOR**

links.jstor.org

#### Citation

Wald, A.; Wolfowitz, J. Tolerance Limits for a Normal Distribution. Ann. Math. Statist. 17 (1946), no. 2, 208--215. doi:10.1214/aoms/1177730981. https://projecteuclid.org/euclid.aoms/1177730981