What Is a Tolerance Interval and How Is It Different from a Confidence Interval?

If you have ever needed to say, “I am 95% confident that at least 99% of my product will fall within these limits,” you are talking about a statistical tolerance interval — not a confidence interval. This article explains what a tolerance interval is, how to compute it, and where it fits in quality engineering, based on the standard methods in Hahn & Meeker and ISO 16269-6.

What It Is

A statistical tolerance interval is an interval that, with a stated level of confidence, contains at least a specified proportion of a population. It is used to capture individual values, not just the mean.

The two key numbers are:

• Coverage (p): the proportion of the population you want the interval to contain (e.g., 99%).
• Confidence (γ): how sure you are that the interval actually achieves that coverage (e.g., 95%).

For example, a “95/99 tolerance interval” means: with 95% confidence, at least 99% of the population lies within the computed limits.

This is fundamentally different from a confidence interval for the mean, which describes uncertainty about the average — not about individual observations.

How It Works / Formula or Steps

For normally distributed data, a two-sided tolerance interval is calculated as:

\[
\bar{x} \pm k \cdot s
\]

Where:

• \(\bar{x}\) = sample mean
• \(s\) = sample standard deviation
• \(k\) = tolerance factor, which depends on the sample size \(n\), the desired coverage \(p\), and the confidence level \(\gamma\)

The factor \(k\) is obtained from tables (Hahn & Meeker) or computed using the noncentral t-distribution. For a one-sided interval, the formula is:

\[
\bar{x} + k_1 \cdot s \quad \text{(upper bound)}
\]

The key steps are:

1. Collect a random sample of size \(n\).
2. Compute the sample mean and standard deviation.
3. Choose the desired coverage (e.g., 99%) and confidence (e.g., 95%).
4. Look up or calculate the tolerance factor \(k\).
5. Compute the interval limits.

For non-normal data, ISO 16269-6 provides nonparametric methods based on order statistics, which require larger sample sizes.

A Worked Illustrative Example

Example data (illustrative only): Suppose you measure the breaking strength of 25 parts. The sample mean is 500 N and the sample standard deviation is 20 N. You want a two-sided tolerance interval that covers 99% of the population with 95% confidence.

For \(n = 25\), \(p = 0.99\), and \(\gamma = 0.95\), the tolerance factor \(k\) is approximately 3.615 (from standard tables in Hahn & Meeker).

The interval is:

\[
500 \pm 3.615 \times 20 = 500 \pm 72.3
\]

So the 95/99 tolerance interval is approximately (427.7 N, 572.3 N).

Interpretation: With 95% confidence, at least 99% of all parts will have breaking strength between 427.7 N and 572.3 N.

Common Pitfalls

• Confusing tolerance intervals with confidence intervals. A confidence interval for the mean shrinks toward a point as \(n\) grows; a tolerance interval approaches the true population range (e.g., ±3σ for 99% coverage).
• Using too small a sample. Tolerance intervals require adequate sample size; small samples produce very wide intervals.
• Ignoring normality assumptions. If the data are not normal, use nonparametric methods from ISO 16269-6 instead.

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