Statistical Tolerance Interval: Coverage and Confidence in One Interval
What Is a Tolerance Interval?
A statistical tolerance interval guarantees, at a stated confidence level, that at least p% of the population falls within the computed limits - fundamentally different from a confidence interval, which describes the estimation precision of a parameter such as the mean. Tolerance intervals describe the coverage of future individual values. For example, from n sample values you can state with 95% confidence that 95% of the population lies in the interval; this also supports product-specification validation by checking whether the process satisfies future-individual coverage requirements.
When to Use It
Use it whenever you must draw conclusions about the range that contains most products from a sample: process capability assessment, supplier quality agreements (e.g., 95/95 requirements), tolerance setting and verification, and design tolerance allocation. It applies to variable data and generally assumes approximate normality (or normality after a suitable transformation). In design it can also allocate part tolerances to processes and test whether sample evidence supports the tolerance settings.
One-Sided or Two-Sided?
When only one boundary matters (lower strength limit, upper impurity limit), use a one-sided interval for tighter limits; when both bounds matter (dimensions), use two-sided, which splits into equal-tail and unequal-tail versions. The tool uses the normal approximation with tabulated K factors: the smaller the sample, or the higher the coverage and confidence, the larger the K factor and the wider the interval. Two-sided intervals distribute risk equally to the tails by default; sample data must come from the same process with independent sampling.
How to Use It
Enter the sample data and the target coverage (e.g., 95%) and confidence (e.g., 95%), choose one- or two-sided, and get the tolerance bounds. Note that tiny samples give very wide intervals - n of at least 30 is recommended; heavily skewed data should be transformed first or handled with a nonparametric method. State the distribution assumption and the K-factor source in the output for review and audit.