Tolerance intervals in Python

The task: state bounds that contain at least 99% of production, with 95% confidence. That is a tolerance interval. It is routinely confused with two other intervals that answer different questions:

• a confidence interval brackets a parameter (the mean),
• a prediction interval brackets the next single observation,
• a tolerance interval brackets a chosen proportion of the whole population. Specification work in pharma and medical devices means this one.

Usage

import shewhart as sw

r = sw.tolerance_interval(df, value="potency", coverage=0.99, confidence=0.95)
r.stats["lower"], r.stats["upper"]

Distribution-free, using only the sample extremes:

r = sw.tolerance_interval(df, value="potency", coverage=0.90,
                          confidence=0.95, method="nonparametric")

The nonparametric route needs a minimum sample size that depends only on n; if the data cannot support the requested confidence, the error message says exactly how many observations would.

Where the numbers come from

The normal method uses Howe's k2 approximation, the same formula the NIST/SEMATECH e-Handbook documents (section 7.2.6.3). The handbook's published factor for n = 43, 90% coverage, 99% confidence is k2 = 2.217; shewhart reproduces it in the validation suite. The nonparametric method is Wilks' classic result for the (min, max) interval.

The normal method is sensitive to the normality assumption precisely because it reaches into the far tails. Check normality first (a capability study reports an Anderson-Darling note), and prefer the nonparametric method when in doubt and the sample is large enough.