norm-tol-int

Exact algorithm for computing two-sided statistical tolerance intervals under a normal distribution assumption using Python.

Methods

The function tolerance_factor computes (by Gauss-Kronod quadrature) the exact tolerance factor k for the two-sided coverage-content and (1-alpha)-confidence tolerance interval

TI = [Xmean - k * S, Xmean + k * S]

where Xmean = mean(X), S = std(X), X = [X_1,...,X_n] is a random sample of size n from the distribution N(mu,sig2) with unknown mean mu and variance sig2.

The algorithm is a Python port of the MATLAB algorithm ToleranceFactor, contributed to the MATLAB Central File Exchange by Viktor Witkovsky. The port attempts to preserve the basic function structure of the algorithm so comparisons back against the MATLAB code are easier to conduct.

For more details on statistical tolerance intervals the technical background on how to compute them, see the following references:

• Krishnamoorthy K, Mathew T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley & Sons, Inc., Hoboken, New Jersey. ISBN: 978-0-470-38026-0, 512 pages.
• Meeker, William Q.; Hahn, Gerald J.; Escobar, Luis A.. Statistical Intervals: A Guide for Practitioners and Researchers (Wiley Series in Probability and Statistics). Wiley.
• Witkovsky V. On the exact two-sided tolerance intervals for univariate normal distribution and linear regression. Austrian Journal of Statistics 43(4), 2014, 279-92. http:// ajs.data-analysis.at/index.php/ajs/article/viewFile/vol43-4-6/35
• ISO 16269-6:2014: Statistical interpretation of data - Part 6: Determination of statistical tolerance intervals.
• Janiga I., Garaj I.: Two-sided tolerance limits of normal distributions with unknown means and unknown common variability. MEASUREMENT SCIENCE REVIEW, Volume 3, Section 1, 2003, 75-78.

Example

The notebook example.ipynb provides a very brief application example.

Environment

The file environment.yml can be used to produce a conda environment suitable for running the example notebook and the unit tests.

Unit Tests

The algorithm accurately reproduces tables of two-sided normal tolerance interval factors from standard sources, including the complete set of tables in ISO 16269-6:2014 Annex F. The unit tests included here represent a sampling of that reproduction for brevity.

To run all the unit tests, invoke the following:

python -m unittest discover -v

License

MIT License