Kolmogorov-Smirnov Tests: Fitted Parameters Change the Reference Problem

The statistical question

The classical one-sample Kolmogorov-Smirnov statistic compares the empirical distribution with a fully specified continuous reference cumulative distribution. If location, scale or other reference parameters are estimated from the tested data, the ordinary critical-value calculation no longer applies unchanged. The fitted-reference case needs an appropriate adjusted method or calibration. A reported statistic is meaningful only with that reference basis.

Primary reference: NIST Engineering Statistics Handbook.

An illustrative review example

In a fictional review, team A specifies a normal reference using an independently established mean and spread. Team B instead fits the mean and spread from the 60 readings being tested. Both spreadsheets may display a K-S statistic, but the same unadjusted reference cutoff cannot be assumed valid in both cases. Merely standardizing the tested readings using their own sample estimates does not make the fitted-reference issue disappear.

Prepare the evidence for discussion

Ask the analyst to identify the distribution, parameter values, their data source and the exact test implementation. Retain the original observations and any treatment of rounding, ties or censoring for their review. Do not treat a nonrejection as proof of normality or future production consistency. For a machinery trial, state which later calculation needs the distributional assumption and how uncertainty about that assumption is handled.

Specify the reference before the comparison

This blank worksheet is for your own project. It contains no H M machine trial result.

Customer Questions

Primary References

These references support the technical principles discussed in this guide. The worked examples and review questions are educational.

• Primary reference: NIST Engineering Statistics Handbook