Kolmogorov-Smirnov Test

The K-S test is a critical validation tool in quality analytics. Before computing Cpk or setting control limits, engineers need to verify that the assumed distribution actually fits the data. A poor fit means every downstream calculation is unreliable.

Unlike the chi-square goodness-of-fit test, the K-S test does not require binning the data into arbitrary intervals, making it more sensitive to subtle distributional differences. It works with continuous data of any sample size, though its power increases with more observations.

In practice, the K-S test is used both for validating parametric fits (e.g., "Is this data really normal?") and for comparing two empirical distributions (e.g., "Has the process changed between these two batches?"). The two-sample variant is especially useful for process drift detection.

EntropyStat uses the K-S test as its primary validation mechanism — but in reverse compared to traditional usage. Instead of testing whether data fits a pre-assumed distribution (like normal or Weibull), EntropyStat uses the K-S test to validate the quality of the EGDF fit itself.

The workflow is: fit the EGDF to data, then apply the K-S test between the fitted EGDF and the empirical CDF. A high p-value confirms that the entropy-based distribution accurately captures the data's true shape. This is a strictly more powerful validation strategy because the EGDF can represent any distribution form — if the K-S test passes, the fit is genuinely good, not just "good for a normal" or "good for a Weibull."

The K-S statistic also guides EntropyStat's automatic Scale parameter optimization. The Scale parameter controls EGDF smoothness, and the optimal Scale is the one that minimizes the K-S statistic — finding the balance between overfitting (too jagged) and underfitting (too smooth).