sqrtKS

sqrtKS is not a universally defined term in statistics, but in many contexts it can refer to the square root of the Kolmogorov–Smirnov statistic used in goodness-of-fit testing. The Kolmogorov–Smirnov (KS) statistic D_n measures the maximum difference between the empirical distribution F_n of a sample and a reference distribution F. It is defined as D_n = sup_x |F_n(x) - F(x)| and takes values in [0,1]. The derived quantity sqrtKS = sqrt(D_n) is simply the square root of this distance and serves as a monotone transformation of D_n.

Computation and interpretation: To obtain sqrtKS, compute the KS statistic D_n from the data and take its square root. The result lies in [0,1], with smaller values indicating closer agreement between the sample distribution and the reference distribution. Because sqrt is monotone, the ranking of distances is preserved, so sqrtKS conveys the same ordering as D_n but on a different scale.

Usage considerations: The standard KS test for a goodness-of-fit p-value relies on the distribution of sqrt(n)·D_n (or related scaling), not on sqrt(D_n) alone. As a result, sqrtKS does not have a conventional, widely used p-value by itself. It may appear in exploratory analyses or custom procedures as a diagnostic measure or as a simplified feature, but its statistical interpretation depends on the specific context and any additional scaling or modeling applied.

Relation to other tests: Other goodness-of-fit measures, such as the Cramér–von Mises or Anderson–Darling statistics, address distributional discrepancies in different ways and have their own p-value calculations. sqrtKS remains a simple, derived distance measure rather than a standalone, standardized test statistic.