# How do I calculate the effect size for the Kolmogorov-Smirnov Z statistic?

is there a way of calculating an effect size for the Kolmogorov-Smirnov Z statistic (in SPSS or by hand)? Or should I stick to the Mann-Whitney test, even though my group sizes are less than n=25?

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First of all, I wouldn't place much trust in the K-S test with samples <25 or >300. But otherwise, interesting: how to quantify on a unidimensional, 0-to-1 scale? On the one hand, it's not hard to imagine that when a distribution shows a perfect fit to the hypothesized one, the effect size would be 1. But what about radically poor fits--how would one distinguish between the poor fits to the normal of, say, a U-shaped distribution, a uniform, or some multi-humped polynomial one? Also, there's a question of scale: it seems Z would have to be infinite for the effect size to be 0. –  rolando2 Mar 3 '11 at 18:10
1. Yes. $D = Z/\sqrt{n}$ for the one-sample test. $D = Z/\sqrt{\frac{n_1 n_2}{n_1 + n_2}}$ for the two-sample test. $D$ should also be the "Most Extreme Differences - Absolute" entry in the output graphic (double-click the table shown in the SPSS output viewer). $Z$ might be labeled "Test Statistic," "Kolmogorov-Smirnov Z," or something else depending on which test and version of SPSS you're using.
$D$ is defined as the supremum of absolute differences in the two ECDFs; $Z$ is defined as $\sqrt{n_1 n_2 / (n_1+n_2)} D$. I was just explaining how to compute $D$ from SPSS output. Hope that clarifies. –  alexkchavez Mar 3 '11 at 20:24