# Null hypothesis rejection with Kolmogorov-Smirnov

I want to compare two samples with a Kolmogorov-Smirnov test. Wikipedia states the null hypothesis is rejected at $\alpha$ if:

$\sqrt( \frac{nn'}{n+n'}) D_{nn'} > K_\alpha$

where $n$ and $n'$ are the sizes of samples, D the KS-statistic and $K_\alpha$ the critical value (probably everyone here already knows). I wonder about the sample sizes: according to this formula every null hypothesis is rejected, if the samples are just large enough.

Could anybody enlighten me, what I am misunderstanding?

• What makes you say that every null is rejected if the samples are large enough? As the sample size increases we also expect Dnn' to decrease. – Dason Nov 1 '12 at 14:49

It is correct that every result will result in rejecting the null if the sample is large enough. This is true, not just for test using KS, but for any significance test. On one level, this is very reasonable: As sample size increases, our estimates get more precise. At some point, they will get precise enough that any difference $d$ will be large enough to be different from $0$ at a level $\alpha$ .