Here is an example. I have two data sets as shown below as histograms:
I want to examine whether these data sets are drawn from the same underlying distribution. To do this, I am using the 2-sample Kolmogorov-Smirnov test. This test works by computing the empirical cumulative distribution function for each data set and then measuring the maximum distance between the two ECDFs. Here are the ECDFs:
Looking at it visually, this looks like a no-brainer: these are from the same distribution! The histograms look very similar and the maximum difference between the two ECDFs is tiny. But, to my surprise, the KS test rejects the null-hypothesis! The P-value is very small (p = 0.0011) suggesting that the two data sets are actually very likely drawn from different distributions.
What's going on here? Am I missing something? Is the KS test the wrong test to use?
Any help is appreciated.