In my textbook it is said, that if the hypothesis of same population distributions is true, then $\sqrt{n}D\rightarrow K$ in distribution, where $D$ is the Kolmogorov-Smirnov statistic, and $K$ is Kolmogorov distribution. So, we are interested in the value of $\sqrt{n}D$. So why does the tables for Kolmogorov distribution have also entries for $n$, and then only for $n$ larger than $50$, it is given in value$/\sqrt{n}$ units? To me, it seems that we only need one row, instead of one row for each $n$. For example, this table:


  • $\begingroup$ Plug the smaller values of $n$ into the formulas, like $1.63/\sqrt{n},$ and compare what you get with what's in the table. $\endgroup$ – whuber May 28 at 19:47

The arrow in the expression $\sqrt{n}D\rightarrow K$ means that $K$ is the asymptotic distribution of $\sqrt{n}D$, and you can consider it to be a good approximation of your test statistic's distribution only when $n$ is big enough. Hence the need for a table that considers the distribution of $D$ when $n$ is little.

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