# Explain the meaning of $\alpha$ in a Kolmogorov-Smirnov two-sample test

The Wikipedia article on the two-sample Kolmogorov-Smirnov test states that:

The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is

$$D_{n,n'}=\sup_x |F_{1,n}(x)-F_{2,n'}(x)|$$

where $F_{1,n}$ and $F_{2,n'}$ are the empirical distribution functions of the first and the second sample respectively. The null hypothesis is rejected at level $\alpha$ if

$$D_{n,n'}>c(\alpha)\sqrt{\frac{n + n'}{n n'}}.$$

It is not clear to me the meaning of the $\alpha$ level. Where does it come from and what does it mean statistically?

## 1 Answer

The level $\alpha$ is the "significance level" of the test, the rate of Type I error, the probability of detecting a difference under the assumptions of the null hypothesis (that the two samples are drawn from the same distribution).

• No problem, I deleted the comment. It's like when we are 5% confident to believe in H1 ;). Commented Nov 5, 2013 at 14:07