📏 The Kolmogorov-Smirnov test

It is a statistical test used to test whether a sample of data comes from a particular theoretical distribution (such as the normal). In essence, it compares the empirical cumulative distribution function of the data with the expected theoretical one.

How to Interpret It?

Null Hypothesis (H0): the distribution of the data follows the specified theoretical distribution (or the two distributions are the same).

Alternative Hypothesis (H1): the distribution of the data does not follow the specified theoretical distribution (or the two distributions are different).

The test calculates a test statistic (D) that measures the maximum vertical distance between the two cumulative distribution functions. The greater this distance, the less likely it is that the data comes from the specified theoretical distribution.

The p-value associated with the D statistic indicates the probability of obtaining a value of D at least as large as the observed one, assuming that the null hypothesis is true.

If p-value < α (significance level, typically 0.05): the null hypothesis is rejected, concluding that the data do not come from the specified theoretical distribution (or that the two distributions are different).

If p-value ≥ α: there is insufficient evidence to reject the null hypothesis.