In statistics, Kolmogorov-Smirnov(K-S) test is a non-parametric test of the equality of the continuous, one-dimensional (univariate) probability distributions.
K-S test compares the two cumulative distributions and returns the maximum difference between them.
One-sample K-S test or goodness of fit test was developed by Andrey Nikolayevich Kolmogorov in 1933. Its purpose is to compare the overall shapes of two sample distributions.
Two-sample K-S test was developed by Nikolai Smirnov in 1939. Its purpose is to compare one sample to a known statistical distribution.
Parametric and non-parametric statistics
We can separate the statistical tests into two: Parametric and non-parametric tests.
Parametric tests are suitable for normally distributed data and they have more statistical power than nonparametric ones.
- Independent samples t-test and paired samples t-test
Nonparametric tests or distribution-free tests are suitable for any continuous data. They can be used with a very small sample size and outliers.
- Kolmogorov-Smirnov test
- Mann-Whitney-U test
- Kruskal-Wallis test
One-sample K-S test:
If we are comparing one sample distribution with a known sample, the null hypothesis is:
The sample does not come from a different distribution
Two-sample K-S test:
If we are comparing two sample distributions, the null hypothesis is:
Two samples are from the same distribution.
PDF, CDF, and ECDF
Probability Density Function(PDF)is a relative likelihood that the value of the random variable would equal that sample.
Cumulative Distribution Function(CDF): A hypothetical model of a distribution