Kolmogorov–Smirnov test

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the Kolmogorov–Smirnov test (also K–S test or KS test) is a <a href="/facts/Nonparametric_statistics/svTDJHTb">nonparametric test</a> of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a>. It can be used to test whether a <a href="/facts/Random_sample/URCwKNge">sample</a> came from a given reference probability distribution (one-sample K–S test), or to test whether two samples came from the same distribution (two-sample K–S test). Intuitively, it provides a method to qualitatively answer the question "How likely is it that we would see a collection of samples like this if they were drawn from that probability distribution?" or, in the second case, "How likely is it that we would see two sets of samples like this if they were drawn from the same (but unknown) probability distribution?".
It is named after <a href="/facts/Andrey_Kolmogorov/bEKwtrs6">Andrey Kolmogorov</a> and <a href="/facts/Nikolai_Smirnov_(mathematician)/1Jdh2Aff">Nikolai Smirnov</a>.
The Kolmogorov–Smirnov statistic quantifies a <a href="/facts/Metric_(mathematics)/RkUptSo8">distance</a> between the <a href="/facts/Empirical_distribution_function/frWIF9F9">empirical distribution function</a> of the sample and the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> of the reference distribution, or between the empirical distribution functions of two samples. The <a href="/facts/Null_distribution/ErtrUfrQ">null distribution</a> of this statistic is calculated under the <a href="/facts/Null_hypothesis/C8NwSE9J">null hypothesis</a> that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see Section 2), purely discrete or mixed (see Section 2.2). In the two-sample case (see Section 3), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted. 
The two-sample K–S test is one of the most useful and general <a href="/facts/Nonparametric_statistics/svTDJHTb">nonparametric methods</a> for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.
The Kolmogorov–Smirnov test can be modified to serve as a <a href="/facts/Goodness_of_fit/qlTfUIfS">goodness of fit</a> test. In the special case of testing for <a href="/facts/Normal_distribution/UapjjPyQ">normality</a> of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic (see Test with estimated parameters). Various studies have found that, even in this corrected form, the test is less <a href="/facts/Power_of_a_test/5KU54nW1">powerful</a> for testing normality than the <a href="/facts/Shapiro%25E2%2580%2593Wilk_test/6yu3fq4K">Shapiro–Wilk test</a> or <a href="/facts/Anderson%25E2%2580%2593Darling_test/A6C7nvQn">Anderson–Darling test</a>. However, these other tests have their own disadvantages. For instance the Shapiro–Wilk test is known not to work well in samples with many identical values.

Kolmogorov–Smirnov test open-in-new

Kolmogorov–Smirnov test