Kolmogorov-Smirnov test

In statistics, the Kolmogorov–Smirnov test (often called the K-S test) is used to determine whether two underlying one-dimensional probability distributions differ, or whether an underlying probability distribution differs from a hypothesized distribution, in either case based on finite samples.The one-sample KS test compares the empirical distribution function with the cumulative distribution function specified by the null hypothesis. The main applications are testing goodness of fit with the normal and uniform distributions. For normality testing, minor improvements made by Lilliefors lead to the Lilliefors test. In general the Shapiro-Wilk test or Anderson-Darling test are more powerful alternatives to the Lilliefors test for testing normality.
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The Kolmogorov-Smirnov one-sample test for normality is based on the maximum difference between the sample cumulative distribution and the hypothesized cumulative distribution. If the D statistic is significant, then the hypothesis that the respective distribution is normal should be rejected. For many software programs, the probability values that are reported are based on those tabulated by Massey (1951); those probability values are valid when the mean and standard deviation of the normal distribution are known a-priori and not estimated from the data. However, usually those parameters are computed from the actual data. In that case, the test for normality involves a complex conditional hypothesis ("how likely is it to obtain a D statistic of this magnitude or greater, contingent upon the mean and standard deviation computed from the data"), and the Lilliefors probabilities should be interpreted (Lilliefors, 1967). Note that in recent years, the Shapiro-Wilks' W test has become the preferred test of normality because of its good power properties as compared to a wide range of alternative tests.