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Kolmogorov-Smirnov Test

A goodness-of-fit test for any Distribution. The test relies on the fact that the value of the sample cumulative density function is asymptotically normally distributed.

To apply the Kolmogorov-Smirnov test, calculate the cumulative frequency (normalized by the sample size) of the observations as a function of class. Then calculate the cumulative frequency for a true distribution (most commonly, the Normal Distribution). Find the greatest discrepancy between the observed and expected cumulative frequencies, which is called the ``D-Statistic.'' Compare this against the critical D-Statistic for that sample size. If the calculated D-Statistic is greater than the critical one, then reject the Null Hypothesis that the distribution is of the expected form. The test is an R-Estimate.

See also Anderson-Darling Statistic, D-Statistic, Kuiper Statistic, Normal Distribution, R-Estimate


Boes, D. C.; Graybill, F. A.; and Mood, A. M. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill, 1974.

Knuth, D. E. §3.3.1B in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley, pp. 45-52, 1981.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Kolmogorov-Smirnov Test.'' In Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 617-620, 1992.

© 1996-9 Eric W. Weisstein