Kolmogorov-Smirnov Test

A goodness-of-fit test for any statistical 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.

REFERENCES:

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

DeGroot, M. H. Ch. 9 in Probability and Statistics, 3rd ed. Reading, MA: Addison-Wesley, 1991.

Knuth, D. E. §3.3.1B in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 45-52, 1998.

 Neal, D. K. "Goodness of Fit Tests for Normality." Mathematica Educ. Res. 5, 23-30, 1996. https://library.wolfram.com/infocenter/Articles/1379/.

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.