A non-parametric test for the null hypothesis that a random sample has been drawn from a specified distribution (either discrete or continuous). There are several similar tests, each involving a comparison of the sample distribution function with that hypothesized. For example, let the sample values, in increasing order, be x₍₁₎, x₍₂₎,…, x_(n). Let the hypothesized probability of a value less than or equal to x_(j) be p_j. Let u_j and v_j be defined byThe test statistic is the largest of the absolute magnitudes of these 2n differences. A two-sample version of the test compares the two sample distribution functions. In the single-sample case, approximate critical values are at the 5% level and at the 1% level. The test was introduced by Kolmogorov in 1933, and further developed by Smirnov in 1939.

As described, the test refers to a fully prescribed distribution. However, by using special tables of critical values and estimating unknown parameters from the sample data, its use has been extended to testing for exponential, extreme-value, logistic, normal (the Lilliefors test), and Weibull distributions with unspecified parameters.

As an example, to test the hypothesis that the values 0.273, −1.184, 1.456, −0.655, −0.323, −0.733, −1.600, 0.819, 0.081, 0.971 have been drawn from a standard normal distribution the results given in the table are obtained.

The value of the test statistic is 0.144, which is much less than 1.36/=0.41: so the null hypothesis is acceptable.