Kolmogorov-Smirnov Critical Values

Each critical value is the exact solution to P(D_n < d) = 1 − α, where the exact finite-sample distribution of D_n comes from the Marsaglia-Tsang-Wang matrix-power method — the same algorithm widely used in statistical software — rather than the large-sample approximation many textbook tables rely on throughout their whole range.

The exact computation involves a matrix whose size grows with n, so beyond n = 100 the well-established large-sample approximation noted below the table becomes both fast to compute and highly accurate, and is what this site (and most software) uses at that point.

It works for any fully-specified reference distribution — one where you are not estimating parameters from the same sample. If you estimated parameters like the mean and standard deviation from your data first (a Lilliefors-type test), the critical values are different and smaller than the ones shown here.

The KS test compares the entire cumulative distribution shape directly using the raw, continuous data, while the chi-square test first bins the data into categories and compares counts, which loses some information but works for any type of data, continuous or categorical.