Nonparametric Tests

Two-Sample Kolmogorov-Smirnov Table: Exact Critical Values of D

Exact critical values for the two-sample Kolmogorov-Smirnov test of whether two samples come from the same distribution.

Equal n to 40
Unequal n₁, n₂ to 20
Exact calculator

View the table Open the calculator

Two-Sample KS Table

Three tables: equal sample sizes, then unequal sizes at α = 0.05 and 0.01. Cells: critical D. Reject H₀ if D ≥ the cell.

Equal sample sizes (n₁ = n₂ = n)

Equal sample sizes (n₁ = n₂ = n)
Two-sided α → n ↓	0.1	0.05	0.02	0.01	0.001
3	1.000	—	—	—	—
4	1.000	1.000	—	—	—
5	0.800	1.000	1.000	1.000	—
6	0.834	0.834	1.000	1.000	—
7	0.715	0.858	0.858	0.858	1.000
8	0.625	0.750	0.750	0.875	1.000
9	0.667	0.667	0.778	0.778	0.889
10	0.600	0.700	0.700	0.800	0.900
11	0.546	0.637	0.728	0.728	0.819
12	0.500	0.584	0.667	0.667	0.834
13	0.539	0.539	0.616	0.693	0.770
14	0.500	0.572	0.572	0.643	0.786
15	0.467	0.534	0.600	0.600	0.734
16	0.438	0.500	0.563	0.625	0.688
17	0.471	0.471	0.530	0.589	0.706
18	0.445	0.500	0.556	0.556	0.667
19	0.422	0.474	0.527	0.527	0.632
20	0.400	0.450	0.500	0.550	0.650
21	0.381	0.429	0.477	0.524	0.620
22	0.410	0.410	0.500	0.500	0.591
23	0.392	0.435	0.479	0.479	0.566
24	0.375	0.417	0.459	0.500	0.584
25	0.360	0.400	0.440	0.480	0.560
26	0.347	0.385	0.424	0.462	0.539
27	0.334	0.371	0.445	0.445	0.556
28	0.358	0.393	0.429	0.465	0.536
29	0.345	0.380	0.414	0.449	0.518
30	0.334	0.367	0.400	0.434	0.500
31	0.323	0.355	0.388	0.420	0.517
32	0.313	0.344	0.407	0.407	0.500
33	0.304	0.364	0.394	0.425	0.485
34	0.324	0.353	0.383	0.412	0.471
35	0.315	0.343	0.372	0.400	0.486
36	0.306	0.334	0.362	0.389	0.473
37	0.298	0.325	0.379	0.379	0.460
38	0.290	0.316	0.369	0.395	0.448
39	0.283	0.308	0.359	0.385	0.462
40	0.275	0.325	0.350	0.375	0.450

Unequal sample sizes, two-sided α = 0.05

Unequal sample sizes, two-sided α = 0.05
Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.934	0.938	0.942	0.945	0.948	0.950
3	—	—	—	1.000	1.000	1.000	0.875	0.889	0.900	0.910	0.834	0.847	0.858	0.800	0.813	0.824	0.834	0.790	0.800
4	—	—	1.000	1.000	0.834	0.858	0.875	0.778	0.750	0.750	0.750	0.750	0.750	0.734	0.750	0.706	0.695	0.698	0.750
5	—	1.000	1.000	1.000	0.800	0.800	0.750	0.778	0.800	0.710	0.717	0.693	0.658	0.734	0.675	0.648	0.667	0.643	0.650
6	—	1.000	0.834	0.800	0.834	0.715	0.709	0.723	0.667	0.652	0.667	0.667	0.643	0.634	0.625	0.608	0.667	0.615	0.600
7	—	1.000	0.858	0.800	0.715	0.858	0.715	0.667	0.658	0.624	0.631	0.616	0.643	0.591	0.572	0.572	0.572	0.572	0.565
8	1.000	0.875	0.875	0.750	0.709	0.715	0.750	0.639	0.600	0.603	0.625	0.597	0.572	0.559	0.625	0.567	0.556	0.540	0.550
9	1.000	0.889	0.778	0.778	0.723	0.667	0.639	0.667	0.589	0.596	0.584	0.556	0.556	0.556	0.542	0.536	0.556	0.521	0.517
10	1.000	0.900	0.750	0.800	0.667	0.658	0.600	0.589	0.700	0.546	0.550	0.539	0.529	0.534	0.525	0.524	0.512	0.495	0.550
11	1.000	0.910	0.750	0.710	0.652	0.624	0.603	0.596	0.546	0.637	0.546	0.525	0.533	0.510	0.506	0.498	0.490	0.489	0.487
12	1.000	0.834	0.750	0.717	0.667	0.631	0.625	0.584	0.550	0.546	0.584	0.520	0.512	0.517	0.500	0.491	0.500	0.474	0.484
13	1.000	0.847	0.750	0.693	0.667	0.616	0.597	0.556	0.539	0.525	0.520	0.539	0.490	0.493	0.486	0.476	0.471	0.462	0.462
14	1.000	0.858	0.750	0.658	0.643	0.643	0.572	0.556	0.529	0.533	0.512	0.490	0.572	0.467	0.474	0.467	0.461	0.455	0.450
15	0.934	0.800	0.734	0.734	0.634	0.591	0.559	0.556	0.534	0.510	0.517	0.493	0.467	0.534	0.475	0.455	0.456	0.446	0.450
16	0.938	0.813	0.750	0.675	0.625	0.572	0.625	0.542	0.525	0.506	0.500	0.486	0.474	0.475	0.500	0.456	0.445	0.438	0.438
17	0.942	0.824	0.706	0.648	0.608	0.572	0.567	0.536	0.524	0.498	0.491	0.476	0.467	0.455	0.456	0.471	0.435	0.437	0.430
18	0.945	0.834	0.695	0.667	0.667	0.572	0.556	0.556	0.512	0.490	0.500	0.471	0.461	0.456	0.445	0.435	0.500	0.416	0.423
19	0.948	0.790	0.698	0.643	0.615	0.572	0.540	0.521	0.495	0.489	0.474	0.462	0.455	0.446	0.438	0.437	0.416	0.474	0.422
20	0.950	0.800	0.750	0.650	0.600	0.565	0.550	0.517	0.550	0.487	0.484	0.462	0.450	0.450	0.438	0.430	0.423	0.422	0.450

Unequal sample sizes, two-sided α = 0.01

Unequal sample sizes, two-sided α = 0.01
Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	1.000	1.000
3	—	—	—	—	—	—	—	1.000	1.000	1.000	1.000	1.000	1.000	0.934	0.938	0.942	0.945	0.948	0.950
4	—	—	—	—	1.000	1.000	1.000	1.000	0.900	0.910	0.917	0.924	0.858	0.867	0.875	0.883	0.834	0.843	0.850
5	—	—	—	1.000	1.000	1.000	0.875	0.889	0.900	0.819	0.834	0.800	0.800	0.800	0.800	0.800	0.778	0.748	0.800
6	—	—	1.000	1.000	1.000	0.858	0.834	0.834	0.800	0.819	0.834	0.770	0.762	0.767	0.750	0.716	0.778	0.729	0.734
7	—	—	1.000	1.000	0.858	0.858	0.858	0.778	0.758	0.767	0.715	0.715	0.786	0.715	0.688	0.706	0.691	0.685	0.665
8	—	—	1.000	0.875	0.834	0.858	0.875	0.764	0.750	0.728	0.709	0.693	0.679	0.675	0.688	0.648	0.653	0.645	0.650
9	—	1.000	1.000	0.889	0.834	0.778	0.764	0.778	0.700	0.708	0.695	0.667	0.667	0.667	0.653	0.648	0.667	0.626	0.617
10	—	1.000	0.900	0.900	0.800	0.758	0.750	0.700	0.800	0.700	0.667	0.647	0.643	0.667	0.625	0.624	0.600	0.595	0.650
11	—	1.000	0.910	0.819	0.819	0.767	0.728	0.708	0.700	0.728	0.652	0.637	0.624	0.619	0.603	0.589	0.596	0.584	0.578
12	—	1.000	0.917	0.834	0.834	0.715	0.709	0.695	0.667	0.652	0.667	0.609	0.620	0.600	0.605	0.584	0.584	0.571	0.584
13	—	1.000	0.924	0.800	0.770	0.715	0.693	0.667	0.647	0.637	0.609	0.693	0.572	0.590	0.582	0.575	0.560	0.559	0.550
14	—	1.000	0.858	0.800	0.762	0.786	0.679	0.667	0.643	0.624	0.620	0.572	0.643	0.586	0.563	0.564	0.556	0.557	0.543
15	—	0.934	0.867	0.800	0.767	0.715	0.675	0.667	0.667	0.619	0.600	0.590	0.586	0.600	0.555	0.557	0.545	0.534	0.534
16	—	0.938	0.875	0.800	0.750	0.688	0.688	0.653	0.625	0.603	0.605	0.582	0.563	0.555	0.625	0.526	0.535	0.527	0.525
17	—	0.942	0.883	0.800	0.716	0.706	0.648	0.648	0.624	0.589	0.584	0.575	0.564	0.557	0.526	0.589	0.536	0.514	0.515
18	—	0.945	0.834	0.778	0.778	0.691	0.653	0.667	0.600	0.596	0.584	0.560	0.556	0.545	0.535	0.536	0.556	0.515	0.506
19	1.000	0.948	0.843	0.748	0.729	0.685	0.645	0.626	0.595	0.584	0.571	0.559	0.557	0.534	0.527	0.514	0.515	0.527	0.493
20	1.000	0.950	0.850	0.800	0.734	0.665	0.650	0.617	0.650	0.578	0.584	0.550	0.543	0.534	0.525	0.515	0.506	0.493	0.550

Tip: click any value to highlight its row and column.

How to read this table

Compute the empirical distribution function of each sample and D, the largest vertical distance between them.
For equal sample sizes use the first table; otherwise use the table for your α with n₁ as the row and n₂ as the column.
Reject H₀ if D is greater than or equal to the table value.

Worked example

Time-to-onset data from two sites with 10 patients each give D = 0.70. Row n = 10, α = 0.05 of the equal-size table gives 0.700, so the two distributions differ at the 5% level. With equal samples D can only take values in steps of 1/n, which is why the critical values jump.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

D = max_x |F₁(x) − F₂(x)| · large samples: D_crit ≈ c(α) √((n₁ + n₂)/(n₁n₂)), c(0.05) = 1.358

Values come from the exact null distribution of D, computed by counting the lattice paths that stay inside the band |i/n₁ − j/n₂| < d. Each entry is the smallest attainable D with P(D ≥ d) ≤ α, rounded up to three decimals.

Frequently asked questions

What does the two-sample KS test detect?

Any difference between the two distributions: location, spread or shape. For a shift in location alone, the Mann-Whitney U test is usually more powerful.

What about ties?

The exact values assume continuous data. With many ties the test is conservative.

Two-Sample Kolmogorov-Smirnov Table: Exact Critical Values of D

Two-Sample KS Table

Equal sample sizes (n₁ = n₂ = n)

Unequal sample sizes, two-sided α = 0.05

Unequal sample sizes, two-sided α = 0.01

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

Related calculators and tutorials