Mann–Whitney U Critical Values Table

Find exact Mann–Whitney U critical values for two independent samples. Choose the sample sizes and significance level, read the table, and learn how to compare the observed U statistic with the critical value.

Mann–Whitney U Critical Values — Two-Tailed Test

Rows show the first sample size n₁ and columns show the second sample size n₂. Each cell gives the largest U value that meets the selected exact two-tailed significance level. The table assumes independent observations and no tied pooled ranks.

Reject the two-sided null when U = min(U₁,U₂) is at or below the listed cutoff. A dash (—) means no attainable rejection threshold; it is not zero.

Two-tailed α = 0.05

Exact two-tailed Mann–Whitney U critical values, α = 0.05, no ties
n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

Two-tailed α = 0.01

Exact two-tailed Mann–Whitney U critical values, α = 0.01, no ties
n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	0	0
3	—	—	—	—	—	—	—	0	0	0	1	1	1	2	2	2	2	3	3
4	—	—	—	—	0	0	1	1	2	2	3	3	4	5	5	6	6	7	8
5	—	—	—	0	1	1	2	3	4	5	6	7	7	8	9	10	11	12	13
6	—	—	0	1	2	3	4	5	6	7	9	10	11	12	13	15	16	17	18
7	—	—	0	1	3	4	6	7	9	10	12	13	15	16	18	19	21	22	24
8	—	—	1	2	4	6	7	9	11	13	15	17	18	20	22	24	26	28	30
9	—	0	1	3	5	7	9	11	13	16	18	20	22	24	27	29	31	33	36
10	—	0	2	4	6	9	11	13	16	18	21	24	26	29	31	34	37	39	42
11	—	0	2	5	7	10	13	16	18	21	24	27	30	33	36	39	42	45	48
12	—	1	3	6	9	12	15	18	21	24	27	31	34	37	41	44	47	51	54
13	—	1	3	7	10	13	17	20	24	27	31	34	38	42	45	49	53	57	60
14	—	1	4	7	11	15	18	22	26	30	34	38	42	46	50	54	58	63	67
15	—	2	5	8	12	16	20	24	29	33	37	42	46	51	55	60	64	69	73
16	—	2	5	9	13	18	22	27	31	36	41	45	50	55	60	65	70	74	79
17	—	2	6	10	15	19	24	29	34	39	44	49	54	60	65	70	75	81	86
18	—	2	6	11	16	21	26	31	37	42	47	53	58	64	70	75	81	87	92
19	0	3	7	12	17	22	28	33	39	45	51	57	63	69	74	81	87	93	99
20	0	3	8	13	18	24	30	36	42	48	54	60	67	73	79	86	92	99	105

All displayed cutoffs use the exact no-tie permutation distribution, not a normal approximation. The quick lookup supplies separate directional one-sided cutoffs and attained probabilities.

Find a Mann–Whitney U Critical Value

Enter sample sizes and select the tail convention. The complete static table remains visible without JavaScript.

How to Use the Mann–Whitney U Table

First determine the sizes of your two independent samples. Calculate the Mann–Whitney U statistic using the pooled ranks, then locate the row and column for n₁ and n₂. Choose the correct significance level and test direction. For a two-sided test using the smaller U statistic, compare U with the exact critical value. Reject the null hypothesis when U is at or below the critical value.

Mann–Whitney U Formula

The Mann–Whitney U statistic measures the relative ordering of observations from two independent groups. Rank the combined observations, calculate the rank sum for one group, and use that sum to obtain U. The two group-specific U statistics add to n₁n₂.

U₁ = R₁ − n₁(n₁+1)/2; U₂ = n₁n₂ − U₁, where R₁ is the pooled rank sum for sample 1. For the two-sided reference, U = min(U₁,U₂). Reject when U ≤ Ucritical.

One-Tailed vs. Two-Tailed Mann–Whitney Tests

A two-tailed test looks for a difference in either direction, while a one-tailed test evaluates a direction specified before examining the data. The selected alternative determines which tail probability and which group-specific statistic to use. Critical values must be read using the correct convention.

With the rank-sum definition above, small U₁ supports sample 1 being lower; small U₂ supports sample 1 being higher. For a one-sided lower alternative compare U₁ ≤ c. For a higher alternative compare U₂ ≤ c, equivalently U₁ ≥ n₁n₂ − c. Never use min(U₁,U₂) for both directions at the one-sided α.

What Does a Dash Mean in the U Table?

The exact Mann–Whitney distribution is discrete. With very small samples, even the most extreme possible ordering may not reach a chosen significance level. A dash indicates that no critical U threshold can satisfy that level for the specified test convention.

No exact critical value is available at this significance level for these sample sizes. A dash is not a zero. A listed zero is an attainable cutoff and has a different meaning.

Assumptions and Limitations

Use two independent samples with independent observations and appropriate ordinal or continuous measurements. These exact tables assume no tied pooled ranks and exchangeable group labels under the null of identical distributions.

Tied observations require an adjusted exact method or another appropriate procedure. A Mann–Whitney test does not universally test equality of medians without additional assumptions about the distributions, such as comparable shapes. Lookup is exact only within the documented 2–20 range for each sample; there is no silent normal approximation.

Worked Examples

Example 1: Two-Tailed Test at α = 0.05

Suppose n₁ = 5 and n₂ = 5, with no tied ranks. The exact two-tailed critical value at α = 0.05 is U = 2. If the observed smaller U is 2, the result reaches the critical region. If the observed U is 3, it does not.

Example 2: No Attainable Critical Value

For n₁ = 3 and n₂ = 3, a two-tailed exact test at α = 0.05 has no attainable critical value. Even the most extreme rank separation cannot produce a two-sided p-value at or below 0.05. The table therefore displays a dash.

Example 3: Larger Sample Sizes

For n₁ = 8 and n₂ = 10, the exact two-tailed critical value at α = 0.05 is U = 17. An observed smaller U of 14 falls in the rejection region, provided the assumptions of the exact table apply.

Frequently Asked Questions

What is a Mann–Whitney U table?

It is a reference table of critical U values used when comparing two independent samples with the Mann–Whitney test. The critical value depends on both sample sizes and the significance level.

How do I read the Mann–Whitney U table?

Find the first sample size in the rows and the second in the columns, then read the cell for the correct significance level. Compare your U statistic with the listed critical value using the stated decision rule.

What does U mean in the table?

For the two-sided table, U is the smaller of the two group-specific Mann–Whitney U statistics. Other sources may use different notation, so the statistic definition must always be checked.

Can I use this table for paired samples?

No. The Mann–Whitney U test is designed for independent groups. Paired observations may require the Wilcoxon signed-rank test instead.

Why are some critical values missing?

The exact null distribution is discrete. For certain small sample sizes, no possible U value reaches the requested significance level.

What is the difference between one-tailed and two-tailed U tests?

A one-tailed test examines a preselected direction, while a two-tailed test examines differences in either direction. The significance threshold and correct statistic convention depend on the choice.

Is the Mann–Whitney U test the same as the Wilcoxon rank-sum test?

They are equivalent formulations of the same independent-samples rank test. They should not be confused with the Wilcoxon signed-rank test for paired observations.

Does the Mann–Whitney test always compare medians?

Not necessarily. Its interpretation depends on the distributions and assumptions. A location or median interpretation generally requires additional conditions, such as comparable distribution shapes.

Sources and Methodology

Counts are generated by the inversion recurrence F(m,n,u) = F(m−1,n,u−n) + F(m,n−1,u), with boundary distributions concentrated at zero. Counts total C(m+n,m). BigInt arithmetic counts the actual rejection event min(U₁,mn−U₁) ≤ c for two tails or the selected directional U ≤ c for one tail. The largest cutoff with probability ≤ α is chosen using exact integer cross-multiplication. An independent rank-subset generator verifies every distribution for m,n = 2–20.

Source reconciliation: the Portland State formula packet lists the conservative value 13 for n₁ = 4, n₂ = 20 at two-tailed α = 0.05. Under this page’s largest-attainable-cutoff convention, the value is 14: its exact rejection probability is 241/5313 ≈ 0.04536, while cutoff 15 gives 295/5313 ≈ 0.05552. Independent rank counting and the Clinical Biostats table agree on 14.

Reviewed: October 29, 2026. Reference data is generated into static HTML at build time. Optional lookups and clipboard actions run locally; no measurements are sent to a server.

Without JavaScript, all formulas, tables, examples, and FAQ remain available. Lookup and Copy actions require JavaScript.

n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	0	0
3	—	—	—	—	—	—	—	0	0	0	1	1	1	2	2	2	2	3	3
4	—	—	—	—	0	0	1	1	2	2	3	3	4	5	5	6	6	7	8
5	—	—	—	0	1	1	2	3	4	5	6	7	7	8	9	10	11	12	13
6	—	—	0	1	2	3	4	5	6	7	9	10	11	12	13	15	16	17	18
7	—	—	0	1	3	4	6	7	9	10	12	13	15	16	18	19	21	22	24
8	—	—	1	2	4	6	7	9	11	13	15	17	18	20	22	24	26	28	30
9	—	0	1	3	5	7	9	11	13	16	18	20	22	24	27	29	31	33	36
10	—	0	2	4	6	9	11	13	16	18	21	24	26	29	31	34	37	39	42
11	—	0	2	5	7	10	13	16	18	21	24	27	30	33	36	39	42	45	48
12	—	1	3	6	9	12	15	18	21	24	27	31	34	37	41	44	47	51	54
13	—	1	3	7	10	13	17	20	24	27	31	34	38	42	45	49	53	57	60
14	—	1	4	7	11	15	18	22	26	30	34	38	42	46	50	54	58	63	67
15	—	2	5	8	12	16	20	24	29	33	37	42	46	51	55	60	64	69	73
16	—	2	5	9	13	18	22	27	31	36	41	45	50	55	60	65	70	74	79
17	—	2	6	10	15	19	24	29	34	39	44	49	54	60	65	70	75	81	86
18	—	2	6	11	16	21	26	31	37	42	47	53	58	64	70	75	81	87	92
19	0	3	7	12	17	22	28	33	39	45	51	57	63	69	74	81	87	93	99
20	0	3	8	13	18	24	30	36	42	48	54	60	67	73	79	86	92	99	105

n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

n₁ \ n₂	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	0	0
3	—	—	—	—	—	—	—	0	0	0	1	1	1	2	2	2	2	3	3
4	—	—	—	—	0	0	1	1	2	2	3	3	4	5	5	6	6	7	8
5	—	—	—	0	1	1	2	3	4	5	6	7	7	8	9	10	11	12	13
6	—	—	0	1	2	3	4	5	6	7	9	10	11	12	13	15	16	17	18
7	—	—	0	1	3	4	6	7	9	10	12	13	15	16	18	19	21	22	24
8	—	—	1	2	4	6	7	9	11	13	15	17	18	20	22	24	26	28	30
9	—	0	1	3	5	7	9	11	13	16	18	20	22	24	27	29	31	33	36
10	—	0	2	4	6	9	11	13	16	18	21	24	26	29	31	34	37	39	42
11	—	0	2	5	7	10	13	16	18	21	24	27	30	33	36	39	42	45	48
12	—	1	3	6	9	12	15	18	21	24	27	31	34	37	41	44	47	51	54
13	—	1	3	7	10	13	17	20	24	27	31	34	38	42	45	49	53	57	60
14	—	1	4	7	11	15	18	22	26	30	34	38	42	46	50	54	58	63	67
15	—	2	5	8	12	16	20	24	29	33	37	42	46	51	55	60	64	69	73
16	—	2	5	9	13	18	22	27	31	36	41	45	50	55	60	65	70	74	79
17	—	2	6	10	15	19	24	29	34	39	44	49	54	60	65	70	75	81	86
18	—	2	6	11	16	21	26	31	37	42	47	53	58	64	70	75	81	87	92
19	0	3	7	12	17	22	28	33	39	45	51	57	63	69	74	81	87	93	99
20	0	3	8	13	18	24	30	36	42	48	54	60	67	73	79	86	92	99	105