This table presents upper one-sided and two-sided critical values for the Mann-Whitney
![$U$](img1.png)
statistic. Lower one-sided critical values are computed from the upper one-sided critical value (at the same significance level) as
![$(M \cdot N) - U$](img2.png)
. The Wilcoxon two-sample statistic,
![$W$](img3.png)
, is related to
![$U$](img1.png)
by the equation
![$W = (M \cdot N) + (M \cdot (M + 1)/2) - U$](img4.png)
, where
![$W$](img3.png)
is the sum of the ranks of the sample of size
![$M$](img5.png)
in the combined sample.
Table 1:
Critical values for the Mann-Whitney (Wilcoxon) statistic.
|
One-sided ![$\alpha$](img6.png) |
|
|
.10 |
.05 |
.025 |
.01 |
.005 |
.001 |
|
Two-sided ![$\alpha$](img6.png) |
|
|
.20 |
.10 |
.05 |
.02 |
.01 |
.002 |
![$n$](img7.png) |
![$m$](img8.png) |
|
|
|
|
|
|
3 |
2 |
6 |
-- |
-- |
-- |
-- |
-- |
3 |
3 |
8 |
9 |
-- |
-- |
-- |
-- |
4 |
2 |
8 |
-- |
-- |
-- |
-- |
-- |
4 |
3 |
11 |
12 |
-- |
-- |
-- |
-- |
4 |
4 |
13 |
15 |
16 |
-- |
-- |
-- |
5 |
2 |
9 |
10 |
-- |
-- |
-- |
-- |
5 |
3 |
13 |
14 |
15 |
-- |
-- |
-- |
5 |
4 |
16 |
18 |
19 |
20 |
-- |
-- |
5 |
5 |
20 |
21 |
23 |
24 |
25 |
-- |
6 |
2 |
11 |
12 |
-- |
-- |
-- |
-- |
6 |
3 |
15 |
16 |
17 |
-- |
-- |
-- |
6 |
4 |
19 |
21 |
22 |
23 |
24 |
-- |
6 |
5 |
23 |
25 |
27 |
28 |
29 |
-- |
6 |
6 |
27 |
29 |
31 |
33 |
34 |
-- |
7 |
2 |
13 |
14 |
-- |
-- |
-- |
-- |
7 |
3 |
17 |
19 |
20 |
21 |
-- |
-- |
7 |
4 |
22 |
24 |
25 |
27 |
28 |
-- |
7 |
5 |
27 |
29 |
30 |
32 |
34 |
-- |
7 |
6 |
31 |
34 |
36 |
38 |
39 |
42 |
7 |
7 |
36 |
38 |
41 |
43 |
45 |
48 |
8 |
2 |
14 |
15 |
16 |
-- |
-- |
-- |
8 |
3 |
19 |
21 |
22 |
24 |
-- |
-- |
8 |
4 |
25 |
27 |
28 |
30 |
31 |
-- |
8 |
5 |
30 |
32 |
34 |
36 |
38 |
40 |
8 |
6 |
35 |
38 |
40 |
42 |
44 |
47 |
8 |
7 |
40 |
43 |
46 |
49 |
50 |
54 |
8 |
8 |
45 |
49 |
51 |
55 |
57 |
60 |
9 |
1 |
9 |
-- |
-- |
-- |
-- |
-- |
9 |
2 |
16 |
17 |
18 |
-- |
-- |
-- |
9 |
3 |
22 |
23 |
25 |
26 |
27 |
-- |
9 |
4 |
27 |
30 |
32 |
33 |
35 |
-- |
9 |
5 |
33 |
36 |
38 |
40 |
42 |
44 |
9 |
6 |
39 |
42 |
44 |
47 |
49 |
52 |
9 |
7 |
45 |
48 |
51 |
54 |
56 |
60 |
9 |
8 |
50 |
54 |
57 |
61 |
63 |
67 |
9 |
9 |
56 |
60 |
64 |
67 |
70 |
74 |
10 |
1 |
10 |
-- |
-- |
-- |
-- |
-- |
10 |
2 |
17 |
19 |
20 |
-- |
-- |
-- |
10 |
3 |
24 |
26 |
27 |
29 |
30 |
-- |
10 |
4 |
30 |
33 |
35 |
37 |
38 |
40 |
10 |
5 |
37 |
39 |
42 |
44 |
46 |
49 |
10 |
6 |
43 |
46 |
49 |
52 |
54 |
57 |
10 |
7 |
49 |
53 |
56 |
59 |
61 |
65 |
10 |
8 |
56 |
60 |
63 |
67 |
69 |
74 |
10 |
9 |
62 |
66 |
70 |
74 |
77 |
82 |
10 |
10 |
68 |
73 |
77 |
81 |
84 |
90 |
11 |
1 |
11 |
-- |
-- |
-- |
-- |
-- |
11 |
2 |
19 |
21 |
22 |
-- |
-- |
-- |
11 |
3 |
26 |
28 |
30 |
32 |
33 |
-- |
11 |
4 |
33 |
36 |
38 |
40 |
42 |
44 |
11 |
5 |
40 |
43 |
46 |
48 |
50 |
53 |
11 |
6 |
47 |
50 |
53 |
57 |
59 |
62 |
11 |
7 |
54 |
58 |
61 |
65 |
67 |
71 |
11 |
8 |
61 |
65 |
69 |
73 |
75 |
80 |
11 |
9 |
68 |
72 |
76 |
81 |
83 |
89 |
11 |
10 |
74 |
79 |
84 |
88 |
92 |
98 |
11 |
11 |
81 |
87 |
91 |
96 |
100 |
106 |
12 |
1 |
12 |
-- |
-- |
-- |
-- |
-- |
12 |
2 |
20 |
22 |
23 |
-- |
-- |
-- |
12 |
3 |
28 |
31 |
32 |
34 |
35 |
-- |
12 |
4 |
36 |
39 |
41 |
43 |
45 |
48 |
12 |
5 |
43 |
47 |
49 |
52 |
54 |
58 |
12 |
6 |
51 |
55 |
58 |
61 |
63 |
68 |
12 |
7 |
58 |
63 |
66 |
70 |
72 |
77 |
12 |
8 |
66 |
70 |
74 |
79 |
81 |
87 |
12 |
9 |
73 |
78 |
82 |
87 |
90 |
96 |
12 |
10 |
81 |
86 |
91 |
96 |
99 |
106 |
12 |
11 |
88 |
94 |
99 |
104 |
108 |
115 |
12 |
12 |
95 |
102 |
107 |
113 |
117 |
124 |
13 |
1 |
13 |
-- |
-- |
-- |
-- |
-- |
13 |
2 |
22 |
24 |
25 |
26 |
-- |
-- |
13 |
3 |
30 |
33 |
35 |
37 |
38 |
-- |
13 |
4 |
39 |
42 |
44 |
47 |
49 |
51 |
13 |
5 |
47 |
50 |
53 |
56 |
58 |
62 |
13 |
6 |
55 |
59 |
62 |
66 |
68 |
73 |
13 |
7 |
63 |
67 |
71 |
75 |
78 |
83 |
13 |
8 |
71 |
76 |
80 |
84 |
87 |
93 |
13 |
9 |
79 |
84 |
89 |
94 |
97 |
103 |
13 |
10 |
87 |
93 |
97 |
103 |
106 |
113 |
13 |
11 |
95 |
101 |
106 |
112 |
116 |
123 |
13 |
12 |
103 |
109 |
115 |
121 |
125 |
133 |
13 |
13 |
111 |
118 |
124 |
130 |
135 |
143 |
14 |
1 |
14 |
-- |
-- |
-- |
-- |
-- |
14 |
2 |
23 |
25 |
27 |
28 |
-- |
-- |
14 |
3 |
32 |
35 |
37 |
40 |
41 |
-- |
14 |
4 |
41 |
45 |
47 |
50 |
52 |
55 |
14 |
5 |
50 |
54 |
57 |
60 |
63 |
67 |
14 |
6 |
59 |
63 |
67 |
71 |
73 |
78 |
14 |
7 |
67 |
72 |
76 |
81 |
83 |
89 |
14 |
8 |
76 |
81 |
86 |
90 |
94 |
100 |
14 |
9 |
85 |
90 |
95 |
100 |
104 |
111 |
14 |
10 |
93 |
99 |
104 |
110 |
114 |
121 |
14 |
11 |
102 |
108 |
114 |
120 |
124 |
132 |
14 |
12 |
110 |
117 |
123 |
130 |
134 |
143 |
14 |
13 |
119 |
126 |
132 |
139 |
144 |
153 |
14 |
14 |
127 |
135 |
141 |
149 |
154 |
164 |
15 |
1 |
15 |
-- |
-- |
-- |
-- |
-- |
15 |
2 |
25 |
27 |
29 |
30 |
-- |
-- |
15 |
3 |
35 |
38 |
40 |
42 |
43 |
-- |
15 |
4 |
44 |
48 |
50 |
53 |
55 |
59 |
15 |
5 |
53 |
57 |
61 |
64 |
67 |
71 |
15 |
6 |
63 |
67 |
71 |
75 |
78 |
83 |
15 |
7 |
72 |
77 |
81 |
86 |
89 |
95 |
15 |
8 |
81 |
87 |
91 |
96 |
100 |
106 |
15 |
9 |
90 |
96 |
101 |
107 |
111 |
118 |
15 |
10 |
99 |
106 |
111 |
117 |
121 |
129 |
15 |
11 |
108 |
115 |
121 |
128 |
132 |
141 |
15 |
12 |
117 |
125 |
131 |
138 |
143 |
152 |
15 |
13 |
127 |
134 |
141 |
148 |
153 |
163 |
15 |
14 |
136 |
144 |
151 |
159 |
164 |
174 |
15 |
15 |
145 |
153 |
161 |
169 |
174 |
185 |
16 |
1 |
16 |
-- |
-- |
-- |
-- |
-- |
16 |
2 |
27 |
29 |
31 |
32 |
-- |
-- |
16 |
3 |
37 |
40 |
42 |
45 |
46 |
-- |
16 |
4 |
47 |
50 |
53 |
57 |
59 |
62 |
16 |
5 |
57 |
61 |
65 |
68 |
71 |
75 |
16 |
6 |
67 |
71 |
75 |
80 |
83 |
88 |
16 |
7 |
76 |
82 |
86 |
91 |
94 |
101 |
16 |
8 |
86 |
92 |
97 |
102 |
106 |
113 |
16 |
9 |
96 |
102 |
107 |
113 |
117 |
125 |
16 |
10 |
106 |
112 |
118 |
124 |
129 |
137 |
16 |
11 |
115 |
122 |
129 |
135 |
140 |
149 |
16 |
12 |
125 |
132 |
139 |
146 |
151 |
161 |
16 |
13 |
134 |
143 |
149 |
157 |
163 |
173 |
16 |
14 |
144 |
153 |
160 |
168 |
174 |
185 |
16 |
15 |
154 |
163 |
170 |
179 |
185 |
197 |
16 |
16 |
163 |
173 |
181 |
190 |
196 |
208 |
17 |
1 |
17 |
-- |
-- |
-- |
-- |
-- |
17 |
2 |
28 |
31 |
32 |
34 |
-- |
-- |
17 |
3 |
39 |
42 |
45 |
47 |
49 |
51 |
17 |
4 |
50 |
53 |
57 |
60 |
62 |
66 |
17 |
5 |
60 |
65 |
68 |
72 |
75 |
80 |
17 |
6 |
71 |
76 |
80 |
84 |
87 |
93 |
17 |
7 |
81 |
86 |
91 |
96 |
100 |
106 |
17 |
8 |
91 |
97 |
102 |
108 |
112 |
119 |
17 |
9 |
101 |
108 |
114 |
120 |
124 |
132 |
17 |
10 |
112 |
119 |
125 |
132 |
136 |
145 |
17 |
11 |
122 |
130 |
136 |
143 |
148 |
158 |
17 |
12 |
132 |
140 |
147 |
155 |
160 |
170 |
17 |
13 |
142 |
151 |
158 |
166 |
172 |
183 |
17 |
14 |
153 |
161 |
169 |
178 |
184 |
195 |
17 |
15 |
163 |
172 |
180 |
189 |
195 |
208 |
17 |
16 |
173 |
183 |
191 |
201 |
207 |
220 |
17 |
17 |
183 |
193 |
202 |
212 |
219 |
232 |
18 |
1 |
18 |
-- |
-- |
-- |
-- |
-- |
18 |
2 |
30 |
32 |
34 |
36 |
-- |
-- |
18 |
3 |
41 |
45 |
47 |
50 |
52 |
54 |
18 |
4 |
52 |
56 |
60 |
63 |
66 |
69 |
18 |
5 |
63 |
68 |
72 |
76 |
79 |
84 |
18 |
6 |
74 |
80 |
84 |
89 |
92 |
98 |
18 |
7 |
85 |
91 |
96 |
102 |
105 |
112 |
18 |
8 |
96 |
103 |
108 |
114 |
118 |
126 |
18 |
9 |
107 |
114 |
120 |
126 |
131 |
139 |
18 |
10 |
118 |
125 |
132 |
139 |
143 |
153 |
18 |
11 |
129 |
137 |
143 |
151 |
156 |
166 |
18 |
12 |
139 |
148 |
155 |
163 |
169 |
179 |
18 |
13 |
150 |
159 |
167 |
175 |
181 |
192 |
18 |
14 |
161 |
170 |
178 |
187 |
194 |
206 |
18 |
15 |
172 |
182 |
190 |
200 |
206 |
219 |
18 |
16 |
182 |
193 |
202 |
212 |
218 |
232 |
18 |
17 |
193 |
204 |
213 |
224 |
231 |
245 |
18 |
18 |
204 |
215 |
225 |
236 |
243 |
258 |
19 |
1 |
18 |
19 |
-- |
-- |
-- |
-- |
19 |
2 |
31 |
34 |
36 |
37 |
38 |
-- |
19 |
3 |
43 |
47 |
50 |
53 |
54 |
57 |
19 |
4 |
55 |
59 |
63 |
67 |
69 |
73 |
19 |
5 |
67 |
72 |
76 |
80 |
83 |
88 |
19 |
6 |
78 |
84 |
89 |
94 |
97 |
103 |
19 |
7 |
90 |
96 |
101 |
107 |
111 |
118 |
19 |
8 |
101 |
108 |
114 |
120 |
124 |
132 |
19 |
9 |
113 |
120 |
126 |
133 |
138 |
146 |
19 |
10 |
124 |
132 |
138 |
146 |
151 |
161 |
19 |
11 |
136 |
144 |
151 |
159 |
164 |
175 |
19 |
12 |
147 |
156 |
163 |
172 |
177 |
188 |
19 |
13 |
158 |
167 |
175 |
184 |
190 |
202 |
19 |
14 |
169 |
179 |
188 |
197 |
203 |
216 |
19 |
15 |
181 |
191 |
200 |
210 |
216 |
230 |
19 |
16 |
192 |
203 |
212 |
222 |
230 |
244 |
19 |
17 |
203 |
214 |
224 |
235 |
242 |
257 |
19 |
18 |
214 |
226 |
236 |
248 |
255 |
271 |
19 |
19 |
226 |
238 |
248 |
260 |
268 |
284 |
20 |
1 |
19 |
20 |
-- |
-- |
-- |
-- |
20 |
2 |
33 |
36 |
38 |
39 |
40 |
-- |
20 |
3 |
45 |
49 |
52 |
55 |
57 |
60 |
20 |
4 |
58 |
62 |
66 |
70 |
72 |
77 |
20 |
5 |
70 |
75 |
80 |
84 |
87 |
93 |
20 |
6 |
82 |
88 |
93 |
98 |
102 |
108 |
20 |
7 |
94 |
101 |
106 |
112 |
116 |
124 |
20 |
8 |
106 |
113 |
119 |
126 |
130 |
139 |
20 |
9 |
118 |
126 |
132 |
140 |
144 |
154 |
20 |
10 |
130 |
138 |
145 |
153 |
158 |
168 |
20 |
11 |
142 |
151 |
158 |
167 |
172 |
183 |
20 |
12 |
154 |
163 |
171 |
180 |
186 |
198 |
20 |
13 |
166 |
176 |
184 |
193 |
200 |
212 |
20 |
14 |
178 |
188 |
197 |
207 |
213 |
226 |
20 |
15 |
190 |
200 |
210 |
220 |
227 |
241 |
20 |
16 |
201 |
213 |
222 |
233 |
241 |
255 |
20 |
17 |
213 |
225 |
235 |
247 |
254 |
270 |
20 |
18 |
225 |
237 |
248 |
260 |
268 |
284 |
20 |
19 |
237 |
250 |
261 |
273 |
281 |
298 |
20 |
20 |
249 |
262 |
273 |
286 |
295 |
312 |