Nonparametric Tests

Sign Test Table: Critical Values for n = 5 to 100

Critical values for the sign test, the simplest test for paired data, based only on the direction of each difference.

n = 5 to 100
Exact binomial values
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Sign Test Table

Rows: number of non-tied pairs n · Columns: α · Cells: largest count of the less frequent sign that is significant

Sign Test Table (Critical Values)
One-tailed α →	0.05	0.025	0.01	0.005
Two-tailed α → n ↓	0.1	0.05	0.02	0.01
5	0	—	—	—
6	0	0	—	—
7	0	0	0	—
8	1	0	0	0
9	1	1	0	0
10	1	1	0	0
11	2	1	1	0
12	2	2	1	1
13	3	2	1	1
14	3	2	2	1
15	3	3	2	2
16	4	3	2	2
17	4	4	3	2
18	5	4	3	3
19	5	4	4	3
20	5	5	4	3
21	6	5	4	4
22	6	5	5	4
23	7	6	5	4
24	7	6	5	5
25	7	7	6	5
26	8	7	6	6
27	8	7	7	6
28	9	8	7	6
29	9	8	7	7
30	10	9	8	7
31	10	9	8	7
32	10	9	8	8
33	11	10	9	8
34	11	10	9	9
35	12	11	10	9
36	12	11	10	9
37	13	12	10	10
38	13	12	11	10
39	13	12	11	11
40	14	13	12	11
41	14	13	12	11
42	15	14	13	12
43	15	14	13	12
44	16	15	13	13
45	16	15	14	13
46	16	15	14	13
47	17	16	15	14
48	17	16	15	14
49	18	17	15	15
50	18	17	16	15
51	19	18	16	15
52	19	18	17	16
53	20	18	17	16
54	20	19	18	17
55	20	19	18	17
56	21	20	18	17
57	21	20	19	18
58	22	21	19	18
59	22	21	20	19
60	23	21	20	19
61	23	22	20	20
62	24	22	21	20
63	24	23	21	20
64	24	23	22	21
65	25	24	22	21
66	25	24	23	22
67	26	25	23	22
68	26	25	23	22
69	27	25	24	23
70	27	26	24	23
71	28	26	25	24
72	28	27	25	24
73	28	27	26	25
74	29	28	26	25
75	29	28	26	25
76	30	28	27	26
77	30	29	27	26
78	31	29	28	27
79	31	30	28	27
80	32	30	29	28
81	32	31	29	28
82	33	31	30	28
83	33	32	30	29
84	33	32	30	29
85	34	32	31	30
86	34	33	31	30
87	35	33	32	31
88	35	34	32	31
89	36	34	33	31
90	36	35	33	32
91	37	35	33	32
92	37	36	34	33
93	38	36	34	33
94	38	37	35	34
95	38	37	35	34
96	39	37	36	34
97	39	38	36	35
98	40	38	37	35
99	40	39	37	36
100	41	39	37	36

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

How to read this table

Record whether each paired difference is positive or negative, and drop ties (zero differences). n is the number of pairs left.
Count how many times the less frequent sign occurs.
Reject H₀ if that count is less than or equal to the table value.

Worked example

Twenty patients rate two formulations and 5 prefer the standard one (15 prefer the new one, no ties). Row n = 20, two-tailed α = 0.05 gives 5. Because 5 ≤ 5, the preference is significant at the 5% level.

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

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Formula and how the values were computed

P(X ≤ k) = Σ_i=0^k C(n, i) (½)ⁿ

Each value is the largest k with P(X ≤ k) ≤ α, where X is binomial with n trials and p = 0.5, so the true significance level never exceeds α.

Frequently asked questions

Sign test or Wilcoxon signed-rank test?

The Wilcoxon signed-rank test also uses the size of each difference, so it is usually more powerful. The sign test needs fewer assumptions and works when only the direction of change is known.

Sign Test Table: Critical Values for n = 5 to 100

Sign Test Table

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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