Nonparametric Tests

Wilcoxon Signed-Rank Test Table: Critical Values of T

Exact critical values for the Wilcoxon signed-rank test, the nonparametric alternative to the paired t-test.

n = 5 to 50
Exact values
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Wilcoxon Signed-Rank Table

Rows: number of non-zero differences n · Columns: α · Cells: largest T that is significant. Reject H₀ if T ≤ the cell.

Wilcoxon Signed-Rank 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	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37
21	67	58	49	42
22	75	65	55	48
23	83	73	62	54
24	91	81	69	61
25	100	89	76	68
26	110	98	84	75
27	119	107	92	83
28	130	116	101	91
29	140	126	110	100
30	151	137	120	109
31	163	147	130	118
32	175	159	140	128
33	187	170	151	138
34	200	182	162	148
35	213	195	173	159
36	227	208	185	171
37	241	221	198	182
38	256	235	211	194
39	271	249	224	207
40	286	264	238	220
41	302	279	252	233
42	319	294	266	247
43	336	310	281	261
44	353	327	296	276
45	371	343	312	291
46	389	361	328	307
47	407	378	345	322
48	426	396	362	339
49	446	415	379	355
50	466	434	397	373

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

How to read this table

Compute each paired difference and drop any that are zero. n is the number of differences left.
Rank the absolute differences, add up the ranks of the positive and of the negative differences, and let T be the smaller sum (for a two-sided test).
Reject H₀ if T is less than or equal to the table value.
A dash (—) means no result can be significant at that α with so few pairs.

Worked example

Twelve patients have blood pressure measured before and after treatment, with no zero differences, and the smaller rank sum is T = 10. Row n = 12, two-tailed α = 0.05 gives 13. Because 10 ≤ 13, the change is significant at the 5% level.

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

Reject H₀ if T ≤ T_{α, n}, where P(T ≤ T_{α, n}) ≤ α

Critical values come from the exact null distribution of T, obtained by counting all 2ⁿ sign assignments. Each value is the largest T with P(T ≤ t) ≤ α (one-tailed). The table assumes no tied absolute differences.

Frequently asked questions

Some published tables differ by one. Why?

Tables differ in whether the listed value is the largest T with P(T ≤ t) ≤ α (used here, so the test never exceeds α) or the nearest value. This table always keeps the true significance level at or below α.

What about ties and zeros?

Zero differences are dropped before ranking; tied absolute differences get the average of their ranks. With many ties the exact table is only approximate, and a normal approximation with a tie correction is often used.

What if n is larger than 50?

The calculator above handles n up to 100 exactly. For large n, T is approximately normal with mean n(n + 1)/4 and variance n(n + 1)(2n + 1)/24.

Wilcoxon Signed-Rank Test Table: Critical Values of T

Wilcoxon Signed-Rank Table

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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