TABLE: Distribution function of T, the Wilcoxon signed-rank test

Sample Size

t	2	3	4	5	6	7	8	9	10	11	12
1	0.5000	0.2500	0.1250	0.0625	0.0313	0.0157	0.0079	0.0040	0.0020	0.0010	0.0005
2	0.3750	0.1875	0.0938	0.0469	0.0235	0.0118	0.0059	0.0030	0.0015	0.0008
3	0.6250	0.3125	0.1563	0.0782	0.0391	0.0196	0.0098	0.0049	0.0025	0.0013
4	0.4375	0.2188	0.1094	0.0547	0.0274	0.0137	0.0069	0.0035	0.0018
5	0.5625	0.3125	0.1563	0.0782	0.0391	0.0196	0.0098	0.0049	0.0025
6	0.4063	0.2188	0.1094	0.0547	0.0274	0.0137	0.0069	0.0035
7	0.50000	0.2813	0.1485	0.0743	0.0372	0.0186	0.0093	0.0047
8	0.3438	0.1875	0.0977	0.0489	0.0245	0.0123	0.0062
9	0.4219	0.2344	0.125	0.0645	0.0323	0.0162	0.0081
10	0.5000	0.2891	0.1563	0.0821	0.0420	0.0210	0.0105
11	0.3438	0.1915	0.1016	0.0528	0.0269	0.0135
12	0.4063	0.2305	0.125	0.0655	0.0337	0.0171
13	0.4688	0.2735	0.1504	0.0801	0.0416	0.0213
14	0.5313	0.3204	0.1797	0.0967	0.0508	0.0262
15	.	0.3711	0.2129	0.1163	0.0616	0.0320
16	0.4219	0.2481	0.1377	0.0738	0.0386
17	0.4727	0.2852	0.1612	0.0875	0.0462
18	0.5274	0.3262	0.1875	0.1031	0.0550
19	0.3672	0.2159	0.1202	0.0647
20	0.4102	0.2461	0.1392	0.0757
21	0.4551	0.2784	0.1602	0.0882
22	0.5000	0.3125	0.1827	0.1019
23	0.3477	0.2066	0.1167
24	0.3848	0.2325	0.1331
25	0.4229	0.2598	0.1507
26	0.4610	0.2886	0.1697
27	0.5000	0.3189	0.1902
28	0.3501	0.2120
29	0.3824	0.2349
30	0.4156	0.2593
31	0.4493	0.2847
32	,	0.4830	0.3111
33	0.5171	0.3387
34	0.3667
35	0.3956
36	0.4251
37	0.4549
38	0.4849
39	0.5152

STATISTICS

Nonparametric tests

TABLE: WILCOXON SIGNED RANK TEST (CI % = 95%)

Critical values:  

Wilcoxon Signed-Ranks Test Critical values
Number (n)	2 sided	1 sided
6	0	2
7	2	3
8	3	5
9	5	8
10	8	10
11	10	13
12	13	17
13	17	21
14	21	25
15	25	30
16	29	35
17	34	41
18	40	47
19	46	53
20	52	60
21	58	67
22	65	75
23	73	83
24	81	91
25	89	100
Critical values: Wilcoxon Signed-Ranks test p=0.05 (CI% = 95%). Significant, if the calculated values presented in this table [ the sum of the positive ranks or the negative ranks ] is too small.

FORMULA:
If n (> 8), Z₊ then the distribution is approximately normal. Mean and variance :

Sum of Z values:

 

Test statistic W is

 

Suppose w = teststatistic : Reject the Nulhypothesis if

Prob (W > w) < 0.05	(1 sided)
Prob | (W | > | w | ) = 2 * Prob (W > w) < 0.05	(2 sided)

The Wilcoxon rank test is more powerful than the Sign test

Selected Critical Values of the Wilcoxon Statistic (W) for alpha = .1 and .05
Number of Pairs	Alpha Level
		.1	.05
5	0	-
6	2	0
7	3	2
8	5	3
9	8	5
10	10	8
12	17	13
15	30	25
20	60	52
25	100	89
Note: W is significant at the chosen alpha level if it is smaller than the critical value given in the  table.

1

---