Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.

Wilcoxon Test for Paired Differences

In order to compare two matched samples we have two major possibilities: the t test for matched pairs if the differences of the pairs are normally distributed, and the signed rank test of Wilcoxon if the differences are not normally distributed.

The Wilcoxon test can be used to check whether the differences of matched pairs are distributed symmetrically around the median (provided that the median of the differences is zero). If the null hypothesis has to be rejected then either the medians are not equal, or the samples origin from two different distributions.

In order to perform the test one has first to calculate the pairwise differences between the N observations of the two groups. Pairs whose difference is zero will be discarded. The remaining M differences have to be sorted and ranked according to their absolute values. The ranks of tied differences have to be averaged. Next, the ranks of all positive and of all negative differences are summed up. The smaller of the two sums gives the test statistic W.

The null hypothesis has to be rejected if the statistic W is less than the critical limit Wα. The critical limits for two-tailed tests are listed in the following table (taken from McCornack 1965 ):

M α=0.1 α=0.05% α=0.02% α=0.01% α=0.001%
6 2 0 0 0 0
7 3 2 0 0 0
8 5 3 1 0 0
9 8 5 3 1 0
10 10 8 5 3 0
11 13 10 7 5 0
12 17 13 9 7 1
13 21 17 12 9 2
14 25 21 15 12 4
15 30 25 19 15 6
16 35 29 23 19 8
17 41 34 27 23 11
18 47 40 32 27 14
19 53 46 37 32 18
20 60 52 43 37 21
21 67 58 49 42 25
22 75 65 55 48 30
23 83 73 62 54 35
24 91 81 69 61 40
25 100 89 76 68 45
26 110 98 84 75 51
27 119 107 92 83 57
28 130 116 101 91 64
29 140 126 110 100 71
30 151 137 120 109 78
31 163 147 130 118 86
32 175 159 140 128 94
33 187 170 151 138 102
34 200 182 162 148 111
35 213 195 173 159 120
36 227 208 185 171 130
37 241 221 198 182 140
38 256 235 211 194 150
39 271 249 224 207 161
40 286 264 238 220 172
41 302 279 252 233 183
42 319 294 266 247 195
43 336 310 281 261 207
44 353 327 296 276 220
45 371 343 312 291 233
46 389 361 328 307 246
47 407 378 345 322 260
48 426 396 362 339 274
49 446 415 379 355 289
50 466 434 397 373 304
51 486 453 416 390 319
52 507 473 434 408 335
53 529 494 454 427 351
54 550 514 473 445 368
55 573 536 493 465 385
56 595 557 514 484 402
57 618 579 535 504 420
58 642 602 556 525 438
59 666 625 578 546 457
60 690 648 600 567 476
61 715 672 623 589 495
62 741 697 646 611 515
63 767 721 669 634 535
64 793 747 693 657 556
65 820 772 718 681 577
66 847 798 742 705 599
67 875 825 768 729 621
68 903 852 793 754 643
69 931 879 819 779 666
70 960 907 846 805 689
71 990 936 873 831 712
72 1020 964 901 858 736
73 1050 994 928 884 761
74 1081 1023 957 912 786
75 1112 1053 986 940 811
76 1144 1084 1015 968 836
77 1176 1115 1044 997 862
78 1209 1147 1075 1026 889
79 1242 1179 1105 1056 916
80 1276 1211 1136 1086 943
81 1310 1244 1168 1116 971
82 1345 1277 1200 1147 999
83 1380 1311 1232 1178 1028
84 1415 1345 1265 1210 1057
85 1451 1380 1298 1242 1086
86 1487 1415 1332 1275 1116
87 1524 1451 1366 1308 1146
88 1561 1487 1400 1342 1177
89 1599 1523 1435 1376 1208
90 1638 1560 1471 1410 1240
91 1676 1597 1507 1445 1271
92 1715 1635 1543 1480 1304
93 1755 1674 1580 1516 1337
94 1795 1712 1617 1552 1370
95 1836 1752 1655 1589 1404
96 1877 1791 1693 1626 1438
97 1918 1832 1731 1664 1472
98 1960 1872 1770 1702 1507
99 2003 1913 1810 1740 1543
100 2045 1955 1850 1779 1578

Last Update: 2012-10-08