Wilcoxon Test Calculator - Signed-Rank and Mann-Whitney U

Wilcoxon signed-rank and rank-sum tests, with the p-value and the typical shift

Pain scores for ten patients before and after a course of physical therapy: 7 6 8 5 7 9 6 8 7 6 and then 4 5 6 5 3 7 5 6 4 6. Paste both lists, choose paired, and the calculator returns p = 0.0078, V = 36 and a typical change of -2 points. Eight patients improved, two stayed the same, nobody got worse, and that pattern alone is what the test weighs.

The same tool runs the other Wilcoxon test, the rank-sum test for two separate groups, known in most software as the Mann-Whitney U test. Both work on ranks instead of raw values, so a single wild measurement cannot drag the answer around. Up to 50 pairs or 60 values in total the p-value is exact, counted over every possible arrangement of the ranks; above that it uses the normal approximation with the usual corrections.

A t-test and a rank test on the same data, before and after

Before: a t-test on skewed timings

One slow participant at 190 seconds pushes group A's mean up by 16.75 seconds
The standard deviation balloons with it, and Welch's t falls to 1.67
p = 0.1387: "no difference", although 63 of the 64 cross-group comparisons point the same way

After: the rank-sum test

190 seconds is simply the largest rank, the same rank 70 seconds would get
W = 63 of a possible 64, typical shift -12.5 seconds
p = 0.0003, the same value the data give without the outlier

The numbers come from checkout times for two page designs, eight people each: design A 48 52 61 45 58 190 50 55, design B 38 41 44 36 47 40 43 39. Replace 190 with 56 and Welch's t-test gives p = 0.0002. The rank test does not move.

The form, from test type to alpha

Test - paired (signed-rank) when the same people, pages or samples were measured twice; independent (rank-sum, Mann-Whitney) when the two lists come from different subjects. This decides the whole test, not just a detail of it.
First series - the before values, or group 1. Separate numbers with spaces, commas or new lines, or paste a spreadsheet column. Use a dot for decimals.
Second series - the after values, or group 2. In paired mode the order must match the first list, subject for subject, and the lengths must be equal.
Alternative hypothesis - two-sided, or one-sided with the second series higher or lower. One-sided only when the direction was written down before the data existed.
Significance level - 0.05 by default, 0.01 for many comparisons, 0.10 for a first look.
Read the result - p and the verdict first, then the typical shift, the statistic in R's notation, the effect size r and, in paired mode, every pair with its signed rank.

How the rank arithmetic gets to a p-value

    Paired: d = after - before, drop d = 0, rank |d|, T+ = sum of ranks where d > 0

    Independent: rank all values together, U1 = R1 - n1(n1 + 1) / 2, U2 = n1 n2 - U1

    p = share of all equally likely rank arrangements at least as extreme as yours

The pain scores, by hand. Changes: -3, -1, -2, 0, -4, -2, -1, -2, -3, 0. Two zeros drop out, leaving 8 pairs. Ranking the sizes 1, 1, 2, 2, 2, 3, 3, 4 gives midranks 1.5, 1.5, 4, 4, 4, 6.5, 6.5, 8. Every change is negative, so T+ = 0 and V = 36. Of the 2⁸ = 256 ways to attach signs, exactly one gives T+ = 0 and one gives the mirror image, so p = 2/256 = 0.0078.

The checkout times, by hand. Pooled and ranked, design B holds ranks 1 to 7 and 9 (rank 8 is design A's 45 seconds). R2 = 37, so U2 = 37 - 36 = 1 and U1 = 64 - 1 = 63. Only one pair out of 64 has a B time longer than an A time. Among the 12,870 ways to split sixteen ranks into two groups of eight, so few are that lopsided that p = 0.0003.

The typical shift. For paired data it is the median of all averages (d_i + d_j) / 2 with i ≤ j, the Hodges-Lehmann estimate; for two groups, the median of all n1 n2 differences between a group 2 value and a group 1 value. It answers "by how much", which the p-value does not.

What one outlier does to each test

Five data sets, each run through this calculator and through the matching t-test (paired t, or Welch's t for two groups). The last column is what the rank test protects you from.

Data	t-test p	Wilcoxon p	Typical shift	What happened
Pain scores, 10 pairs	0.0019	0.0078	-2 points	clean data, both agree; the rank test cannot go below 2/2⁸
Page load, 8 pages, all faster	below 0.0001	0.0078	-0.35 s	both significant
Same, one page broke: 2.6 s to 9.8 s	0.5450	0.1641	-0.3 s	t-test's mean change flips to +0.6 s slower; the rank test still sees 7 of 8 pages faster
Checkout times, 8 vs 8, no outlier	0.0002	0.0003	-12 s	both significant
Same, one participant at 190 s	0.1387	0.0003	-12.5 s	the t-test loses the effect, the rank test keeps it

The price of that protection shows in the first row. With clean, roughly normal data a t-test extracts a little more from the same numbers, and in tiny samples the rank test has a hard floor. That is the whole trade: some efficiency on well-behaved data in exchange for answers that one bad measurement cannot overturn.

The smallest p each sample size can ever reach

Because the p-value counts arrangements, it has a floor: the case where every change points the same way, or every value of one group beats every value of the other. Below these sizes, 0.05 two-sided is out of reach no matter how strong the effect is.

Paired, nonzero pairs	Smallest two-sided p	Two groups	Smallest two-sided p
4	0.1250	3 and 3	0.1000
5	0.0625	3 and 5	0.0357
6	0.0313	4 and 4	0.0286
7	0.0156	4 and 6	0.0095
8	0.0078	5 and 5	0.0079

Pairs with no change do not count toward the left column. Ten pain scores with two unchanged patients behave like a study of eight, which is why the first example sits exactly on the 0.0078 floor.

Rank tests: the questions that keep coming up

Is the Wilcoxon rank-sum test the same as the Mann-Whitney U test?

Yes, two names for one test. Wilcoxon wrote it with rank sums, Mann and Whitney with U counts, and U = R - n(n + 1)/2 turns one into the other. For design A and B above, R1 = 99 and U1 = 63; either one gives p = 0.0003.

When should I use Wilcoxon instead of a t-test?

When the data are ordinal (ratings, pain scales, Likert items), clearly skewed (times, incomes, counts), or small with a value you cannot vouch for. With large, symmetric, clean samples the t-test is slightly more powerful and reports a mean difference, which some readers want.

Does the rank-sum test compare medians?

Only if both groups have the same shape and differ by a shift. In general it tests whether a value from one group tends to be larger than a value from the other, which the calculator prints as the probability of superiority. In the checkout data that probability is 1.6% for design B beating design A.

Why does my V differ from another program's T or W+?

Programs pick different sums. R's wilcox.test reports V, the sum of ranks where series 1 minus series 2 is positive; SPSS and many textbooks report the smaller of the two sums, T. The calculator shows both sums, so any of them can be read off. The p-value is the same whichever you use.

What happens to ties and to pairs with no change?

Pairs with a change of exactly zero are dropped, as in Wilcoxon's original test, and the result says how many. Equal values share the average of their ranks. The exact p-value is then computed over those midranks, where R would switch to a normal approximation and print a warning, so small samples with ties can differ from R in the third decimal.

What effect size should I report?

The calculator gives r = |z| / √n, with n the number of nonzero pairs or the total number of values, and the Hodges-Lehmann shift in the original units. Common rough bands for r are 0.1 small, 0.3 medium, 0.5 large. Some texts divide by the total number of measurements in paired designs instead, which gives a smaller r for the same data.

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