How to use the wilcoxon signed-rank test calculator
- Choose whether you will paste pairs (two values per line, such as before and after) or differences (one number per line).
- Paste the data. Each difference is the first value minus the second value. Pairs where the two values are equal give a zero difference and are left out of the test.
- Choose the alternative: the two sets differ, the first values tend to be greater, or the first values tend to be smaller. Open the advanced option if you want a continuity correction for the normal approximation.
- Read the p value, the sums of the positive and negative ranks, and the table of ranks. The page states whether the p value is exact or an approximation.
Formula
rank the absolute differences, giving ties the average of their ranks
T+ = sum of ranks of positive differences, T- = sum of ranks of negative differences
T+ + T- = n(n + 1) / 2
normal approximation: z = (T+ - n(n + 1)/4) / sqrt(n(n + 1)(2n + 1)/24 - sum(t^3 - t)/48)
matched-pairs rank-biserial correlation = (T+ - T-) / (n(n + 1) / 2)
n is the number of nonzero differences and t is the size of each group of tied absolute differences. With no ties and n up to 50 the p value is exact, from counting all sign patterns.
Twelve pairs with no ties
Twelve pairs give the differences -0.8, 1.1, 2.3, -0.4, 1.9, 0.7, 3.2, -1.5, 2.2, 0.9, 1.4 and -0.3. The ranks of the positive differences add up to T+ = 63 and those of the negative differences to T- = 15, out of a total of 78. The two-sided exact p value is 0.0640, the one-sided value for "the first values are greater" is 0.0320, and the value for "smaller" is 0.9739.
With ties and a zero, the exact method no longer applies. Ten differences of 15, -7, 5, 20, 0, -9, 17, -12, 5 and -10 leave nine nonzero values with a tie at 5. T+ is 27, T- is 18, and the tie-corrected normal approximation gives z = 0.534 and a two-sided p value of 0.5936. Adding the continuity correction changes it to 0.6353.
When to use a rank test
The signed-rank test looks only at the order of the sizes of the differences and at their signs, so a single wild value cannot dominate it. That makes it a common choice for small paired samples, scores on an arbitrary scale and data with outliers. It does assume that the differences are roughly symmetric around their center.
If the differences are close to a bell curve, a paired t test uses more of the information and has somewhat higher power. For two separate groups, use the Mann-Whitney U test calculator instead. To plan how many pairs you need, try the statistical power calculator, and to turn a statistic into a p value see the p value calculator.
Ties and zeros
A zero difference carries no sign, so the standard method drops it and reduces n. Tied absolute differences share the average of their ranks. Ties make the exact count of sign patterns invalid, so the page switches to the normal approximation with a variance that is corrected for the ties.
Assumptions and limits
- The pairs are independent of each other, and each pair is a measurement on the same unit.
- The differences are roughly symmetric around their center. The test does not need a bell curve.
- Rounding is applied to differences at nine decimals so that equal differences really tie.
- Up to 2,000 pairs are accepted. The exact method is used for up to 50 nonzero differences with no ties.
Frequently asked questions
What is the test statistic W?
For a two-sided test W is the smaller of T+ and T-. For a one-sided test the page shows T+, the sum of ranks of the positive differences. Both statistics carry the same information because they add up to a fixed total.
Why is my p value different from another program?
Programs differ in how they treat zeros and ties, whether they use a continuity correction, and when they switch from the exact to the approximate method. This page drops zeros, corrects for ties and leaves the continuity correction off unless you turn it on.
What does the rank-biserial value tell me?
It is an effect size between -1 and 1. It is the share of the rank total that favors positive differences minus the share that favors negative ones. A value near 0 means the two sides balance.
Can I paste data with commas inside numbers?
Use a point for decimals and separate the two values of a pair with a comma, a tab or spaces. Thousands separators are not accepted because they would be read as the end of a value.