Wilcoxon Signed-Rank Test Calculator: Paired Data Without Assuming Normality

A rank-based test for paired data, as an alternative to the paired t-test.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Set how many pairs you have and enter the before and after values.

Enter the significance level.

The calculator finds the differences, ranks their sizes and gives the rank sums, the exact and normal p-values and the effect size.

Formulas used

  • W^+ = \sum\text{ranks of positive differences}
  • E(W^+) = \frac{m(m + 1)}{4}
  • z = \frac{|W^+ - E| - 0.5}{\sigma_W}

Worked example

With the default numbers in the calculator above:

  1. The differences (after minus before), with zero differences left out: 2, 3, 1, 3, 1, 3, 1, 3 (8 used out of 8 pairs)
  2. Sum of the ranks of the positive differences (the ranks are of the sizes of the differences, tied sizes share the average rank): 36
  3. Sum of the ranks of the negative differences: 0 (the two add up to 36)
  4. The test statistic W (the smaller of the two): 0
  5. Exact two-tailed p-value: the share of all 256 sign patterns that are as extreme as yours: 0.0078125
  6. Normal approximation: z with continuity and tie corrections: 2.4936
  7. Two-tailed p-value by the normal approximation: 0.012644
  8. Median of the differences: 2.5
  9. Rank-biserial correlation (effect size, from -1 to 1): 1

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: After (or second measurement) for pair 2 13; After (or second measurement) for pair 7 10.

  1. The differences (after minus before), with zero differences left out: 2, -2, 1, 3, 1, 3, -3, 3 (8 used out of 8 pairs)
  2. Sum of the ranks of the positive differences (the ranks are of the sizes of the differences, tied sizes share the average rank): 26
  3. Sum of the ranks of the negative differences: 10 (the two add up to 36)
  4. The test statistic W (the smaller of the two): 10
  5. Exact two-tailed p-value: the share of all 256 sign patterns that are as extreme as yours: 0.3125
  6. Normal approximation: z with continuity and tie corrections: 1.066
  7. Two-tailed p-value by the normal approximation: 0.28642
  8. Median of the differences: 1.5
  9. Rank-biserial correlation (effect size, from -1 to 1): 0.4444

Common questions

When do I use the Wilcoxon signed-rank test?

For paired data when the differences are not normally distributed, or the sample is small.

What happens to a pair with no difference?

It is left out, because it has no sign and does not count.

How is it different from the Mann-Whitney test?

The Wilcoxon signed-rank test is for paired samples, and the Mann-Whitney test is for two independent groups.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any statistics text (non-parametric tests, the Wilcoxon signed-rank test).

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