Advanced Wilcoxon Signed Rank Calculator

Calculated Result

Calculator Inputs

Before Values

Enter comma, space, tab, or newline separated paired values.

After Values

The number of entries must match the first sample.

Alternative Hypothesis

Significance Level (α)

Continuity Correction

Zero Difference Handling

Quick Example Loader

Large screens use three columns, medium screens use two, and mobile screens use one.

Example Data Table

Pair	Before	After	Difference
1	10	12	2
2	12	15	3
3	15	16	1
4	14	15	1
5	18	19	1
6	20	23	3
7	19	18	-1
8	17	20	3

You can paste this example into the calculator to verify the workflow and compare the output against ranked paired differences.

Formula Used

The Wilcoxon signed rank test evaluates whether the median paired difference is zero. For each pair, compute the difference d_i = after_i − before_i.

Ignore zero differences when using the classic method. Rank the absolute nonzero differences from smallest to largest. When ties occur, assign the average rank.

Then compute:

W⁺ = sum of positive ranks
W⁻ = sum of negative ranks
Test statistic W = min(W⁺, W⁻) for a two-sided test

For the normal approximation, this page calculates:

μ_W = n(n + 1) / 4

σ_W = √[n(n + 1)(2n + 1) / 24]

z = (W⁺ − μ_W − c) / σ_W

Here, c is the optional continuity correction. The calculator also reports an effect size using r = |z| / √n.

How to Use This Calculator

Enter the first paired dataset in the Before Values box.
Enter the second paired dataset in the After Values box.
Choose the alternative hypothesis matching your test question.
Set the significance level and continuity correction preference.
Select the zero handling method if your paired differences contain zeros.
Press the calculate button to view the summary, detail table, and graph.
Use the CSV or PDF buttons to export the current results.

Frequently Asked Questions

1. When should I use the Wilcoxon signed rank test?

Use it for paired measurements when differences are not safely treated as normally distributed. It is common for before-and-after studies, repeated measures, and matched observational data.

2. What is the main assumption of this test?

The paired differences should come from a distribution that is roughly symmetric around the median. The observations must also be meaningfully paired and measured on at least an ordinal scale.

3. Why are zero differences handled separately?

A zero difference has no sign and contributes no directional rank. Many implementations remove zeros, while some retain them in sample accounting through the Pratt approach.

4. What does a small p value mean here?

A small p value suggests the paired differences are unlikely under the null hypothesis of zero median difference. It provides evidence of a systematic shift between paired values.

5. What is the difference between W+, W−, and W?

W+ sums positive ranks, W− sums negative ranks, and W is often the smaller of the two for two-sided testing. Together they show direction and magnitude of ranked change.

6. Does this calculator use exact or approximate p values?

This implementation reports the normal approximation with optional continuity correction. That makes it practical for general use, especially when sample size is moderate or larger.

7. What does the reported effect size r indicate?

The effect size summarizes how strong the paired shift appears relative to sample size. Larger values suggest a stronger practical difference, not merely statistical significance.

8. Can I use decimals or negative values?

Yes. The calculator accepts integers, decimals, and negative values. It ranks absolute paired differences, so the original sign only determines whether each rank contributes positively or negatively.