Paired Rank Test Calculator

Calculator Input

Sample A values

Enter one list using commas, spaces, or line breaks.

Sample B values

Each value must align with the matching pair above.

Label for first sample

Label for second sample

Alternative hypothesis

Alpha level

Decimal places

The test ignores zero differences, ranks the absolute non-zero differences, and evaluates whether positive and negative signed ranks balance under the null hypothesis.

Example Data Table

Pair	Before	After	Difference (Before − After)
1	82	79	3
2	75	73	2
3	91	88	3
4	68	70	-2
5	77	74	3
6	84	81	3
7	79	77	2
8	88	85	3

Use the example button above to populate the calculator instantly. This sample is useful for checking data entry, rank assignment, and output formatting.

Formula Used

This calculator applies the Wilcoxon signed-rank paired rank test.

1. Compute the paired difference for each observation:

d_i = A_i − B_i

2. Remove any pairs where d_i = 0.

3. Rank the absolute differences |d_i| from smallest to largest. Tied absolute differences receive average ranks.

4. Add ranks for positive and negative differences separately:

W+ = Σ positive ranks

W− = Σ absolute value of negative ranks

5. For a two-sided test, the test statistic is:

T = min(W+, W−)

6. When exact enumeration is unavailable, the normal approximation is used:

μ = (Σ ranks) / 2

σ = √[(Σ rank²) / 4]

z = (W+ − μ ± continuity correction) / σ

The p value is then compared with the selected alpha level to determine whether the paired distributions show a significant shift.

How to Use This Calculator

Enter the first matched dataset in the first values box.
Enter the second matched dataset in the second values box.
Make sure both lists have the same number of observations.
Choose the alternative hypothesis that matches your question.
Select the alpha level for your significance decision.
Set the decimal places you want in the output.
Click the calculate button to produce the result section.
Review the decision, p value, ranks table, and graph.
Use the CSV or PDF buttons to export the current output.

Frequently Asked Questions

1) What does this paired rank test measure?

It checks whether paired observations tend to shift in one direction. The method ranks absolute paired differences and compares positive ranks against negative ranks.

2) When should I use this instead of a paired t test?

Use it when paired differences are not normally distributed, contain outliers, or are measured on an ordinal scale. It is a robust nonparametric alternative.

3) What happens to zero differences?

Pairs with zero difference are excluded from rank assignment. They do not support either direction, so the effective sample size becomes the number of non-zero pairs.

4) What are W+ and W−?

W+ is the sum of ranks for positive paired differences. W− is the sum of ranks for negative paired differences. Their balance drives the test result.

5) Why might the exact p value be unavailable?

Exact enumeration is simplest for smaller samples without tied absolute differences. If ties appear or the sample is larger, the calculator switches to a normal approximation.

6) What does the effect size r mean?

It summarizes the strength of the paired shift. Larger values indicate a stronger change. This page labels the result as negligible, small, moderate, or large.

7) Can I paste values with spaces or line breaks?

Yes. The calculator accepts commas, spaces, semicolons, and line breaks. It converts the pasted text into numeric paired lists automatically.

8) What does the graph show?

The Plotly chart displays signed ranks for each pair. Positive bars support higher first-sample values, while negative bars support higher second-sample values.