When should I use the Wilcoxon signed-rank test instead of a paired t-test?

Use it when your differences are not normally distributed, your sample is small (n < 30), or the data are ordinal. The t-test is more powerful when normality holds; the Wilcoxon test is safer when it does not.

What does the W statistic represent?

W is the sum of the ranks associated with the smaller-signed group (positive or negative differences). Under the null hypothesis of no effect, W+ and W− are expected to be roughly equal.

How are ties handled?

Pairs with zero difference are excluded from the analysis. Pairs with equal absolute differences receive the average of the ranks they would have occupied.

Is the normal approximation accurate for small samples?

For n ≤ 10 the Z approximation can be imprecise. The tool flags this and the p-value should be treated as approximate. For very small samples, consult exact critical-value tables.

What is a one-sample Wilcoxon test?

When you have a single column of data, you can test it against a hypothesized median M0. Internally the tool computes differences as (observation − M0) and then runs the standard signed-rank procedure.

What does a p-value of 0.03 mean?

It means there is a 3% probability of observing a W statistic as extreme as yours (or more extreme) if the null hypothesis (median difference = 0) were true. At alpha = 0.05 you would reject the null.

Wilcoxon Signed-Rank Test Calculator

Compute the Wilcoxon signed-rank W statistic and p-value for paired data or a one-sample test against a hypothesized median.

Enter Data

Test Type

Before / Group 1

After / Group 2

Significance Level (α)

Enter your data and click Run Wilcoxon Test to see results.

n (used)

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W statistic

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Z score

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p-value (2-tail)

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Signed-Rank Details

Pairs dropped (d=0) —

Tie groups —

W+ (positive ranks) —

W− (negative ranks) —

Expected W (H0) —

Variance of W —

Ranked Differences Table

#	Difference (d)	\|d\|	Rank	Signed Rank

Summary

The Wilcoxon signed-rank test is a non-parametric alternative to the paired t-test. It tests whether the median difference between paired observations (or between a single sample and a hypothesized median) is zero, without assuming the data follow a normal distribution. Enter two paired columns or a single column with a target median, and the calculator returns the W statistic, Z approximation, and two-tailed p-value.

How it works

Enter paired sample values in the two columns (Before / After), or enter one column and set a hypothesized median for a one-sample test.
Differences between each pair are computed; zero differences are dropped per standard procedure.
Absolute differences are ranked from smallest to largest, with ties receiving average ranks.
Positive and negative signed ranks are summed separately to produce W+ and W−.
The smaller of W+ and W− is the test statistic W. For n > 10 a normal approximation with continuity correction gives a Z score and two-tailed p-value.
Interpret: if p < your chosen alpha (commonly 0.05), reject the null that the median difference is zero.

Use cases

Comparing before-and-after measurements (weight, blood pressure, test scores) without assuming normality.
Analyzing small paired samples where the t-test assumptions are questionable.
Testing whether a sample median differs from a known or hypothesized value.
Non-parametric alternative when paired-t-test residuals are skewed or contain outliers.
Clinical trials comparing matched patient pairs on an ordinal outcome.
Quality-control studies assessing process improvement across matched units.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu