Wilcoxon Signed-Rank Table

What Is a Wilcoxon Signed-Rank Table?

The Wilcoxon signed-rank table provides critical values for the Wilcoxon signed-rank test. This non-parametric test compares paired observations—like before/after measurements or matched pairs—without assuming normality.

It's the non-parametric alternative to the paired t-test.

When to Use This Test

Choose the Wilcoxon signed-rank test when:

• You have paired or matched data
• The differences are ordinal or non-normally distributed
• Outliers are present
• Sample size is small (n < 30)
• You want to test whether the median difference equals zero

How the Test Works

1. Calculate the difference for each pair: d_i = X₂ - X₁
2. Exclude any pairs with d = 0 (ties with zero)
3. Rank the absolute differences (1 = smallest)
4. Assign signs to ranks based on the sign of each difference
5. Sum the positive ranks (W⁺) and negative ranks (W⁻)
6. Test statistic: W = min(W⁺, W⁻)

How to Read the Table

1. Determine n (number of pairs after excluding zeros)
2. Select your α level and test type (one-tailed or two-tailed)
3. Find the critical W value

Decision: If your calculated W ≤ critical W, reject H₀.

Example: 15 pairs, α = 0.05 (two-tailed):

• Critical W = 25
• If calculated W = 18, reject H₀ (significant difference)

Worked Example

Data (Before vs. After treatment):

| Pair | Before | After | Diff | |Diff| | Rank | Signed Rank | |------|--------|-------|------|--------|------|-------------| | 1 | 85 | 82 | -3 | 3 | 2 | -2 | | 2 | 90 | 91 | +1 | 1 | 1 | +1 | | 3 | 78 | 72 | -6 | 6 | 4 | -4 | | 4 | 92 | 88 | -4 | 4 | 3 | -3 | | 5 | 75 | 68 | -7 | 7 | 5 | -5 |

Calculation:

• W⁺ = 1 (sum of positive ranks)
• W⁻ = 2 + 4 + 3 + 5 = 14 (sum of negative ranks)
• W = min(1, 14) = 1

From table (n = 5, α = 0.05, two-tailed): Critical W = 1

• Since W = 1 ≤ 1, we reject H₀ at the boundary

Handling Ties

Ties among differences (same |d| values):

• Assign the average rank to tied values
• Continue ranking subsequent values normally

Ties with zero (d = 0):

• Exclude these pairs from the analysis
• Reduce n accordingly

Normal Approximation (n > 25)

For larger samples, use:

$\mu_W = \frac{n(n+1)}{4}$

$\sigma_W = \sqrt{\frac{n(n+1)(2n+1)}{24}}$

$z = \frac{W - \mu_W}{\sigma_W}$

Compare z to standard normal critical values.

Effect Size

The matched-pairs rank-biserial correlation:

$r = \frac{|W^+ - W^-|}{W^+ + W^-}$

Or using z:

$r = \frac{z}{\sqrt{n}}$

Comparison with Paired t-Test

Assumptions

The Wilcoxon signed-rank test assumes:

1. Paired observations that are dependent
2. Random sampling from the population
3. At least ordinal level of measurement
4. Symmetric distribution of differences around the median

If the symmetry assumption is violated, consider the sign test (less powerful but more robust).

One-Tailed vs. Two-Tailed

• Two-tailed: Tests H₀: median difference = 0 vs. H₁: median difference ≠ 0
• One-tailed: Tests in a specific direction (median difference > 0 or < 0)

Use the appropriate column in the table for your hypothesis.

Related Tools

• Wilcoxon Signed-Rank Calculator — Run the test
• Paired t-Test Calculator — Parametric alternative
• Sign Test Calculator — More robust alternative
• Mann-Whitney U Test — For independent groups

Summary

The Wilcoxon signed-rank table enables hypothesis testing for paired data without assuming normality. By comparing your calculated W statistic (the smaller of the positive and negative rank sums) to the critical value, you can determine whether paired observations show a systematic difference. This test preserves more information than the sign test by considering the magnitude of differences, making it a powerful non-parametric alternative to the paired t-test.