Question 1

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

Accepted Answer

Use the Wilcoxon signed-rank test when: (1) the difference scores are clearly non-normal (highly skewed, heavy-tailed), (2) you have outliers that would distort the mean, (3) your data is ordinal (e.g., Likert scale ratings), or (4) the sample is very small (n < 15) and normality cannot be verified. If differences appear roughly normal, the paired t-test is preferred because it has slightly more statistical power.

Question 2

What is the null hypothesis of the Wilcoxon signed-rank test?

Accepted Answer

H₀: The median difference between pairs is zero — equivalently, the distribution of differences is symmetric around zero. H₁: The median difference ≠ 0 (two-tailed). The test also assumes the distribution of differences is symmetric (but not necessarily normal). It is not simply a test of the mean — it is a test about the location of the distribution.

Question 3

What is the W-statistic?

Accepted Answer

W is calculated as: (1) Compute differences d_i = x_i − y_i; (2) Exclude zeros; (3) Rank |d_i| from 1 (smallest) to n (largest), averaging tied ranks; (4) Attach the original sign; (5) W = sum of all signed ranks (equivalently, W⁺ − W⁻). Under H₀, W has mean 0 and known variance n(n+1)(2n+1)/6. Large |W| provides evidence against H₀.

Question 4

How accurate is the normal approximation?

Accepted Answer

The normal approximation z = W / √[n(n+1)(2n+1)/6] is accurate for n ≥ 10 to 20. This calculator uses a continuity correction: z = (|W| − 0.5) / σ_W, which improves accuracy. For n < 10, use exact critical values from a Wilcoxon table. The approximation is also less accurate with many ties — this calculator applies a tie correction to σ_W.

Question 5

Wilcoxon signed-rank test vs Wilcoxon rank-sum test — what is the difference?

Accepted Answer

The signed-rank test (this page) is for paired data (same subjects measured twice). The rank-sum test (Mann-Whitney U test) is for two independent groups. Both are non-parametric alternatives to t-tests but for different study designs. Use the signed-rank test for paired/repeated-measures designs, and the rank-sum test for independent group comparisons.

Question 6

What should I do with tied differences?

Accepted Answer

Tied absolute differences are assigned the average of the ranks they would occupy. Zero differences (d_i = 0) are dropped from the analysis and n is reduced accordingly. With many ties or many zeros, the test may lose power. This calculator automatically handles ties using average ranks and applies Kendall's tie correction to the standard error.

Feature	Wilcoxon Signed-Rank	Paired t-Test
Normality required?	No	Yes (or n ≥ 30)
Tests	Median difference = 0	Mean difference = 0
Sensitive to outliers?	Resistant	Sensitive
Works with ordinal data?	Yes	No
Statistical power	Slightly lower (for normal data)	Slightly higher (for normal data)
Best for	Non-normal, skewed, ordinal, outlier-prone data	Normal continuous data

Wilcoxon Signed-Rank Test Calculator

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Wilcoxon Signed-Rank vs Paired t-Test

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