Written by Dackohn; last reviewed November 2026.
The sign test uses only the direction of each paired difference. The Wilcoxon signed-rank test also uses the size of the differences. Wilcoxon is usually more powerful but assumes the differences are roughly symmetric. The sign test assumes almost nothing, so it suits skewed differences and direction-only data.
Worked example
Ten people’s sleep scores change by +3, −1, +5, +6, −2, +8, +4, +9, +7, +10.
Sign test. Eight rise and two fall. If rises and falls were equally likely, the two-sided probability of a split at least this lopsided is 2 × P(X ≤ 2), with X following Binomial(10, 0.5): 2 × 0.0547 = p = .109. Not significant.
Wilcoxon signed-rank test. Rank the absolute changes. The two falls are the two smallest, so they get ranks 1 and 2, and T = 3. The exact p = .010. Significant.
The sign test treats a +10 and a +1 identically. Wilcoxon notices that the falls were tiny and the rises large. That is extra information, and it comes with an extra assumption.
Comparing the two
| Sign test | Wilcoxon signed-rank | |
|---|---|---|
| Uses | Direction only | Direction and rank of size |
| Assumes | Independent pairs | Independent pairs and roughly symmetric differences |
| Needs | Only better or worse | Differences that can be ordered |
| Power | Lower, usually | Higher, usually |
| Tests | Median difference = 0 | Symmetric around 0 |
Which should I choose?
- Use Wilcoxon when you have numeric differences that look roughly symmetric. This is the usual choice.
- Use the sign test when differences are strongly skewed, when you only have directions (“preferred A”), or when the sizes cannot be trusted.
- Use a paired t-test when differences are roughly normal.
Choose before you look at the results. Trying each test and reporting the best is a form of p-hacking.
Small samples
With very few pairs, the sign test can never reach significance. Six out of 6 in the same direction gives p = .031, but 5 out of 5 gives p = .063. And 9 out of 10 gives p = .021. Wilcoxon has a similar limit: with 5 pairs, its smallest two-sided p is also .063.
Handling zeros
Both tests usually drop pairs with a difference of exactly zero and reduce n. If many pairs are tied, report how many, since it says something about the result.
Try it yourself
Sign Test Calculator →Exact p-values from differences or counts. Wilcoxon Signed-Rank Test →Exact p-values and an effect size.Frequently asked questions
Is the sign test the same as the Wilcoxon test?
No. The sign test uses only directions, while Wilcoxon also uses the ranked sizes of the differences.
Which is more powerful?
Usually Wilcoxon, because it uses more information, as long as its symmetry assumption holds reasonably well.
When is the sign test better?
When the differences are strongly skewed or heavy-tailed, or when only directions are available.
Related reading: Wilcoxon signed-rank table · Paired vs independent samples · Parametric vs nonparametric tests
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.