Introduction
When you measure the same subjects under two conditions or at two time points, you have paired data. The classic approach is the paired t-test, which compares the mean of the differences to zero. But what if the differences are not normally distributed? That is where the Wilcoxon signed-rank test comes in as the nonparametric alternative.
Both tests address the same research question: Is there a systematic difference between two related measurements? Yet they differ in their assumptions, what they compare, and how sensitive they are to outliers.
This article guides you through both tests on the same dataset, shows you how to check assumptions, and provides clear decision rules. You can run either analysis with our Paired T-Test Calculator or explore other comparison tools on StatMate.
Quick Comparison Table
| Feature | Paired T-Test | Wilcoxon Signed-Rank Test | |---------------------------|-----------------------------------|--------------------------------------| | Type | Parametric | Nonparametric | | Data requirement | Continuous (interval/ratio) | At least ordinal | | Tests | Mean difference = 0 | Symmetric distribution of differences around 0 | | Assumes normality | Yes (of the differences) | No | | Sensitive to outliers | Yes | No (uses ranks) | | Power (when normal) | Higher | ~95% of paired t-test | | Power (when non-normal) | May be lower | Often higher | | Effect size | Cohen's d | Rank-biserial correlation r | | Output | t-statistic, p-value | W (or T) statistic, p-value | | Sample size needed | ~15-20+ pairs recommended | ~6+ pairs (works with small n) |
When to Use the Paired T-Test
Choose the paired t-test when:
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The differences between pairs are approximately normally distributed. This is the key assumption. It is NOT about the raw scores being normal; it is about the difference scores.
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The sample size is large enough (n > 25-30). The paired t-test is robust to non-normality with larger samples due to the central limit theorem.
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There are no severe outliers in the differences. A single extreme difference can heavily influence the mean and inflate or deflate the t-statistic.
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You care about the magnitude of the mean difference. The paired t-test gives you a direct estimate of the average change, which is often clinically or practically meaningful.
When to Use the Wilcoxon Signed-Rank Test
Choose Wilcoxon when:
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The differences are not normally distributed and the sample is small (n < 25-30). Skewed or heavy-tailed difference distributions favor Wilcoxon.
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Outliers are present in the differences and cannot be removed.
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Data are ordinal. For example, pain rated on a 0-10 scale before and after treatment.
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You have a very small sample (as few as 6 pairs), where testing normality is unreliable.
Example Dataset
A physical therapist measures grip strength (in kg) before and after a 6-week rehabilitation program for 15 patients recovering from wrist surgery.
| Patient | Before | After | Difference (After - Before) | |---------|--------|-------|-----------------------------| | 1 | 28.5 | 34.2 | 5.7 | | 2 | 32.1 | 37.8 | 5.7 | | 3 | 25.3 | 30.1 | 4.8 | | 4 | 30.7 | 36.5 | 5.8 | | 5 | 22.0 | 26.9 | 4.9 | | 6 | 35.4 | 41.2 | 5.8 | | 7 | 27.8 | 32.0 | 4.2 | | 8 | 31.5 | 38.3 | 6.8 | | 9 | 29.0 | 33.7 | 4.7 | | 10 | 26.2 | 31.8 | 5.6 | | 11 | 33.9 | 39.5 | 5.6 | | 12 | 24.7 | 29.4 | 4.7 | | 13 | 28.3 | 34.9 | 6.6 | | 14 | 30.0 | 35.1 | 5.1 | | 15 | 27.1 | 32.6 | 5.5 |
Descriptive Statistics
| Measure | Value | |-------------------------|-------| | Mean difference | 5.43 | | SD of differences | 0.72 | | Median difference | 5.60 | | Minimum difference | 4.20 | | Maximum difference | 6.80 | | n (pairs) | 15 |
All differences are positive, indicating every patient improved. The mean improvement is 5.43 kg.
Step 1: Check Assumptions
Normality of the Differences
Apply the Shapiro-Wilk test to the 15 difference scores:
| Test | Statistic | p-value | |--------------|-----------|---------| | Shapiro-Wilk | 0.957 | 0.643 |
The p-value is 0.643 (well above 0.05), so we do not reject normality. The differences appear normally distributed.
Check for Outliers
Examine the differences for outliers using the 1.5 x IQR rule:
| Statistic | Value | |----------|-------| | Q1 | 4.80 | | Q3 | 5.75 | | IQR | 0.95 | | Lower fence | 4.80 - 1.425 = 3.375 | | Upper fence | 5.75 + 1.425 = 7.175 |
All differences fall between 3.375 and 7.175. No outliers are present.
Conclusion: Both assumptions for the paired t-test are met. In practice, this means the paired t-test is the more powerful and appropriate choice. We will run both tests for comparison.
Running the Paired T-Test
Hypotheses
- H0: The mean difference in grip strength (After - Before) equals 0 (mu_d = 0).
- H1: The mean difference in grip strength does not equal 0 (mu_d != 0).
Calculation
t = (Mean difference - 0) / (SD / sqrt(n))
t = 5.43 / (0.72 / sqrt(15))
t = 5.43 / (0.72 / 3.873)
t = 5.43 / 0.186
t = 29.19
Degrees of freedom: df = n - 1 = 14
Results
| Statistic | Value | |------------------|----------| | t | 29.19 | | df | 14 | | p-value (two-tailed) | < 0.001 | | Mean difference | 5.43 kg | | 95% CI | [5.03, 5.83] | | Cohen's d | 7.54 |
Interpretation: The rehabilitation program produced a statistically significant increase in grip strength, t(14) = 29.19, p < .001. On average, grip strength increased by 5.43 kg (95% CI [5.03, 5.83]). Cohen's d = 7.54 indicates an extremely large effect.
Running the Wilcoxon Signed-Rank Test
Procedure
- Calculate the differences (already done).
- Remove any zero differences (none in this case).
- Rank the absolute differences from smallest to largest.
- Assign each rank the sign of the original difference.
- Sum the positive ranks (W+) and negative ranks (W-).
Ranking
| Patient | Difference | |Difference| | Rank | Signed Rank | |---------|------------|-------------|------|-------------| | 7 | 4.2 | 4.2 | 1 | +1 | | 9 | 4.7 | 4.7 | 2.5 | +2.5 | | 12 | 4.7 | 4.7 | 2.5 | +2.5 | | 3 | 4.8 | 4.8 | 4 | +4 | | 5 | 4.9 | 4.9 | 5 | +5 | | 14 | 5.1 | 5.1 | 6 | +6 | | 15 | 5.5 | 5.5 | 7 | +7 | | 10 | 5.6 | 5.6 | 8.5 | +8.5 | | 11 | 5.6 | 5.6 | 8.5 | +8.5 | | 1 | 5.7 | 5.7 | 10.5 | +10.5 | | 2 | 5.7 | 5.7 | 10.5 | +10.5 | | 4 | 5.8 | 5.8 | 12.5 | +12.5 | | 6 | 5.8 | 5.8 | 12.5 | +12.5 | | 13 | 6.6 | 6.6 | 14 | +14 | | 8 | 6.8 | 6.8 | 15 | +15 |
Test Results
| Statistic | Value | |-----------|----------| | W+ (positive ranks) | 120.0 | | W- (negative ranks) | 0.0 | | W (test statistic) | 120.0 | | p-value (two-tailed)| < 0.001 | | Rank-biserial r | 1.000 |
Interpretation: The Wilcoxon signed-rank test confirmed a statistically significant increase in grip strength after rehabilitation, W = 120.0, p < .001. The rank-biserial correlation of 1.000 indicates that every patient improved (all ranks are positive).
Side-by-Side Comparison
| Aspect | Paired T-Test | Wilcoxon Signed-Rank | |--------------------|-------------------------|--------------------------| | Test statistic | t(14) = 29.19 | W = 120.0 | | p-value | < 0.001 | < 0.001 | | Effect size | Cohen's d = 7.54 | r = 1.000 | | Central tendency | Mean diff = 5.43 kg | Median diff = 5.60 kg | | Confidence interval| [5.03, 5.83] | Not standard (bootstrap) | | Conclusion | Significant improvement | Significant improvement |
Both tests agree strongly. The paired t-test provides a more informative result with a confidence interval for the mean difference, while Wilcoxon confirms the finding without requiring normality.
Scenario Where Tests Diverge
To illustrate when the choice matters, consider an alternative dataset with an outlier. Suppose patient 8's difference was 25.0 instead of 6.8 (perhaps a data entry error or a genuine extreme responder):
| Statistic | Paired T-Test (with outlier) | Wilcoxon (with outlier) | |-----------------------|-----------------------------|--------------------------| | Mean difference | 6.65 | (not affected by magnitude) | | t-statistic | 5.30 | W = 120.0 | | p-value | < 0.001 | < 0.001 | | Without outlier: t | 29.19 | W = 105.0 |
The paired t-test's t-statistic drops dramatically from 29.19 to 5.30 because the outlier inflates the standard deviation. The Wilcoxon test is barely affected because it uses ranks. In a borderline case, this could change the t-test's significance while Wilcoxon remains robust.
Decision Framework
Ask these questions in order:
1. Are the data at least ordinal and paired?
- No: These tests do not apply.
- Yes: Continue.
2. Are the difference scores approximately normally distributed?
Check with the Shapiro-Wilk test and a histogram of the differences.
- Yes and n >= 15: Use the paired t-test for maximum power.
- Borderline (n > 30): Use the paired t-test (robust due to CLT).
- No (skewed, outliers, or n < 15): Use the Wilcoxon signed-rank test.
3. Are there influential outliers in the differences?
- Yes: Use Wilcoxon (or remove the outlier with justification and use the paired t-test).
- No: Stick with your choice from step 2.
4. Is the data ordinal (not continuous)?
- Yes: Use Wilcoxon regardless of normality.
- No: Follow the decision from steps 2-3.
Effect Size Interpretation
Cohen's d (Paired T-Test)
| d Value | Interpretation | |---------|---------------| | 0.20 | Small | | 0.50 | Medium | | 0.80 | Large |
Cohen's d is calculated as the mean difference divided by the standard deviation of the differences.
Rank-Biserial Correlation r (Wilcoxon)
| r Value | Interpretation | |---------|---------------| | 0.10 | Small | | 0.30 | Medium | | 0.50 | Large |
The rank-biserial r is calculated as (W+ - W-) / (W+ + W-) or equivalently from the W statistic divided by the total number of ranks.
Reporting Examples
Paired T-Test Report
A paired-samples t-test was conducted to evaluate the impact of a 6-week rehabilitation program on grip strength. There was a statistically significant increase from pre-treatment (M = 28.83, SD = 3.58) to post-treatment (M = 34.27, SD = 3.89), t(14) = 29.19, p < .001 (two-tailed). The mean increase was 5.43 kg (95% CI [5.03, 5.83]), with a large effect size (d = 7.54).
Wilcoxon Signed-Rank Report
A Wilcoxon signed-rank test determined that grip strength was statistically significantly higher post-treatment (Mdn = 34.20) than pre-treatment (Mdn = 28.50), W = 120.0, p < .001. The rank-biserial correlation was 1.00, indicating that all 15 participants showed improvement.
Try It Yourself
Analyze your paired data with our interactive tools:
- Paired T-Test Calculator for parametric paired comparisons
- T-Test Calculator for both independent and paired samples
- Normality Test Calculator to check your difference scores
FAQ
How many pairs do I need for the Wilcoxon signed-rank test?
The Wilcoxon test can be used with as few as 6 pairs, though power will be low. For reasonable power to detect a medium effect, aim for at least 15-20 pairs. With fewer than 6 pairs, exact p-values should be used rather than the normal approximation.
Does the paired t-test require both variables to be normal?
No. The paired t-test requires that the differences (variable 1 minus variable 2) are normally distributed. The individual variables do not need to be normal. For example, if both variables are right-skewed but the differences are symmetric, the paired t-test is appropriate.
Can I use the Wilcoxon test if the differences are symmetric but not normal?
Yes. In fact, the Wilcoxon test assumes that the difference distribution is symmetric around the median under the null hypothesis. If the differences are symmetric but have heavy tails (e.g., a uniform or Laplace distribution), Wilcoxon is a good choice because it is robust to heavy tails.
What if I have ties in the data?
Ties (identical values in the differences) are handled by assigning average ranks. Zero differences are excluded before ranking. Most software handles ties automatically. With many ties (common in ordinal data), exact p-values or permutation-based p-values are preferred over the normal approximation.
Can I do a one-tailed test with either method?
Yes. For a one-tailed paired t-test, divide the two-tailed p-value by 2 (if the observed direction matches your hypothesis). For a one-tailed Wilcoxon test, most software offers this option directly. Use one-tailed tests only when you have a strong a priori directional hypothesis stated before data collection.
What is the sign test, and how does it relate to Wilcoxon?
The sign test is an even simpler nonparametric alternative that only considers the direction (positive or negative) of each difference, ignoring magnitude. Wilcoxon signed-rank test uses both direction and magnitude (via ranks), making it more powerful than the sign test. Use the sign test only when you cannot assume the differences are symmetric.
How do I handle missing data in paired designs?
If one measurement in a pair is missing, both measurements are excluded from both tests. This is an inherent limitation of paired designs. If missingness is substantial, consider using a mixed-effects model that can incorporate all available data points rather than deleting incomplete pairs.