Wilcoxon Signed Rank Test Calculator

Wilcoxon Signed-Rank Test

Sample 1 (e.g. Before / Condition A) Separate values with commas, spaces, or newlines

Please enter valid numeric data for Sample 1.

Sample 2 (e.g. After / Condition B) Must match the number of observations in Sample 1

Please enter valid numeric data for Sample 2.

Hypothesis Direction / Tail Type

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Non-Parametric Paired Test

What Is the Wilcoxon Signed-Rank Test?

The Wilcoxon Signed-Rank Test is a non-parametric statistical hypothesis test used to compare two related, matched, or paired samples (such as pre-test vs. post-test measurements).

It serves as the direct non-parametric equivalent of the Paired Samples T-Test. Unlike the paired t-test, it does not assume that difference scores are normally distributed.

When to Use It

Repeated Measures: Evaluating before-and-after treatment outcomes on the same participants.
Matched-Pair Designs: Comparing twins, case-control pairs, or subject pairs matched by baseline covariates.
Ordinal & Skewed Data: Analyzing paired survey Likert scores or heavily skewed continuous difference variables.
Non-Normal Paired Differences: When paired difference scores violate normality assumptions.

The Mathematical Formulas

$$d_i = x_{1,i} - x_{2,i}, \quad |d_i| \text{ ranked for all } d_i \neq 0$$ $$W^+ = \sum_{d_i > 0} R_i, \quad W^- = \sum_{d_i < 0} R_i, \quad W = \min(W^+, W^-)$$ $$\mu_W = \frac{N_r(N_r + 1)}{4}, \quad \sigma_W = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24} - \sum \frac{t_i^3 - t_i}{48}}$$ $$Z = \frac{W^+ - \mu_W}{\sigma_W}, \quad r = \frac{W^+ - W^-}{W^+ + W^-}$$

Where $N_r$ is the number of non-zero paired differences, $t_i$ is the number of ties in tie group $i$, $Z$ is the standard normal approximation test statistic, and $r$ is the matched-pairs rank-biserial correlation effect size.

Worked Example

Scenario: 8 patients record systolic blood pressure Before ($S_1$) and After ($S_2$) medication at $\alpha = 0.05$.

Step 1: Compute Differences ($d_i = S_1 - S_2$):
Differences: $[+7, -2, +7, +5, +5, +2, +5, +8]$. All $N_r = 8$ are non-zero.

Step 2: Rank Absolute Differences ($|d_i|$):
$|2|$ (2 ties) $\to \text{rank } 1.5$; $|5|$ (3 ties) $\to \text{rank } 4.0$; $|7|$ (2 ties) $\to \text{rank } 6.5$; $|8|$ $\to \text{rank } 8.0$.

Step 3: Sum Positive & Negative Ranks:
$W^+ = 1.5 + 4 + 4 + 4 + 6.5 + 6.5 + 8 = 34.5$
$W^- = 1.5 \implies W = \min(34.5, 1.5) = 1.5$.

Step 4: Standardize & P-Value:
$\mu_W = \frac{8 \times 9}{4} = 18.0,\quad \sigma_W = \sqrt{\frac{8(9)(17)}{24} - \frac{36}{48}} = \sqrt{50.25} \approx 7.0887$
$Z = \frac{34.5 - 18.0}{7.0887} = 2.3276 \implies p = 2 \times (1 - \Phi(2.3276)) = 0.0199$.

Conclusion: Since $p = 0.0199 \le 0.05$, reject $H_0$. Blood pressure decreased significantly after treatment.

Computation Complete

Wilcoxon Test Results

Computed Test Statistic & P-Value

p = 0.0199

Reject Null Hypothesis (Statistically Significant at α = 0.05)

Positive Ranks (W⁺) 34.5

Negative Ranks (W⁻) 1.5

Test Stat (W) 1.5

Z-Score 2.3276

Rank-Biserial (r) 0.9167

Valid Pairs (Nᵣ) 8

Wilcoxon Signed Rank Test Calculator

Wilcoxon Signed-Rank Test

Wilcoxon Test Results

Step-by-Step Calculation

Interpretation

Standard Normal Approximation & Critical Region

Perfect for students, researchers, and professionals looking to build real statistical skills.