Wilcoxon Signed-Rank Test Calculator
Calculate non-parametric paired Wilcoxon Signed-Rank statistic ($W = \min(W^+, W^-)$), rank sums, $Z$-score approximation, and $p$-value.
Calculate Paired Wilcoxon Signed-Rank Test
Enter paired before-after or condition values for Group 1 and Group 2.
Calculation Results
Calculated using Wilcoxon signed-rank equations: W = \min(W^+, W^-), \quad \mu_W = \frac{n(n+1)}{4}, \quad \sigma_W = \sqrt{\frac{n(n+1)(2n+1)}{24}}
Quick Summary
The Wilcoxon Signed-Rank Test Calculator evaluates non-parametric paired differences ($W$), positive rank sum ($W^+$), negative rank sum ($W^-$), $Z$-approximation, and $p$-values.
Formula Explanation
W = \min(W^+, W^-)
Z = \frac{W - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n+1)(2n+1)}{24}}}How It Works
The Wilcoxon Signed-Rank test compares paired observations (Before vs After) without assuming normal difference scores. The Wilcoxon Signed-Rank Test Calculator computes absolute difference ranks, summing positive ($W^+$) and negative ($W^-$) ranks.
Step-by-Step Worked Example
Practical Problem: Paired observations ($n = 5$): Before = [120, 135, 128, 140, 132], After = [115, 125, 126, 130, 131]. Test for significant reduction.
- Step 1: Calculate paired differences ($d = \text{Before} - \text{After}$): $d = [+5, +10, +2, +10, +1]$. Zero differences are removed.
- Step 2: Rank absolute differences ($|d|$): $1(1), 2(2), 5(3), 10(4.5), 10(4.5)$.
- Step 3: Assign signs to ranks: All differences are positive ($d > 0$).
- Step 4: Calculate W+ and W-: $W^+ = 1 + 2 + 3 + 4.5 + 4.5 = \mathbf{15.0}$; $W^- = \mathbf{0.0}$.
- Step 5: Compute W stat & p-value ($p = 0.0431$): $W = \min(15, 0) = \mathbf{0.0}$; reject $H_0$ (significant reduction).
Real-World Calculation Examples
Scenario 1: Blood Pressure Trial (n = 5 pairs)
Parameters: W+ = 15.0, W- = 0.0
Result: W = 0.0, p = 0.0431 (Significant BP reduction).
Scenario 2: Pre-Test vs Post-Test Training Scores
Parameters: Skewed employee test score differences
Result: Wilcoxon signed-rank test.
Scenario 3: Product Usability Ratings Before/After Redesign
Parameters: Ordinal 1-7 Likert rating scale
Result: Non-parametric paired rank evaluation.
Scenario 4: Environmental Air Quality (Morning vs Evening)
Parameters: Matched daily air pollution readings
Result: Wilcoxon test output.
Key Benefits of Using This Calculator
Non-Parametric Paired Testing
Valid for non-normal paired differences and ordinal repeated measures.
Separate W+ and W- Outputs
Displays positive rank sum ($W^+$) and negative rank sum ($W^-$).
Automatic Zero Difference Exclusion
Automatically handles and excludes zero differences ($d_i = 0$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Wilcoxon Signed-Rank test?
The Wilcoxon Signed-Rank test is a non-parametric test comparing two related/paired samples to assess median differences.
When should I use Wilcoxon Signed-Rank instead of Paired t-test?
Use Wilcoxon Signed-Rank when paired difference scores are non-normally distributed or ordinal.
How do I calculate Wilcoxon Signed-Rank test in Excel?
Calculate differences $d_i$, rank $|d_i|$ using RANK.AVG, and sum positive and negative ranks.
What is null hypothesis ($H_0$)?
$H_0$: The median difference between paired observations is zero.
What happens to zero differences ($d_i = 0$)?
Zero differences are discarded, and sample size $n$ is reduced accordingly.
What is sum of all ranks formula?
$W^+ + W^- = \frac{n(n + 1)}{2}$.
How are ties handled in absolute differences?
Tied absolute differences receive average ranks.
Can Wilcoxon Signed-Rank test be used for a single sample?
Yes — it can test if a single sample median differs from a benchmark median $\mu_0$.
What is Sign Test vs Wilcoxon Signed-Rank Test?
The Sign Test considers ONLY difference signs (+/-); Wilcoxon Signed-Rank also incorporates MAGNITUDE ranks, making it more powerful.
Who developed the Wilcoxon Signed-Rank test?
Frank Wilcoxon proposed the test in 1945.