Mann-Whitney U / Wilcoxon Rank-Sum Non-Parametric Z-Test & Rank-Biserial r Calculator

Compute Mann-Whitney U, Z-statistic, Common Language Effect Size (CLES), and Rank-Biserial r.

Mann-Whitney U Statistic
65 U (Z = -1.97)
Normal Approximation Z-Score -1.97 Z
Rank-Biserial Effect Size (r_rb) 0.422 r_rb
Common Language Effect Size (CLES) 71.111 %
Inputs
⚡ Scenarios:
n₁
Sample size of Group 1 (n₁).
n₂
Sample size of Group 2 (n₂).
R₁
Observed sum of pooled ranks for Group 1 (R₁).
Results & Graph
Mann-Whitney U Statistic
65 U (Z = -1.97)
Normal Approximation Z-Score
-1.97 Z
Rank-Biserial Effect Size (r_rb)
0.422 r_rb
Common Language Effect Size (CLES)
71.111 %

Mann-Whitney U Statistic

Graph 1 • Mann-Whitney U Test • Domain Architecture & Live Telemetry Maps Sample Size Group1 N1, Sample Size Group2 N2, Rank Sum Group1 R1 through Mann-Whitney U Test Formula.
Mann-Whitney U Statistic: 65 U (Z = -1.97) Mann-Whitney U Test • Invariants & Metrics Mann-Whitney U Statistic 65 U (Z = -1.97) Normal Approximation Z-Score -1.97 Z Rank-Biserial Effect Size (r 0.422 r_rb Common Language Effect Size 71.111 %
Graph 2 • Mann-Whitney U Statistic Across Sample Size Group1 N1 Sweep (40% → 160%) Evaluates U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁; μ_U = n₁n₂ / 2; σ_U = √[ n₁n₂(n₁ + n₂ + across 7 operating points (current input highlighted).
-169 U (Z = -24.55 U (Z = 119.9 U (Z = -169 6n₁ -100 9n₁ -22 12n₁ 65 15n₁ 109 18n₁ 49 21n₁ 0 24n₁
Graph 3 • Joint Sensitivity: Sample Size Group1 N1 × Sample Size Group2 N2 Compares Mann-Whitney U Statistic when Sample Size Group2 N2 is shifted +25% (solid) vs. −25% (dashed).
-93 U (Z = 23.35 U (Z = 139.7 U (Z = --- Sample Size Group2 N2 @ 75% -5 -30% 58 -15% 125 +0% 127 +15% 70 +30%
Summary: With n₁=15, n₂=15, and R₁=280, Mann-Whitney U = 65 (expected μ_U = 112.5, σ_U = 24.11), normal Z = -1.97, and Rank-Biserial r = 0.422.
Summary: With n₁=15, n₂=15, and R₁=280, Mann-Whitney U = 65 (expected μ_U = 112.5, σ_U = 24.11), normal Z = -1.97, and Rank-Biserial r = 0.422.
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Formula & Mathematical Methodology: Mann-Whitney U Test Formula

U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
μU = n₁n₂ / 2
σU = n₁n₂(n₁ + n₂ + 1) / 12
Z = (U₁ − μU) / σU
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁; μ_U = n₁n₂ / 2; σ_U = √[ n₁n₂(n₁ + n₂ + 1) / 12 ]; Z = (U₁ − μ_U) / σ_U
  • sampleSizeGroup1N1: Sample Size Group1 N1 (n₁)
  • sampleSizeGroup2N2: Sample Size Group2 N2 (n₂)
  • rankSumGroup1R1: Rank Sum Group1 R1 (R₁)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Mann-Whitney U Test

Compute Mann-Whitney U, Z-statistic, Common Language Effect Size (CLES), and Rank-Biserial r.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁; μ_U = n₁n₂ / 2; σ_U = √[ n₁n₂(n₁ + n₂ + 1) / 12 ]; Z = (U₁ − μ_U) / σ_U.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Mann-Whitney U Test via U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁; μ_U = n₁n₂ / 2; σ_U = √[ n₁n₂(n₁ + n₂ + 1) / 12 ]; Z = (U₁ − μ_U) / σ_U.

Frequently Asked Questions

How does the Mann-Whitney U / Wilcoxon Rank-Sum Non-Parametric Z-Test & Rank-Biserial r Calculator compute the solution?

It evaluates U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁; μ_U = n₁n₂ / 2; σ_U = √[ n₁n₂(n₁ + n₂ + 1) / 12 ]; Z = (U₁ − μ_U) / σ_U directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Non-Parametric & Hypothesis Testing

Mann-Whitney U / Wilcoxon Rank-Sum Non-Parametric Z-Test & Rank-Biserial r Calculator: inputs, method, and how to read the result

Compute Mann-Whitney U, Z-statistic, Common Language Effect Size (CLES), and Rank-Biserial r.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Sample Size Group1 N1: Sample size of Group 1 (n₁).
  • Sample Size Group2 N2: Sample size of Group 2 (n₂).
  • Rank Sum Group1 R1: Observed sum of pooled ranks for Group 1 (R₁).

What the calculation includes

With the default values shown, Mann-Whitney U Statistic is 65 U (Z = -1.97). The calculation follows Mann-Whitney U Test Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Sample Size Group1 N1 from 15 n₁ to 17 n₁ changes Mann-Whitney U Statistic from 65 U (Z = -1.97) to 127 U (Z = 0.02). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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