Mann Whitney U Test Calculator

Mann-Whitney U Test

Sample 1 Data (Group A) Separate values with commas, spaces, or newlines

Please enter at least one valid numeric value for Sample 1.

Sample 2 Data (Group B) Separate values with commas, spaces, or newlines

Please enter at least one valid numeric value 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 Analysis

What Is the Mann-Whitney U Test?

The Mann-Whitney U Test (also known as the Wilcoxon Rank-Sum Test) is a non-parametric hypothesis test used to determine whether two independent samples originate from populations with the same distribution. It compares the median ranks rather than the means.

It serves as the non-parametric equivalent of the Two-Sample Independent T-Test, making it ideal for ordinal data or continuous datasets that violate normality and homoscedasticity assumptions.

When to Use It

Non-Normal Data: Evaluating differences between two independent groups with skewed distributions.
Ordinal Variables: Analyzing Likert-scale responses, customer ratings, or competition rankings.
Small Sample Sizes: Where sample size is too modest to reliably satisfy the Central Limit Theorem.
Alternative to Independent T-Test: When the assumptions of normality and equal variances are violated.

The Mathematical Formulas

$$U_1 = n_1 n_2 + \frac{n_1(n_1 + 1)}{2} - R_1$$ $$U_2 = n_1 n_2 + \frac{n_2(n_2 + 1)}{2} - R_2$$ $$U = \min(U_1, U_2), \quad \mu_U = \frac{n_1 n_2}{2}$$ $$\sigma_U = \sqrt{\frac{n_1 n_2}{12}\left( (n_1 + n_2 + 1) - \sum \frac{t_i^3 - t_i}{(n_1 + n_2)(n_1 + n_2 - 1)} \right)}$$ $$Z = \frac{U_1 - \mu_U}{\sigma_U}, \quad r = 1 - \frac{2U_1}{n_1 n_2}$$

Where $R_1$ and $R_2$ represent the sum of ranks for Sample 1 and Sample 2 respectively, $t_i$ represents the number of tied observations in tie group $i$, and $r$ denotes the rank-biserial correlation effect size.

Worked Example

Scenario: Group A has scores $[12, 15, 18, 20, 22]$ ($n_1 = 5$), and Group B has scores $[8, 10, 11, 14, 16]$ ($n_2 = 5$) at $\alpha = 0.05$.

Step 1: Combine & Rank All 10 Values:
$8_{(B)}\to 1,\; 10_{(B)}\to 2,\; 11_{(B)}\to 3,\; 12_{(A)}\to 4,\; 14_{(B)}\to 5,\; 15_{(A)}\to 6,\; 16_{(B)}\to 7,\; 18_{(A)}\to 8,\; 20_{(A)}\to 9,\; 22_{(A)}\to 10$.

Step 2: Calculate Rank Sums:
$R_1 = 4 + 6 + 8 + 9 + 10 = 37$
$R_2 = 1 + 2 + 3 + 5 + 7 = 18$

Step 3: Calculate U Statistics:
$U_1 = 5(5) + \frac{5(6)}{2} - 37 = 25 + 15 - 37 = 3$
$U_2 = 25 + 15 - 18 = 22 \implies U = \min(3, 22) = 3$

Step 4: Standardize & Evaluate P-Value:
$\mu_U = \frac{25}{2} = 12.5,\quad \sigma_U = \sqrt{\frac{25 \times 11}{12}} \approx 4.7871$
$Z = \frac{3 - 12.5}{4.7871} = -1.9845 \implies p = 2 \times \Phi(-1.9845) = 0.0472$

Conclusion: Since $p = 0.0472 \le 0.05$, reject $H_0$. Group A scored significantly higher than Group B.

Computation Complete

Mann-Whitney U Results

Computed Test Statistic & P-Value

p = 0.0472

Reject Null Hypothesis (Statistically Significant at α = 0.05)

Test Stat (U) 3

Rank Sum (R₁) 37

Rank Sum (R₂) 18

Z-Score -1.9845

Effect Size (r) 0.7600

Sample Sizes n₁=5, n₂=5

Mann Whitney U Test Calculator

Mann-Whitney U Test

Mann-Whitney U Results

Step-by-Step Calculation

Interpretation

Standard Normal Approximation & Critical Region

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