Mann-Whitney U Test

The Mann-Whitney U Test Calculator helps you perform non-parametric analysis to compare two independent groups. It determines whether samples originate from the same distribution by analyzing the ranks of the data rather than the raw values. This makes it particularly useful when your data violates the assumptions of the independent t-test, such as normality or equal variances. Common applications include comparing treatment outcomes between two groups, analyzing differences in measurements, or evaluating performance metrics across two departments. Click here to populate the sample data for a quick example.

Calculator

1. Load Your Data

2. Select Columns & Options

Select categorical column (Two groups):

Select numeric column:

Adjust for Ties:

Continuity Correction:

Yates' continuity correction improves accuracy for discrete data. Learn more

Significance Level:

Alternative Hypothesis:

Learn More

Mann-Whitney U Test

Definition

Mann-Whitney U Test (also known as Wilcoxon rank-sum test) is a non-parametric alternative to the independent t-test. It compares two independent groups by analyzing the rankings of the data rather than the raw values.

Formula

U Statistics:

U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1

U_2 = n_1n_2 + \frac{n_2(n_2+1)}{2} - R_2

U = \min(U_1, U_2)

Where:

$n_1, n_2$ = sample sizes
$R_1, R_2$ = sum of ranks for group 1 and group 2
$U$ = the minimum of U₁ and U₂

Standardized Test Statistic:

z = \frac{U - \frac{n_1n_2}{2}}{\sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}}

Correction for Ties:

When ties occur in the data, a correction is applied to the standard deviation:

\sigma_U = \sqrt{\frac{n_1n_2}{N(N-1)} \left(\frac{N^3-N}{12} - \sum_{i=1}^g \frac{t_i^3-t_i}{12}\right)}

$N$ = total sample size ( $n_1 + n_2$ )
$g$ = number of tied groups
$t_i$ = number of tied values in the $i^{th}$ group

Continuity Correction:

z = \frac{U - \frac{n_1n_2}{2} - 0.5}{\sigma_U}

The 0.5 term is the continuity correction, which improves the approximation to the normal distribution.

Effect Size

Effect size r for Mann-Whitney U test:

r = \frac{|z|}{\sqrt{N}}

Where:

$z$ = standardized test statistic
$N$ = total sample size

Interpretation:

Small effect: $|r| \approx 0.1$
Medium effect: $|r| \approx 0.3$
Large effect: $|r| \approx 0.5$

Key Assumptions

Independent Samples: Observations must be independent between and within groups

Ordinal Scale: Data must be at least ordinal (can be ranked)

Similar Shapes: Distributions should have similar shapes (for comparing medians)

Common Pitfalls

Using with paired/dependent samples (use Wilcoxon signed-rank instead)
Interpreting results as comparing means rather than distributions
Not checking for ties in the data when using exact calculations

Example without Ties

Problem Statement

A researcher wants to test if a treatment affects test scores. Students were randomly assigned to either control or treatment group, and their test scores were recorded:

Control group: 45, 47, 43, 44

Treatment group: 52, 48, 54, 50

Sample sizes: $n_1 = n_2 = 4$

Step 1: State Hypotheses

$H_0$ : The distributions of scores are the same for both groups.

$H_1$ : The distributions of scores differ between the two groups.

$\alpha = 0.05$

Step 2: Combine and Rank Data

Group	Value	Rank
Control	43	1
Control	44	2
Control	45	3
Control	47	4
Treatment	48	5
Treatment	50	6
Treatment	52	7
Treatment	54	8

Step 3: Calculate Rank Sums

Control rank sum (R₁): 1 + 2 + 3 + 4 = 10

Treatment rank sum (R₂): 5 + 6 + 7 + 8 = 26

Step 4: Calculate U Statistics

U_1 = (4)(4) + \frac{4(5)}{2} - 10 = 16 + 10 - 10 = 16

U_2 = (4)(4) + \frac{4(5)}{2} - 26 = 16 + 10 - 26 = 0

U = \min(U_1, U_2) = \min(16, 0) = 0

Step 5: Calculate Z-Statistic

z = \frac{U - \frac{n_1n_2}{2}}{\sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}}

z = \frac{0 - \frac{16}{2}}{\sqrt{\frac{16(9)}{12}}} = -2.3094

Step 6: Draw Conclusion

The $p$ -value for this test is $0.0209$ . Since $p$ -value $\lt 0.05$ , we reject $H_0$ . There is sufficient evidence to conclude that the treatment and control groups have different distributions of scores.

Example with Ties

Problem Statement

A researcher wants to compare the effectiveness of two different teaching methods on student performance. Students were randomly assigned to either Method A or Method B, and their test scores were recorded:

Method A: 85, 92, 78, 90, 85, 76, 88

Method B: 79, 85, 81, 89, 84, 82, 85

Note that there are ties in the data: the score 85 appears three times (once in Method A and twice in Method B).

Step 1: State Hypotheses

$H_0$ : The distributions of scores are the same for both teaching methods.

$H_1$ : The distributions of scores differ between the two teaching methods.

Step 2: Combine and Rank Data

Group	Value	Rank
A	76	1
A	78	2
B	79	3
B	81	4
B	82	5
B	84	6
A	85	8
B	85	8
B	85	8
A	88	10
B	89	11
A	90	12
A	92	13

Note: For the three tied values of 85, we assign the average rank of (7+8+9)/3 = 8 to each.

Step 3: Calculate Rank Sums

Method A rank sum (R₁): 1 + 2 + 8 + 10 + 12 + 13 = 46

Method B rank sum (R₂): 3 + 4 + 5 + 6 + 8 + 8 + 11 = 45

Step 4: Calculate U Statistics

U_1 = (6)(7) + \frac{6(7)}{2} - 46 = 42 + 21 - 46 = 17

U_2 = (6)(7) + \frac{7(8)}{2} - 45 = 42 + 28 - 45 = 25

U = \min(U_1, U_2) = \min(17, 25) = 17

Step 5: Apply Correction for Ties

Since we have ties (three values of 85), we need to apply the correction to the standard deviation:

\sigma_U = \sqrt{\frac{n_1n_2}{N(N-1)} \left(\frac{N^3-N}{12} - \sum_{i=1}^g \frac{t_i^3-t_i}{12}\right)}

Where N = 14, and we have one tied group with t = 3 (for the value 85).

\sigma_U = \sqrt{\frac{(6)(7)}{(13)(12)} \left(\frac{13^3-13}{12} - \frac{3^3-3}{12}\right)}

\sigma_U = \sqrt{\frac{42}{156} \left(\frac{2184}{12} - \frac{24}{12}\right)} = \sqrt{\frac{42}{156} \cdot \frac{2160}{12}} = 6.96

Step 6: Calculate Z-Statistic (no continuity correction)

z = \frac{U - \frac{n_1n_2}{2}}{\sigma_U}

z = \frac{17 - \frac{(6)(7)}{2}}{6.96} = \frac{17 - 21}{6.96} = \frac{-4}{6.96} = -0.574

Step 7: Draw Conclusion

The $p$ -value for this test is $0.5660$ (two-sided). Since $p$ -value $\gt 0.05$ , we fail to reject $H_0$ . There is insufficient evidence to conclude that the two teaching methods result in different distributions of test scores.

Step 8: Calculate Effect Size

r = \frac{|z|}{\sqrt{N}} = \frac{|-0.574|}{\sqrt{13}} = 0.159

This represents a small effect size, which aligns with our failure to reject the null hypothesis.

Mann-Whitney U Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Mann-Whitney U Test

Definition

Formula

Effect Size

Key Assumptions

Common Pitfalls

Example without Ties

Problem Statement

Step 1: State Hypotheses

Step 2: Combine and Rank Data

Step 3: Calculate Rank Sums

Step 4: Calculate U Statistics

Step 5: Calculate Z-Statistic

Step 6: Draw Conclusion

Example with Ties

Problem Statement

Step 1: State Hypotheses

Step 2: Combine and Rank Data

Step 3: Calculate Rank Sums

Step 4: Calculate U Statistics

Step 5: Apply Correction for Ties

Step 6: Calculate Z-Statistic (no continuity correction)

Step 7: Draw Conclusion

Step 8: Calculate Effect Size

Verification

Related Calculators

Independent T-Test Calculator

Paired T-Test Calculator

Wilcoxon Signed-Rank Test Calculator

Friedman Test Calculator

Mann-Whitney U Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Mann-Whitney U Test

Definition

Formula

Effect Size

Key Assumptions

Common Pitfalls

Example without Ties

Problem Statement

Step 1: State Hypotheses

Step 2: Combine and Rank Data

Step 3: Calculate Rank Sums

Step 4: Calculate U Statistics

Step 5: Calculate Z-Statistic

Step 6: Draw Conclusion

Example with Ties

Problem Statement

Step 1: State Hypotheses

Step 2: Combine and Rank Data

Step 3: Calculate Rank Sums

Step 4: Calculate U Statistics

Step 5: Apply Correction for Ties

Step 6: Calculate Z-Statistic (no continuity correction)

Step 7: Draw Conclusion

Step 8: Calculate Effect Size

Verification

View Verification Details

Related Calculators

Independent T-Test Calculator

Paired T-Test Calculator

Wilcoxon Signed-Rank Test Calculator

Friedman Test Calculator