Mann-Whitney U Test

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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 Statistic:

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

Where:

$n_1, n_2$ = sample sizes
$R_1$ = sum of ranks for group 1

Standardized Test Statistic:

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

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

Practical Example

Testing if a treatment affects test scores:

Step 1: State the Data

Control group: 45, 47, 43, 44
Treatment group: 52, 48, 54, 50
Sample sizes: $n_1 = n_2 = 4$

Step 2: State Hypotheses

$H_0$ : The distributions are identical
$H_a$ : The distributions differ
$\alpha = 0.05$

Step 3: Calculate Rankings

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

Sum of Control ranks: $R_1 = 1 + 2 + 3 + 4 = 10$

Step 4: Calculate U Statistic

Using the formula:

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

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

Step 5: Calculate Standardized Statistic

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

z = \frac{16 - \frac{16}{2}}{\sqrt{\frac{16(9)}{12}}} = 2.31

Step 6: Draw Conclusion

The $p$ -value for this test is 0.021. 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.

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: $|r| \approx 0.5$

In the example above, the effect size is

r = \frac{|2.31|}{\sqrt{8}} = 0.82

Code Examples

1# Effect size calculation for Mann-Whitney U test
2library(tidyverse)
3
4wilcoxonR <- function(x, y) {
5  n1 <- length(x)
6  n2 <- length(y)
7  
8  is_small_sample <- (n1 + n2) <= 30
9  
10  # Best practices for parameter 'exact'
11  # - If the sample size is small (n1 + n2 <= 30) and there are no ties, set 'exact = TRUE'
12  # - Otherwise, set 'exact = FALSE'
13  
14  if (is_small_sample && !anyDuplicated(c(x, y))) {
15    test <- wilcox.test(x, y, exact = TRUE)  
16  } else {
17    test <- wilcox.test(x, y, exact = FALSE)
18  }
19  
20  # Extract U statistic
21  W <- as.numeric(test$statistic)
22  
23  # Calculate Z-score manually
24  mean_U <- n1 * n2 / 2
25  sd_U <- sqrt((n1 * n2 * (n1 + n2 + 1)) / 12)
26  z <- (W - mean_U) / sd_U
27  
28  # Calculate effect size (r)
29  r <- abs(z) / sqrt(n1 + n2)
30  
31  return(list(effect_size = r, test_details = test))
32}
33
34# Example data
35control <- c(45, 47, 43, 44)
36treatment <- c(52, 48, 54, 50)
37
38# Calculate effect size
39result <- wilcoxonR(control, treatment)
40print(str_glue("Effect size (r): {round(result$effect_size, 4)}"))
41print(result$test_details)

Python

1from scipy.stats import mannwhitneyu
2import numpy as np
3
4control  = [45, 47, 43, 44]
5treatment = [52, 48, 54, 50]
6
7# Perform Mann-Whitney U test
8stat, pvalue = mannwhitneyu(
9    control, 
10    treatment,
11    alternative='two-sided',
12    method='auto' 
13)
14
15print(f'U-statistic: {stat}')
16print(f'p-value: {pvalue:.4f}')
17
18# Effect size (r = Z/sqrt(N))
19n1, n2 = len(control), len(treatment)
20z_score = (stat - (n1*n2/2)) / np.sqrt((n1*n2*(n1+n2+1))/12)
21effect_size = abs(z_score) / np.sqrt(n1 + n2)
22print(f'Effect size (r): {effect_size:.4f}')

Alternative Tests

Consider these alternatives:

Independent t-test: When data is normal and interval/ratio
Mood's Median Test: When only interested in median differences
Kolmogorov-Smirnov Test: When interested in any distributional differences

Related Calculators

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Mann-Whitney U Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Mann-Whitney U Test

Definition

Formula

Key Assumptions

Common Pitfalls

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Rankings

Step 4: Calculate U Statistic

Step 5: Calculate Standardized Statistic

Step 6: Draw Conclusion

Effect Size

Code Examples

Alternative Tests

Related Calculators

Independent T-Test Calculator

Paired T-Test Calculator

Wilcoxon Signed-Rank Test Calculator

Friedman Test Calculator

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