This calculator helps you compare two independent groups using a non-parametric approach that doesn't assume your data follows a normal distribution. Perfect for comparing treatment outcomes, survey responses, or any scenario where you need to know if two groups come from different populations—especially when your data is skewed, ordinal, or violates t-test assumptions.
💡 When to Use: Choose Mann-Whitney U when your data is non-normal, ordinal, or when you want a robust alternative to theindependent t-test. It compares distributions and ranks rather than means.
Ready to compare your groups without parametric assumptions? to see the ranking process in action, or upload your own data to discover if your groups have significantly different distributions.
Yates' continuity correction improves accuracy for discrete data. Learn more
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.
U Statistics:
Where:
Standardized Test Statistic:
Correction for Ties:
When ties occur in the data, a correction is applied to the standard deviation:
Continuity Correction:
The 0.5 term is the continuity correction, which improves the approximation to the normal distribution.
Effect size r for Mann-Whitney U test:
Where:
Interpretation:
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:
: The distributions of scores are the same for both groups.
: The distributions of scores differ between the two groups.
| 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 |
Control rank sum (R₁): 1 + 2 + 3 + 4 = 10
Treatment rank sum (R₂): 5 + 6 + 7 + 8 = 26
The -value for this test is . Since -value , we reject . There is sufficient evidence to conclude that the treatment and control groups have different distributions of scores.
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).
: The distributions of scores are the same for both teaching methods.
: The distributions of scores differ between the two teaching methods.
| 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.
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
Since we have ties (three values of 85), we need to apply the correction to the standard deviation:
Where N = 14, and we have one tied group with t = 3 (for the value 85).
The -value for this test is (two-sided). Since -value , we fail to reject . There is insufficient evidence to conclude that the two teaching methods result in different distributions of test scores.
This represents a small effect size, which aligns with our failure to reject the null hypothesis.
# Effect size calculation for Mann-Whitney U test
library(tidyverse)
wilcoxonR <- function(x, y) {
n1 <- length(x)
n2 <- length(y)
is_small_sample <- (n1 + n2) <= 30
# Best practices for parameter 'exact'
# - If the sample size is small (n1 + n2 <= 30) and there are no ties, set 'exact = TRUE'
# - Otherwise, set 'exact = FALSE'
if (is_small_sample && !anyDuplicated(c(x, y))) {
test <- wilcox.test(x, y, exact = TRUE)
} else {
test <- wilcox.test(x, y, exact = FALSE)
}
# Extract U statistic
W <- as.numeric(test$statistic)
# Calculate Z-score manually
mean_U <- n1 * n2 / 2
sd_U <- sqrt((n1 * n2 * (n1 + n2 + 1)) / 12)
z <- (W - mean_U) / sd_U
# Calculate effect size (r)
r <- abs(z) / sqrt(n1 + n2)
return(list(effect_size = r, test_details = test))
}
# Example data
control <- c(45, 47, 43, 44)
treatment <- c(52, 48, 54, 50)
# Calculate effect size
result <- wilcoxonR(control, treatment)
print(str_glue("Effect size (r): {round(result$effect_size, 4)}"))
print(result$test_details)from scipy.stats import mannwhitneyu
import numpy as np
control = [45, 47, 43, 44]
treatment = [52, 48, 54, 50]
# Perform Mann-Whitney U test
stat, pvalue = mannwhitneyu(
control,
treatment,
alternative='two-sided',
method='auto'
)
print(f'U-statistic: {stat}')
print(f'p-value: {pvalue:.4f}')
# Effect size (r = Z/sqrt(N))
n1, n2 = len(control), len(treatment)
z_score = (stat - (n1*n2/2)) / np.sqrt((n1*n2*(n1+n2+1))/12)
effect_size = abs(z_score) / np.sqrt(n1 + n2)
print(f'Effect size (r): {effect_size:.4f}')This calculator helps you compare two independent groups using a non-parametric approach that doesn't assume your data follows a normal distribution. Perfect for comparing treatment outcomes, survey responses, or any scenario where you need to know if two groups come from different populations—especially when your data is skewed, ordinal, or violates t-test assumptions.
💡 When to Use: Choose Mann-Whitney U when your data is non-normal, ordinal, or when you want a robust alternative to theindependent t-test. It compares distributions and ranks rather than means.
Ready to compare your groups without parametric assumptions? to see the ranking process in action, or upload your own data to discover if your groups have significantly different distributions.
Yates' continuity correction improves accuracy for discrete data. Learn more
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.
U Statistics:
Where:
Standardized Test Statistic:
Correction for Ties:
When ties occur in the data, a correction is applied to the standard deviation:
Continuity Correction:
The 0.5 term is the continuity correction, which improves the approximation to the normal distribution.
Effect size r for Mann-Whitney U test:
Where:
Interpretation:
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:
: The distributions of scores are the same for both groups.
: The distributions of scores differ between the two groups.
| 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 |
Control rank sum (R₁): 1 + 2 + 3 + 4 = 10
Treatment rank sum (R₂): 5 + 6 + 7 + 8 = 26
The -value for this test is . Since -value , we reject . There is sufficient evidence to conclude that the treatment and control groups have different distributions of scores.
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).
: The distributions of scores are the same for both teaching methods.
: The distributions of scores differ between the two teaching methods.
| 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.
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
Since we have ties (three values of 85), we need to apply the correction to the standard deviation:
Where N = 14, and we have one tied group with t = 3 (for the value 85).
The -value for this test is (two-sided). Since -value , we fail to reject . There is insufficient evidence to conclude that the two teaching methods result in different distributions of test scores.
This represents a small effect size, which aligns with our failure to reject the null hypothesis.
# Effect size calculation for Mann-Whitney U test
library(tidyverse)
wilcoxonR <- function(x, y) {
n1 <- length(x)
n2 <- length(y)
is_small_sample <- (n1 + n2) <= 30
# Best practices for parameter 'exact'
# - If the sample size is small (n1 + n2 <= 30) and there are no ties, set 'exact = TRUE'
# - Otherwise, set 'exact = FALSE'
if (is_small_sample && !anyDuplicated(c(x, y))) {
test <- wilcox.test(x, y, exact = TRUE)
} else {
test <- wilcox.test(x, y, exact = FALSE)
}
# Extract U statistic
W <- as.numeric(test$statistic)
# Calculate Z-score manually
mean_U <- n1 * n2 / 2
sd_U <- sqrt((n1 * n2 * (n1 + n2 + 1)) / 12)
z <- (W - mean_U) / sd_U
# Calculate effect size (r)
r <- abs(z) / sqrt(n1 + n2)
return(list(effect_size = r, test_details = test))
}
# Example data
control <- c(45, 47, 43, 44)
treatment <- c(52, 48, 54, 50)
# Calculate effect size
result <- wilcoxonR(control, treatment)
print(str_glue("Effect size (r): {round(result$effect_size, 4)}"))
print(result$test_details)from scipy.stats import mannwhitneyu
import numpy as np
control = [45, 47, 43, 44]
treatment = [52, 48, 54, 50]
# Perform Mann-Whitney U test
stat, pvalue = mannwhitneyu(
control,
treatment,
alternative='two-sided',
method='auto'
)
print(f'U-statistic: {stat}')
print(f'p-value: {pvalue:.4f}')
# Effect size (r = Z/sqrt(N))
n1, n2 = len(control), len(treatment)
z_score = (stat - (n1*n2/2)) / np.sqrt((n1*n2*(n1+n2+1))/12)
effect_size = abs(z_score) / np.sqrt(n1 + n2)
print(f'Effect size (r): {effect_size:.4f}')