Mann-Whitney U Test Calculator

Hypothesis Testing and Statistical Inference

A non-parametric test to determine if two independent samples were drawn from the same distribution.

Examples

Explore some common scenarios for the Mann-Whitney U Test.

New Drug Efficacy

Medical Study

Comparing the recovery times (in days) for patients on a new drug versus a placebo.

Sample A: 6, 7, 7, 8, 9, 10, 11

Sample B: 8, 9, 9, 10, 11, 12, 12

Teaching Method Comparison

Educational Research

Comparing the test scores of students taught with two different methods.

Sample A: 88, 72, 94, 65, 80, 75

Sample B: 91, 85, 79, 97, 88

Website Conversion Rates

A/B Testing

Comparing the number of daily sign-ups from two different website layouts (A and B).

Sample A: 25, 30, 32, 28, 40, 35

Sample B: 18, 22, 25, 20, 15, 21

Crop Yield Analysis

Agricultural Science

Comparing the yield (in kg) from two different types of fertilizer.

Sample A: 150, 155, 160, 148, 152

Sample B: 162, 165, 158, 170, 163, 159

Other Titles
Understanding the Mann-Whitney U Test: A Comprehensive Guide
Dive deep into the concepts, application, and mathematics behind this powerful non-parametric statistical test.

What is the Mann-Whitney U Test?

  • Core Concept: Comparing Distributions, Not Means
  • Why Use a Non-Parametric Test?
  • Null and Alternative Hypotheses Explained
The Mann-Whitney U Test, also known as the Wilcoxon Rank-Sum Test, is a non-parametric statistical test used to determine whether two independent samples have been drawn from the same population (i.e., from populations with the same distribution). Unlike its parametric counterpart, the t-test, it does not assume that the data is normally distributed. This makes it an incredibly versatile and robust tool for data analysis, especially when dealing with skewed data or small sample sizes.
Core Concept: Comparing Distributions, Not Means
The key idea behind the Mann-Whitney U test is to work with ranks instead of the raw data values. All data points from both samples are combined, sorted, and ranked from smallest to largest. The test then checks if the ranks from one sample are systematically higher or lower than the ranks from the other. If there's a significant difference in the sum of ranks, we infer that the underlying distributions of the two groups are different.
Why Use a Non-Parametric Test?
Parametric tests like the t-test rely on strict assumptions, most notably that the data follows a normal distribution. When these assumptions are violated, the results of a parametric test can be misleading. The Mann-Whitney U test is ideal for situations involving: 1. Ordinal data (e.g., satisfaction ratings from 1 to 5). 2. Continuous data that is not normally distributed. 3. Small sample sizes where testing for normality is unreliable.

Step-by-Step Guide to Using the Mann-Whitney U Test Calculator

  • Entering Your Data
  • Choosing a Significance Level (α)
  • Interpreting the Results: U, Z, and P-Value
1. Entering Your Data
In the 'Sample A Data' and 'Sample B Data' fields, input your two independent sets of observations. You can separate numbers with commas, spaces, or newlines. Ensure that you do not mix data points between the two groups.
2. Choosing a Significance Level (α)
The significance level, alpha (α), represents the threshold for statistical significance. It's the probability of rejecting the null hypothesis when it is actually true. A common choice for α is 0.05 (or 5%). If the calculated p-value is less than α, the result is considered statistically significant.
3. Selecting the Alternative Hypothesis
You must specify what you are testing for. A 'Two-Tailed' test checks for any difference between the groups. A 'Right-Tailed' (Group A > Group B) test checks if group A values are significantly larger than group B. A 'Left-Tailed' (Group B > Group A) test checks if group B values are significantly larger than group A.
4. Interpreting the Results
The calculator provides several key outputs: the U-statistic, the Z-score (for larger samples), and the p-value. The most important is the p-value. Compare the p-value to your chosen α. If p < α, you reject the null hypothesis and conclude there's a significant difference. If p ≥ α, you fail to reject the null hypothesis.

Real-World Applications of the Mann-Whitney U Test

  • Medical and Clinical Trials
  • Psychology and Social Sciences
  • Business and A/B Testing
The test's flexibility makes it applicable in numerous fields.
Example: Medical Research
A researcher wants to compare the effectiveness of a new pain reliever against a standard one. They measure patient-reported pain scores (on a scale of 1-10) after treatment. Since pain scores are ordinal and may not be normally distributed, the Mann-Whitney U test is the perfect tool to see if one drug provides significantly lower pain scores than the other.
Example: A/B Testing
A marketing team tests two different website landing pages (A and B) to see which one leads to a higher user engagement time. Since engagement time is often heavily skewed (many users leave quickly, a few stay for a long time), the Mann-Whitney U test can determine if one page has a stochastically greater engagement time without being distorted by outliers.

Mathematical Derivation and Formulas

  • Ranking the Data
  • Calculating the U Statistic
  • Normal Approximation (Z-Score)
The test statistic U is calculated based on the ranks of the combined data.
1. Ranking Procedure
Combine all data from both samples (n1 and n2). Sort the combined data in ascending order. Assign ranks, starting with 1 for the smallest value. If there are ties, each tied value gets the average of the ranks they would have occupied. For example, if the 3rd and 4th values are the same, they both get a rank of (3+4)/2 = 3.5.
2. U Statistic Formulas
Let R1 be the sum of ranks for Sample 1 and R2 be the sum of ranks for Sample 2. Calculate two U values: U1 = n1n2 + (n1(n1 + 1))/2 - R1 and U2 = n1n2 + (n2(n2 + 1))/2 - R2. The test statistic U is the smaller of these two values: U = min(U1, U2).
3. Normal Approximation (for large samples)
When sample sizes are large (e.g., n1 > 20 or n2 > 20), the distribution of U approximates a normal distribution. The Z-score is calculated as: Z = (U - μᵤ) / σᵤ, where the mean μᵤ = (n1 n2) / 2 and the standard deviation σᵤ = sqrt((n1 n2 * (n1 + n2 + 1)) / 12). The p-value is then found from this Z-score.