Wilcoxon Rank Sum Test Calculator

Hypothesis Testing and Statistical Inference

Use this calculator to perform the Wilcoxon Rank Sum Test (also known as the Mann-Whitney U Test) for two independent samples. Enter your data below.

Practical Examples

Explore these common scenarios to see how the Wilcoxon Rank Sum Test is applied.

New Drug Efficacy

Medical Study

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

Sample 1: 7, 8, 8, 9, 10, 12

Sample 2: 9, 11, 12, 13, 14, 15

α: 0.05, Type: two-tailed

Teaching Method Comparison

Education

Comparing the test scores of students taught with a new method versus a traditional method.

Sample 1: 85, 90, 78, 92, 88, 76

Sample 2: 72, 80, 81, 75, 68, 79

α: 0.05, Type: right-tailed

Fertilizer Impact on Crop Yield

Agriculture

Comparing the crop yield (kg/acre) from two different types of fertilizers.

Sample 1: 120, 125, 130, 110, 115, 122, 128

Sample 2: 130, 135, 140, 128, 132, 138, 142

α: 0.01, Type: left-tailed

Anxiety Scores Comparison

Psychology

Comparing anxiety scores between a therapy group and a control group.

Sample 1: 15, 18, 22, 25, 20, 17

Sample 2: 28, 30, 25, 35, 32, 29

α: 0.05, Type: two-tailed

Other Titles
Understanding the Wilcoxon Rank Sum Test: A Comprehensive Guide
A deep dive into the non-parametric method for comparing two independent groups, also known as the Mann-Whitney U Test.

What is the Wilcoxon Rank Sum Test?

  • Core Concepts
  • Non-Parametric Approach
  • Hypotheses
The Wilcoxon Rank Sum Test, also widely known as the Mann-Whitney U Test, is a powerful non-parametric statistical test used to determine whether two independent samples were drawn from populations with the same distribution. It serves as an alternative to the two-sample t-test when the data does not meet the assumption of normality.
Why Use a Non-Parametric Test?
Parametric tests like the t-test assume that the data follows a specific distribution, usually the normal distribution. When this assumption is violated, the results of the t-test can be misleading. The Wilcoxon Rank Sum Test does not require this assumption, making it more robust for ordinal data or for continuous data that is not normally distributed.
Key Concepts
Null Hypothesis (H₀): The two populations are identical. There is no difference in the medians of the two groups.
Alternative Hypothesis (H₁): The two populations are not identical (two-tailed), or one population's median is greater/smaller than the other (one-tailed).
Ranks: The test works by combining both samples, ranking them from lowest to highest, and then summing the ranks for each sample.

Step-by-Step Guide to Using the Calculator

  • Data Entry
  • Setting Parameters
  • Interpreting Results
Our calculator simplifies the process into a few easy steps:
1. Data Entry for Sample 1
In the 'Sample 1 Data' field, enter the numeric values for your first group. Separate each number with a comma. For example: 10, 15, 12, 18.
2. Data Entry for Sample 2
In the 'Sample 2 Data' field, enter the values for your second group, also separated by commas. For example: 20, 22, 19, 25.
3. Set the Significance Level (α)
Choose your desired significance level. This value represents the threshold for statistical significance. A common choice is 0.05, which corresponds to a 95% confidence level.
4. Select the Hypothesis Test Type
Choose between a two-tailed, left-tailed, or right-tailed test based on your research question. A two-tailed test checks for any difference, while a one-tailed test checks for a difference in a specific direction.
5. Calculate and Interpret Results
Click 'Calculate' to see the results. The key output is the p-value. If the p-value is less than your chosen significance level (α), you can reject the null hypothesis and conclude that a statistically significant difference exists between the two groups.

Real-World Applications of the Test

  • Medicine
  • Social Sciences
  • Ecology
The Wilcoxon Rank Sum Test is used across various fields due to its versatility.
In Medicine and Healthcare
To compare the effectiveness of two different treatments where patient outcomes (like pain levels or recovery times) might not be normally distributed.
In Psychology and Social Sciences
To compare survey responses on a Likert scale (e.g., satisfaction ratings from 1 to 5) between two different demographic groups.
In Ecology and Environmental Science
To compare measurements of pollutant levels at two different sites, where the data might be skewed by a few high readings.

The Mathematical Logic Behind the Test

  • Ranking Procedure
  • U Statistic Calculation
  • Normal Approximation
Understanding the calculations can provide deeper insight into your results.
1. Pooling and Ranking
First, the data from both samples are combined into a single set. Each value in this combined set is then assigned a rank from lowest to highest. If there are ties, each tied value gets the average of the ranks they would have occupied.
2. Summing the Ranks
The ranks for each of the original two samples are summed up separately. Let's call these R₁ and R₂.
3. Calculating the U Statistic
The Mann-Whitney U statistic is calculated for each sample: U₁ = R₁ - n₁(n₁+1)/2 and U₂ = R₂ - n₂(n₂+1)/2. The test statistic U is the smaller of U₁ and U₂.
4. Normal Approximation (for large samples)
For larger samples (typically n₁, n₂ > 10), the distribution of U can be approximated by a normal distribution. A Z-score is calculated, which is then used to find the p-value.
Formula: Z = (U - μᵤ) / σᵤ
Where μᵤ = (n₁ n₂) / 2 and σᵤ = √[(n₁ n₂ * (n₁ + n₂ + 1)) / 12].