Sample 1 size (n₁):
Sample 2 size (n₂):
U statistic:
U' statistic:
p-value:
Conclusion:
The Wilcoxon Rank-Sum Test, also known as the Mann-Whitney U Test, is a non-parametric statistical test used to evaluate whether two independent samples come from the same distribution. Instead of comparing means like the traditional t-test, it compares the ranks of values from both samples, making it especially useful when the data does not meet the assumptions of normality.
This test works by combining the values from both groups, ranking them, and then analyzing the sum of the ranks for each group. The resulting U statistic reflects how different the groups are in terms of their overall distribution. A smaller U value typically indicates a larger difference between the groups.
Since it relies on the order or rank of the data rather than the actual values, the Wilcoxon Rank-Sum Test is robust against outliers and skewed data. It is suitable for ordinal data, interval data with non-normal distributions, or continuous data with small sample sizes.
The Wilcoxon Rank-Sum Test is an excellent choice when comparing two independent groups, particularly in situations where the assumptions of a parametric test like the independent samples t-test cannot be satisfied. Below are common scenarios where you might use this test:
For example, in medical research, the Wilcoxon Rank-Sum Test might be used to compare the recovery times of two different treatments. In social sciences, it could be used to analyze survey results between two distinct groups. The key advantage is its flexibility and reliability when standard assumptions cannot be met.
This Wilcoxon Rank-Sum Test Calculator is designed to make it easy for you to perform the test online by simply entering your data and selecting your preferences. Below are the steps to help you get started:
You can enter your sample data using commas, spaces, or new lines. The calculator is flexible and will automatically recognize the format. For example, any of the following formats are accepted:
12, 15, 17, 19, 2112 15 17 19 21
12
15
17
19
21
Make sure to enter numerical values only. Non-numeric inputs will result in an error message.
To ensure the test produces reliable results, each sample must contain at least three values. The two samples should also be independent—that is, the values in one sample should not influence the values in the other.
If either sample contains fewer than three values, the calculator will notify you and ask for more data.
The significance level (denoted as α) is the probability of rejecting the null hypothesis when it is actually true. Common values include:
Select the value that best fits the level of confidence you require for your analysis.
You can choose from three types of alternative hypotheses depending on the question you're trying to answer:
Choose the option that best reflects your research hypothesis. The interpretation of the p-value and conclusion will adjust accordingly.
After entering your data and clicking "Calculate," the Wilcoxon Rank-Sum Test Calculator will display the results in a clear and organized format. Here's how to understand each part of the output:
These represent the number of values in each group:
The sample sizes are important because they are used to calculate the rank sums, U statistics, and the p-value. A balanced number of observations in both groups usually improves the reliability of the test.
The test generates two values:
These statistics measure how different the two samples are in terms of ranking. The smaller the U value, the more evidence there is of a difference between the groups. The calculator automatically uses the smaller of the two for computing the final test statistic.
The p-value tells you the probability of observing the data (or something more extreme) if the null hypothesis were true. Depending on the type of alternative hypothesis you selected, the p-value is calculated accordingly:
A smaller p-value indicates stronger evidence against the null hypothesis. Typically, if the p-value is less than or equal to the chosen significance level (e.g., 0.05), the result is considered statistically significant.
The conclusion tells you whether the result is statistically significant based on your chosen alpha level and hypothesis direction. It is automatically interpreted and presented in plain language:
For example, if your alternative hypothesis was "Sample 1 is greater than Sample 2" and the result is significant, the conclusion will confirm that Sample 1 tends to have higher values. If not significant, the calculator will indicate that the evidence is insufficient to draw that conclusion.
Understanding these components helps you interpret the outcome of your test clearly and apply the findings to your research or decision-making process.
Along with the test results, the calculator also provides a detailed ranking table to show how each value from both samples was ranked. This breakdown helps you understand exactly how the Wilcoxon Rank-Sum Test works under the hood.
All values from both Sample 1 and Sample 2 are combined into a single list and sorted from smallest to largest. Each value is then assigned a rank based on its position in the sorted list. The smallest value gets rank 1, the next gets rank 2, and so on.
These ranks are used instead of the raw values because the Wilcoxon test is based on the order of the data, not the actual numbers. This makes the test more robust, especially when dealing with outliers or non-normal data.
When two or more values are equal, they are considered tied. Instead of assigning each tied value a separate rank, the calculator assigns them the average of the ranks they would have occupied.
For example, if two values are tied for the 4th and 5th positions, each will receive a rank of 4.5. This ensures the fairness and accuracy of the ranking process and helps prevent bias in the final U statistic.
In the ranking table, each value is labeled according to the group it came from:
12This grouping allows you to see how values from each sample are distributed across the rank spectrum. If one sample consistently has higher or lower ranks than the other, it can be a sign of a significant difference, which is exactly what the test is designed to detect.
Reviewing this table can also help you validate your input and better understand the behavior of your data in the context of the test.
No. The Wilcoxon Rank-Sum Test (Mann-Whitney U Test) is designed for comparing two independent samples. If your data consists of matched or paired observations (such as before-and-after measurements on the same subjects), consider using the Wilcoxon Signed-Rank Test instead.
This calculator automatically accounts for ties by assigning tied values their average ranks and adjusting the standard deviation accordingly. However, a large number of ties can slightly affect the accuracy of the p-value. If possible, try to collect more precise data to reduce ties.
Yes. One of the advantages of the Wilcoxon test is that it works well even when the sample sizes are small. The minimum requirement for each sample is at least three values for a meaningful result.
The p-value represents the probability that the observed difference (or a more extreme one) could have occurred by chance under the null hypothesis. A smaller p-value (typically less than 0.05) suggests that the difference between the groups is statistically significant.
Make sure you’ve entered only numeric values, separated by commas, spaces, or new lines. Non-numeric characters or missing values may cause the calculator to display an error message. Check that both samples contain at least three valid numbers.
You can choose from:
Yes. The calculator fully supports decimal values. You can enter data like 12.5, 14.8, 19.3 without any problem.
The Wilcoxon Rank-Sum Test is best suited for ordinal or continuous data that are not normally distributed. It should not be used with categorical data or paired/matched samples.
A non-parametric statistical test used to compare two independent samples to determine whether they come from the same distribution. Also known as the Mann-Whitney U Test.
Another name for the Wilcoxon Rank-Sum Test, commonly used in statistics. Both names refer to the same test.
A type of statistical test that does not assume a specific distribution (like normal distribution) for the data. Suitable for ordinal data or data with outliers and skewness.
Two sets of data are considered independent when the values in one group do not influence or relate to the values in the other group.
The position of a value in a sorted list. In this test, values from both samples are combined and sorted, and each value is assigned a rank based on its order.
When two or more values are equal, they share the average of the ranks they would have received if they were slightly different. This ensures fairness in ranking.
A measure of how different the two samples are based on their ranks. It is one of the main results produced by the Wilcoxon Rank-Sum Test.
The complementary U statistic calculated using the second sample. The smaller of U and U' is typically used for test evaluation.
The probability of obtaining the observed results (or more extreme) under the assumption that the null hypothesis is true. A small p-value suggests that the difference between groups is statistically significant.
The assumption that there is no difference between the two samples. The Wilcoxon test checks whether there is enough evidence to reject this assumption.
The assumption that there is a difference between the two samples. This can be two-sided (any difference), or one-sided (greater or less).
A threshold value (commonly 0.05) used to decide whether to reject the null hypothesis. It represents the maximum acceptable probability of a Type I error (false positive).
This Wilcoxon Rank-Sum Test Calculator is provided for educational and informational purposes only. While every effort has been made to ensure accuracy and usability, the results generated by this tool should not be considered a substitute for professional statistical analysis or expert consultation.
Users are responsible for verifying their input data and interpreting the results appropriately. This calculator is not intended for use in critical decision-making processes in fields such as medicine, finance, or legal matters without further review by qualified professionals.
By using this tool, you acknowledge that the creators and maintainers of this calculator are not liable for any outcomes, decisions, or actions taken based on the information provided. Always consult a statistician or subject-matter expert if you are conducting formal research or making data-driven decisions with significant consequences.