Mann-Whitney U Table: Critical Values & How to Read It

Introduction

Ever wondered whether your Mann-Whitney U result is statistically significant or just a random difference?

The Mann-Whitney U table helps you answer that question by comparing your calculated U statistic with a critical value. It is especially useful when comparing two independent groups without relying on normality assumptions.

For beginners in the USA, statistical tables can seem confusing at first. Terms like sample size, significance level, critical value, and U statistic may feel overwhelming. The good news is that reading the table becomes much easier with a clear process.

This guide explains what a Mann-Whitney U table means and how its critical values work. You will also learn how to find the correct value for your sample sizes and significance level.

By the end, you will know how to read a Mann-Whitney table and make the correct statistical decision with confidence.

What Is a Mann-Whitney U Table?

A Mann-Whitney U table helps you determine whether two independent groups show a statistically significant difference. It works alongside the Mann-Whitney U test, a nonparametric test for comparing two groups.

The test ranks observations instead of comparing their raw values directly. It then calculates a U statistic, which represents the difference between the groups based on their ranks.

To interpret the U statistic, you compare it with a critical value from the appropriate table. The critical value depends mainly on your sample sizes and chosen significance level.

The significance level, often written as α, sets the threshold for deciding whether the result is statistically significant. Common choices include 0.05 and 0.01, depending on your analysis.

Your sample sizes also matter because the critical U value changes when the number of observations in either group changes. You must use the correct sample size for each group when reading the table.

Key terms to remember:

• Mann-Whitney U test: Compares two independent groups using ranked data.
• U statistic: The value calculated from your sample data.
• Critical value: The table value used to evaluate the U statistic.
• Significance level: The chosen threshold for statistical significance.
• Sample sizes: The number of observations in each independent group.

When Is the Mann-Whitney U Test Used?

The Mann-Whitney U test is useful when you need to compare two independent groups and a standard independent-samples t-test may not be suitable.

Researchers often consider it when the data do not meet important assumptions required for a t-test. For example, the outcome may be ordinal, strongly skewed, or otherwise unsuitable for a normal-based comparison.

The test can also work well when your measurements use ranks or ordered categories. It does not require the same distributional assumptions as a traditional t-test.

However, the Mann-Whitney U test still has assumptions. The observations should be independent, and the groups should come from comparable populations when interpreting the test as a location comparison.

Example: A researcher might compare satisfaction ratings from two independent customer groups. If those ratings are ordinal, the Mann-Whitney U test can provide an appropriate comparison.

Mann-Whitney U Critical Values Table

The Mann-Whitney U critical values table lets you quickly find the cutoff for your statistical test. Your sample sizes determine which table cell you should use.

The table below uses α = 0.05 for a two-tailed test. Each cell shows the critical U value for the corresponding sample sizes. A dash means the sample sizes cannot produce significance at this level using the exact critical-value approach.

Group 1 (n₁) ↓ / Group 2 (n₂) →	3	4	5	6	7	8	9	10
3	—	—	0	1	1	2	2	3
4	—	0	1	2	3	4	4	5
5	0	1	2	3	5	6	7	8
6	1	2	3	5	6	8	10	11
7	1	3	5	6	8	10	12	14
8	2	4	6	8	10	13	15	17
9	2	4	7	10	12	15	17	20
10	3	5	8	11	14	17	20	23

How to use the table: Find Group 1’s sample size in the left column. Then move across to Group 2’s sample size. The intersecting value is your critical U value.

For example, if Group 1 has 8 observations and Group 2 has 10, the critical value is 17. You then compare your calculated U with 17. If your calculated U is 17 or less, you reject the null hypothesis at α = 0.05 under this table’s decision rule.

Important: Critical-value tables can differ based on the significance level and whether the test is one-tailed or two-tailed. Always confirm the table’s convention before interpreting your result.

How to Read the Mann-Whitney U Table

Learning how to read the Mann-Whitney table becomes simple once you know what each part represents. You mainly need your two sample sizes, significance level, and calculated U statistic.

Follow these steps to find the correct critical value and interpret your test result.

Step 1: Identify the Sample Size for Each Group

Start by finding the number of observations in both independent groups. Call these values n₁ for Group 1 and n₂ for Group 2.

For example, suppose Group 1 contains 8 observations and Group 2 contains 10 observations. Your sample sizes are n₁ = 8 and n₂ = 10.

Step 2: Determine the Significance Level

Next, identify the significance level used for your test. This value is usually written as α.

A common choice is α = 0.05. However, your research question or study design may require a different level.

Also confirm whether your table uses a one-tailed or two-tailed test. This distinction can change the critical value.

Step 3: Locate the Correct Row and Column

Find the sample size for one group along the table’s rows. Then find the other group’s sample size across the columns.

For n₁ = 8 and n₂ = 10, locate row 8 and column 10. Their intersection gives the required critical U value.

Step 4: Find the Critical U Value

Read the number where the two sample-size positions meet. This number is your critical U value.

Using the example above, the critical value is 17 when using the table shown earlier.

Step 5: Compare U With the Critical Value

Now compare your calculated U statistic with the critical value.

For a standard lower-tail critical-value approach, a calculated U that is equal to or smaller than the critical value indicates statistical significance at the selected level.

Step 6: Interpret the Result

If your calculated U meets the table’s rejection rule, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Always check whether your table uses the same significance level and tail convention as your test.

Pro tip: Write down n₁, n₂, α, calculated U, and critical U before making your final decision. This simple checklist can prevent common table-reading mistakes.

How to Use a Mann-Whitney U Table

Using a Mann-Whitney U table involves more than finding a number. You need to calculate the U statistic, select the correct sample sizes, and use the right significance level.

The process below shows how to move from your raw data to a statistical decision.

Step 1 — Calculate the U Statistic

First, calculate the U statistic from your two independent groups. One common formula is:

Here, n₁ and n₂ represent the group sample sizes. R₁ represents the sum of ranks for Group 1.

You can calculate both U values when needed:

The smaller U value is commonly used when consulting a critical-value table.

Step 2 — Identify the Two Sample Sizes

Count the observations in each independent group. Record them as n₁ and n₂.

For example, Group 1 might have 8 observations, while Group 2 has 10.

Step 3 — Choose the Significance Level

Select your significance level, represented by α. A common choice is 0.05.

Also check whether your hypothesis requires a one-tailed or two-tailed test. Your critical-value table must match this choice.

Step 4 — Find the Critical U Value

Use the two sample sizes to locate the correct table cell. Match the significance level and test direction used by your table.

The resulting number is your critical U value.

Step 5 — Compare U With the Critical Value

Compare your calculated U with the critical value using the decision rule for your table.

For tables using the lower-tail critical U approach, U ≤ critical U indicates a statistically significant result at the selected level.

Step 6 — Make the Statistical Decision

If the calculated U meets the rejection rule, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Always verify the table’s convention before making your final decision.

Mann-Whitney U Table Example

A worked example can make the table much easier to understand. Consider a researcher comparing satisfaction ratings from two independent groups.

Given: The researcher wants to test whether the two groups differ at α = 0.05 using a two-tailed test.

Sample sizes: Group 1 has 8 observations and Group 2 has 10 observations.

Calculated U: After ranking all observations and applying the U formula, suppose the smaller U statistic is 12.

Table lookup: Find n₁ = 8 in the Group 1 row and n₂ = 10 in the Group 2 column.

Critical U: Using the critical-value table above, the critical U value is 17.

Decision: Compare the calculated U with the critical value:

Because the calculated U is less than the critical value, it meets the table’s rejection rule.

Interpretation: The researcher rejects the null hypothesis at the 0.05 significance level. The results provide evidence of a statistically significant difference between the two groups.

Pro tip: Always confirm your table’s significance level and tail convention before interpreting a critical U value.

Mann-Whitney U Critical Value vs. U Statistic

The U statistic and critical U value serve different purposes. Understanding the difference helps prevent mistakes when interpreting your test.

Term	Meaning
U statistic	The value calculated from your sample data and ranks.
Critical U	The cutoff obtained from a Mann-Whitney U critical-value table.
p-value	A probability-based measure used to assess evidence against the null hypothesis.

Think of the U statistic as your calculated result. The critical U provides the comparison point.

The p-value provides another way to make the statistical decision. When using the p-value approach, compare it with your selected significance level rather than relying on the table.

What Does the Mann-Whitney U Table Tell You?

A Mann-Whitney U table gives you a critical value for your chosen sample sizes and significance level. You then compare your calculated U statistic with that cutoff.

For the common lower-tail critical-value approach, the basic rule is:

• U ≤ critical U: Reject the null hypothesis when the table’s setup supports this rule.
• U > critical U: Fail to reject the null hypothesis under that same setup.
• Statistically significant: Your U statistic falls in the table’s rejection region at the selected α level.
• Not statistically significant: Your U statistic does not fall in the rejection region.

For example, suppose your critical U value is 17 and your calculated U is 12. Because 12 is below 17, the result meets the rejection rule for that table.

If your calculated U were 21, it would be above 17. The result would not meet that rejection rule.

The exact interpretation depends on your significance level and test direction. Always follow the convention specified by your critical-value table.

Mann-Whitney U Table vs. P-Value

There are two common ways to determine whether a Mann-Whitney U result is statistically significant. The critical-value approach uses a table, while the p-value approach uses a probability calculated from the test.

With the critical-value approach, you compare the calculated U statistic with the table’s critical U value. Your decision depends on whether U falls within the rejection region.

With the p-value approach, you compare the reported p-value with your chosen significance level, such as α = 0.05. If the p-value is at or below α, you reject the null hypothesis.

This explains why your textbook may provide a U table while statistical software reports a p-value. Both approaches help you make a statistical decision, but they present the evidence differently.

Approach	What You Compare	Decision Basis
Critical-value	U statistic vs. critical U	Rejection region
P-value	p-value vs. α	Probability threshold

Pro tip: If software provides an exact or asymptotic p-value, use that result according to the software’s test settings. Don’t treat a table’s critical value as an exact p-value.

Mann-Whitney U Test Assumptions

The Mann-Whitney U test has several important assumptions. Checking them helps you choose and interpret the test correctly.

• Independent observations: Each observation should be independent of the others.
• Independent groups: The two groups should contain different, unrelated observations.
• Appropriate measurement scale: The outcome should be at least ordinal, allowing observations to be meaningfully ranked.
• Distribution shape: If you interpret the test specifically as a comparison of medians or locations, the group distributions should have similar shapes.

The last assumption deserves special attention. The Mann-Whitney U test does not automatically test whether two group medians differ.

When the distributions have different shapes, a significant result can reflect differences beyond their central locations. Therefore, consider the distribution shapes before describing the result as a median difference.

Pro tip: Don’t describe every significant Mann-Whitney result as a difference in medians. Check the distribution shapes first.

Common Mistakes When Using a Mann-Whitney U Table

A small lookup error can lead to the wrong statistical conclusion. Check these common mistakes before interpreting your result.

1. Using the Wrong Sample Sizes

Use the actual number of observations in each group. Don’t accidentally include missing values or unrelated observations.

2. Looking Up the Wrong Critical Value

Match both sample sizes with the correct row and column. Also verify the table’s significance level.

3. Confusing U Statistic With Critical U

Your U statistic comes from your sample data. The critical U comes from the reference table.

4. Using the Wrong Significance Level

A table for α = 0.05 may produce a different cutoff than one for α = 0.01. Always confirm your selected α.

5. Forgetting One-Tailed vs. Two-Tailed Testing

Check whether your hypothesis requires a one-tailed or two-tailed test. The critical value can depend on this choice.

6. Treating Critical U as an Exact P-Value

A critical U value is a decision cutoff, not an exact probability. A p-value requires a separate calculation or statistical software.

Quick check: Before making your decision, verify n₁, n₂, α, test direction, calculated U, and critical U.

Conclusion

The Mann-Whitney U table provides a simple way to interpret a calculated U statistic. By checking your sample sizes and significance level, you can find the correct critical U value.

Remember to confirm whether your test is one-tailed or two-tailed. Then compare your calculated U with the appropriate critical value and follow the table’s decision rule.

You can also use the p-value approach when statistical software performs the test. Both methods help you determine whether the evidence supports rejecting the null hypothesis.

For accurate results, use the correct sample sizes, significance level, and table convention. If you’re unsure about the calculation, a Mann-Whitney U calculator can help verify your result quickly.