# U-Test Calculator (Mann-Whitney)

Free U-test (Mann-Whitney) calculator. Compare two independent samples without assuming normality. Get U statistics, z-score, p-value, and rank sums.

## What this calculates

Perform the Mann-Whitney U test to compare two independent samples without assuming the data is normally distributed. This nonparametric test is a common alternative to the independent samples t-test.

## Inputs

- **Sample 1 Count** — min 2, max 10 — Number of values in sample 1 (2-10).
- **S1 Value 1** — Sample 1, value 1.
- **S1 Value 2** — Sample 1, value 2.
- **S1 Value 3** — Sample 1, value 3.
- **S1 Value 4** — Sample 1, value 4.
- **S1 Value 5** — Sample 1, value 5.
- **S1 Value 6** — Sample 1, value 6.
- **S1 Value 7** — Sample 1, value 7.
- **S1 Value 8** — Sample 1, value 8.
- **S1 Value 9** — Sample 1, value 9.
- **S1 Value 10** — Sample 1, value 10.
- **Sample 2 Count** — min 2, max 10 — Number of values in sample 2 (2-10).
- **S2 Value 1** — Sample 2, value 1.
- **S2 Value 2** — Sample 2, value 2.
- **S2 Value 3** — Sample 2, value 3.
- **S2 Value 4** — Sample 2, value 4.
- **S2 Value 5** — Sample 2, value 5.
- **S2 Value 6** — Sample 2, value 6.
- **S2 Value 7** — Sample 2, value 7.
- **S2 Value 8** — Sample 2, value 8.
- **S2 Value 9** — Sample 2, value 9.
- **S2 Value 10** — Sample 2, value 10.
- **Significance Level (α)** — options: 0.01, 0.05, 0.10 — The significance level for the hypothesis test.

## Outputs

- **U₁ (Sample 1)** — U statistic for sample 1.
- **U₂ (Sample 2)** — U statistic for sample 2.
- **U (test statistic)** — The smaller of U₁ and U₂, used as the test statistic.
- **Z-Score (approx.)** — Normal approximation z-score with continuity correction.
- **P-Value (two-tailed)** — Approximate two-sided p-value.
- **Rank Sum (Sample 1)** — Sum of ranks for sample 1.
- **Conclusion** — formatted as text — Whether the difference is statistically significant.

## Details

The U-test (Mann-Whitney U test, also called the Wilcoxon rank-sum test) determines whether two independent groups come from the same distribution. It works by ranking all observations together and comparing the sum of ranks between groups.

**How the U-Test Works:**

1. Combine all values from both samples
2. Rank them from smallest to largest (average ties)
3. Calculate U = n1 x n2 + n1(n1+1)/2 - R1
4. For larger samples, approximate with a z-test using continuity correction

**When to Use the U-Test:**

- Your data is ordinal (ranked) rather than interval
- The normality assumption is violated
- You have outliers that would distort a t-test
- Sample sizes are small and you cannot verify normality

**Interpreting Results:**

A small p-value (below your chosen significance level) means the two groups differ significantly. The U statistic itself indicates how much one group tends to have larger values than the other. When U equals half of n1 x n2, the groups are perfectly balanced.

## Frequently Asked Questions

**Q: What is the difference between the U-test and the t-test?**

A: The t-test compares means and assumes the data is normally distributed. The U-test compares the overall distribution using ranks and makes no normality assumption. When normality holds, the t-test has slightly more statistical power. When it does not, the U-test is more reliable and less sensitive to outliers.

**Q: What does the U statistic represent?**

A: U counts the number of times a value from one sample precedes a value from the other sample in the combined ranking. U1 + U2 always equals n1 x n2. When U is near zero, one group consistently outranks the other. When U is near n1 x n2 / 2, the groups are interleaved.

**Q: Can I use the U-test for paired or matched samples?**

A: No. The U-test is for independent samples only. If your observations are paired (before/after measurements on the same subjects), use the Wilcoxon signed-rank test instead. Using the wrong test can produce misleading results because the pairing structure affects the variance estimate.

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Source: https://vastcalc.com/calculators/statistics/u-test
Category: Statistics
Last updated: 2026-04-08