Calculator Form
Example Data Table
| Item | Sample A | Sample B | Note |
|---|---|---|---|
| Raw values | 12, 14, 15, 17, 19, 20 | 8, 10, 11, 13, 16, 18, 21 | Independent groups |
| Sample sizes | n1 = 6 | n2 = 7 | Total N = 13 |
| Rank result | Rank sum A = 49 | U2 = 14 | U1 = 28 |
| Two sided alpha 0.05 | Lower critical U = 6 | Upper critical U = 36 | Observed smaller U = 14 |
Formula Used
Rank all observations from smallest to largest. Average the ranks for tied observations.
Rank sum for group A is R1. The U statistics are:
U1 = R1 - n1(n1 + 1) / 2
U2 = n1 n2 - U1
Mean U = n1 n2 / 2
Variance without ties = n1 n2(n1 + n2 + 1) / 12
Tie corrected variance = n1 n2 / 12 × [(N + 1) - Σ(t³ - t) / (N(N - 1))]
z = (U - Mean U) / Standard deviation U
Effect size r = |z| / √N
How to Use This Calculator
Enter raw values in both sample boxes when data are available. Separate values with commas, spaces, or line breaks.
Leave raw values empty when you already know n1, n2, and rank sum A. Then enter those values directly.
Select alpha and the test direction. Use two sided when either group may differ. Use one sided only when the direction was planned before analysis.
Press calculate. The result appears above the form. Review U values, critical limits, p values, effect measures, and the decision statement.
Use CSV or PDF export to save the result table for records.
Understanding the Mann Whitney U Test
The Mann Whitney U test compares two independent groups. It is useful when data are ordinal, skewed, or not safely normal. The method ranks all observations together. Then it checks whether one group usually receives higher ranks than the other group.
Why Critical Values Matter
Critical values give a decision limit. A small U value can show strong separation between groups. The calculator finds lower and upper rejection limits for selected alpha levels. It also reports exact tail areas when sample sizes are practical. This helps users avoid rough table lookup.
Advanced Ranking Support
You can enter raw values for both samples. The tool then builds pooled ranks and handles equal values with average ranks. You can also enter sample sizes and a known rank sum. That option is helpful when a textbook problem already gives ranks. The result shows U1, U2, mean, variance, z score, and effect size.
Exact and Approximate Methods
Exact distribution is preferred for small samples without ties. It counts all possible rank arrangements. This gives a direct probability for each U value. Larger samples often use a normal approximation. Tie correction and continuity correction improve that estimate. Both views are shown, so you can compare them.
Practical Interpretation
For a two tailed test, compare the smaller U with the lower critical value. For a one tailed test, use the selected direction. A p value below alpha suggests a statistically significant rank difference. The effect size r explains practical strength. Values near zero show weak separation. Larger absolute values show stronger evidence.
Reporting Results
A good report states group sizes, U statistic, p value, alpha level, and test direction. Mention if ties were present. Also note whether exact or normal results guided the conclusion. The CSV and PDF buttons help save the calculation record. Use them for homework, lab notes, audit files, and peer review.
Use With Care
The test compares rank distributions. It is often treated as a median test, but that is safest when group shapes are similar. Check study design first. Samples must be independent. Measurements should be at least ordinal. Outliers are less damaging here than in a mean based test, yet data quality still matters.
FAQs
What is a Mann Whitney U critical value?
It is the rejection limit for the U statistic. If the observed U crosses that limit, the rank difference is significant at the chosen alpha level.
When should I use exact critical values?
Use exact critical values for small independent samples, especially when no tied ranks exist. They avoid normal approximation error.
Can this calculator handle tied values?
Yes. Raw value mode assigns average ranks for ties. The normal variance can use tie correction. Exact critical values still assume no ties.
Which U value should I compare?
For two sided tests, compare the smaller of U1 and U2 with the lower critical value. For one sided tests, use the selected direction.
What does A tends lower mean?
It means the planned alternative expects sample A to have lower ranks than sample B. This makes small U1 values more extreme.
What does rank biserial correlation show?
It shows the direction and strength of rank separation. Positive values favor higher ranks for sample A. Negative values favor sample B.
Why is exact distribution sometimes skipped?
Exact counting can become heavy for large combined sample sizes. Increase the limit if your server can handle the calculation.
Can I export the result?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a simple printable calculation report.