METHOD GUIDE
Mann-Whitney U Test in R
W in the Output, Exact p-Values and Effect Size
Base R runs this comparison through wilcox.test(), whose statistic is labeled W and whose p-value is exact in some calls and approximate in others. Which group W describes and which kind of p-value the header names decide how the result can be worded.
IN SHORT
What wilcox.test() Compares and What Goes in the Report
A Mann-Whitney U test examines whether observations from one of two independent groups tend to exceed those of the other, using the ranks of the pooled outcome values. It suits an ordinal outcome, and a continuous one when a comparison of means is not what the study needs. To run the Mann-Whitney U test in R, call wilcox.test() with the outcome and a two-level grouping factor, read W with the p-value and the method named in the header, and add an effect size. OnlineSPSS, a specialist statistical consultancy for academic, clinical and professional research, carries out and reviews this analysis through an in-house PhD-level statistician.
ANALYSIS SUPPORT
Analysis Support for This Method
Ranks and means answer different questions, and a script can produce either in one line, so which line belongs in the report follows from the question and the outcome scale.
Mann-Whitney U Test
The rank-based comparison of two independent groups, carried out in R or another program, with the shape check, the effect size and the wording agreed for your data.
Independent Samples t-Test
A comparison of two independent group means on a continuous outcome, for when the size of the mean difference and its precision are what the study needs.
BEFORE YOU RUN IT
What to Fix Before Calling wilcox.test()
Whether the result will describe a shift, a tendency to score higher or a median gap is fixed by the design and the outcome scale, not by anything in the call.
Unrelated Cases Each case belongs to one group and is measured once. Paired data take the paired form of the same function, and clustered data need a model that allows for the clustering.
An Outcome That Ranks Ties are frequent on rating scales and change how the p-value is computed. With a continuous outcome, ranks set aside magnitude that a comparison of means would use.
Wording the Claim in Advance A statement about medians is supported only when both distributions have a similar shape. Otherwise write that values in one group tend to be larger, which is what the ranks test.
Exact or Approximate p By its documented default, wilcox.test() computes an exact p only for fewer than 50 observations per group without ties, and otherwise approximates. Decide which to report and read any warning.
IN R
How to Run the Mann-Whitney U Test in R
Base R provides wilcox.test(), while the rank effect size needs an add-on package, whose name and version belong in the methods text.
- Confirm the design: an ordinal or continuous outcome, two independent groups, one row per case.
- Make the grouping variable a factor and set the order of its levels, because W refers to the first level.
- Plot the outcome in each group, for example with a histogram per group, and compare shape and spread.
- Run wilcox.test in R as wilcox.test(outcome ~ group, data = d, conf.int = TRUE); add exact = TRUE or FALSE only when the default is not what you want.
- Read W, the p-value, the method in the header and any warning about ties, then the estimate and its interval.
- Obtain the rank-biserial correlation in R, for example with rank_biserial() in the effectsize package, and keep the plot.
READING THE OUTPUT
W, Warnings and the Method Line in wilcox.test()
The same call can print an exact test or a corrected approximation, and the header line and any warning beneath it decide which p-value you are holding.
| Output element | What it tells you | What to check |
|---|---|---|
| Header line | Wilcoxon rank sum exact test, or Wilcoxon rank sum test with continuity correction | That the report names the same method |
| W | The rank sum of the first group minus its smallest possible value, n1(n1 + 1)/2, which equals that group’s Mann-Whitney U | That the first group is named, since the other U is n1 times n2 minus W |
| p-value | The exact p or the normal-approximation p, according to the header | That an approximate p is not described as exact |
| Warning about ties | R could not compute an exact p because values were tied | That the text states the approximation and why it was used |
| Alternative hypothesis | A location shift not equal to zero, in wording that differs by version | That a shift is read as a median gap only for similar shapes |
| Difference in location | With conf.int = TRUE, a Hodges-Lehmann estimate with an interval | That it is called a shift estimate and its unit is stated |
| Effect size output | A rank-based effect from an add-on package, with its interval | That the sign is checked against the order of the groups |
REPORTING
What to Report Beside W and p
W says little without the group it belongs to and the way its p-value was computed, so both belong in the first sentence.
Write W = ___, n1 = ___ and n2 = ___, the p-value, and whether it is exact or a normal approximation with continuity correction. Name the group W belongs to. Add the rank-biserial r = ___ with its 95% CI [___, ___], noting that it equals twice the probability of superiority minus 1 and that its sign follows the group order. Give the probability of superiority, W divided by n1 times n2, if readers need a plain reading, with ties counted as half. Describe each group with a median and an interquartile range, and show the distributions, so that readers can judge shape for themselves. State the n of each group and any cases excluded. The interval printed by wilcox.test is for a location shift, so mention medians in the conclusion only when the shapes were similar.
COMMON MISTAKES
Labels and Claims That Skew a Rank-Sum Report
Three of the six trace back to labels R prints, the header, W and the warning, and the other three to choices about the claim.
Median Gap Claimed A significant W is reported as a difference in medians. That holds only for distributions of similar shape, and groups with equal medians can still differ in ranks when spreads differ.
Wrong Group for W W belongs to the first factor level. Describing it as the statistic of the higher group, or as the smaller of the two U values, mislabels it unless the level order was checked.
Ties Warning Ignored The warning that an exact p-value cannot be computed means an approximation was used. Reporting the p-value as exact, or leaving the warning out of the log, misdescribes the method.
Interval Misnamed With conf.int = TRUE the interval refers to a location shift under a shift model, not to the difference between medians, so the label in the text should say shift.
Paired Data Run as Groups Siblings, repeated visits or matched cases make observations related while the two-sample call treats them as unrelated. The design shows this, and a paired analysis is the match.
Ranks Chosen After the Fact Moving to ranks because a t-test gave a larger p is a choice made after seeing the data. Make the choice from the outcome scale and the question, and record it in the analysis plan.
W reported in a draft? Independent Statistical Review checks the method line, the p-value you reported and the wording of the conclusion before you submit.
FREQUENTLY ASKED QUESTIONS
Questions About W and the Rank Output in R
W is easy to compute and easy to misattribute, so the first two answers tie it to a group and to a probability before the interval is explained.
Which group does W refer to in wilcox.test()?
W is the Mann-Whitney U of the first group, meaning the first level of the grouping factor. The U of the other group is n1 times n2 minus W, and the two sum to that product. Check the level order with levels() before writing which group scored higher, because reversing the order changes W but not the p-value.
How do I turn W into the probability of superiority?
Divide W by the product of the two group sizes. The result is the probability that a randomly chosen case from the first group exceeds one from the second, with ties counted as half. A value of 0.5 means no tendency in either direction, and the rank-biserial coefficient is twice the value minus 1.
Does wilcox.test() give an interval, and for what?
With conf.int = TRUE it returns a Hodges-Lehmann estimate of the location shift with an interval. That quantity equals a difference in medians only under special conditions, so name it a shift estimate, and report medians and quartiles for each group separately, with the plot that shows their shapes.
Does OnlineSPSS check which group W and the effect size describe?
Yes. An in-house PhD-level statistician can confirm the level order behind W, the sign of the rank effect size and whether a statement about medians is supported, or run the comparison on your data. When the analysis is final and the reporting is the open question, Results Reporting & Interpretation is the service to consider. How the finding bears on your hypothesis is for you to decide.
How do you protect my research data and confidentiality?
Your data and project details are handled confidentially, and project information is not published. Researcher names and institutions are not used publicly without permission. An NDA is available where required and can be arranged before sensitive materials are shared. Read more about confidentiality and data security.
What happens when I request an assessment, and how long does it take?
You share your research question, design and relevant materials, such as the dataset or the output you have, and a statistically trained Research Coordinator reviews the request and responds the same day. The analysis, deliverables, timeline and one fixed project fee are then confirmed before any work begins, so the timeline follows from the agreed scope.
Still have a question? Find more answers in our full FAQ.
NEXT STEP
Not Sure What a Significant
W Lets You Claim?
Whether a significant W supports a statement about medians, a shift or only a tendency depends on the shapes of the two groups, and that judgment can form part of an agreed review of your script and wording.
No Commitment Required
Strictly Confidential
Fixed Project Fee
In-House PhD-Level Statisticians
Same-Day Response
Serving Researchers Since 2014