What this answers

This calculator answers "how large is the effect behind a Mann-Whitney result, expressed on a familiar correlation-like scale?" for situations where you already have a U statistic and the two group sizes, perhaps from a published paper or another software package's output, but want the accompanying effect size.

How it is calculated

The rank-biserial correlation is 1 minus 2 times U divided by the product of the two group sizes. When U is 0 (complete separation favoring one direction), r equals exactly 1; when U equals the maximum possible value (complete separation the other direction), r equals exactly negative 1; and when U sits exactly at the midpoint of its possible range, r equals 0, indicating no rank-based dominance in either direction. This is the same effect size the Mann-Whitney U Test already reports automatically when you have raw data; this calculator exists for when you only have U and the sample sizes, not the original data.

Worked example

For a U statistic of 0 with 5 observations in each group, the maximum possible U is 25, and r comes out to exactly 1, the strongest possible rank-biserial correlation, indicating every single observation in one group ranked higher than every observation in the other.

Assumption audit

Calculated from your data: the rank-biserial correlation directly from your supplied U and group sizes, and whether U falls within its valid range of 0 to n1 times n2.

Evidence to review: where your U statistic came from; if it was computed with a tie correction or a continuity correction applied differently than this site's own Mann-Whitney engine, the resulting r may not match exactly.

You must verify: that the U value you are entering is the smaller of the two possible U statistics (U for group 1 versus group 2), the standard convention this formula assumes.

What this result does not mean

A rank-biserial correlation describes the direction and strength of rank dominance between two groups, not a difference in means or medians directly, and not a claim about causation between group membership and the outcome.

Limitations

This calculator performs a direct algebraic conversion only; it does not recompute U from raw data or provide a p-value, since a p-value requires the original data or additional distributional assumptions this simple conversion does not have access to. Use the Mann-Whitney U Test directly when you have the raw data.