Calculator Input
This calculator measures ordinal dominance between two independent groups using the rank-biserial correlation and related Mann-Whitney statistics.
Example Data Table
| Observation | Sample Group A | Sample Group B | Comment |
|---|---|---|---|
| 1 | 18 | 12 | Group A exceeds Group B. |
| 2 | 22 | 14 | Strong separation remains visible. |
| 3 | 24 | 16 | Ordinal dominance continues. |
| 4 | 27 | 21 | Difference narrows slightly. |
| 5 | 29 | 23 | Ranks still favor Group A. |
| 6 | 31 | 25 | Useful for non-normal data. |
| 7 | 35 | 26 | Pairwise wins accumulate. |
| 8 | 39 | 30 | Likely positive rank-biserial value. |
You can replace these values with your own independent samples and rerun the calculator instantly.
Formula Used
U₁ = R₁ - n₁(n₁ + 1) / 2R₁ is the sum of average ranks for Group A.
rrb = 2U₁ / (n₁n₂) - 1rrb = (Wins - Losses) / (n₁n₂)
CLES = (Wins + 0.5 × Ties) / (n₁n₂)This estimates the probability that a random Group A observation exceeds a random Group B observation.
Var(U) = (n₁n₂ / 12) × [(N + 1) - Σ(t³ - t) / (N(N - 1))]The p-value shown uses a normal approximation with tie correction and continuity adjustment.
Positive values mean Group A generally ranks higher. Negative values mean Group B generally ranks higher. Values near zero suggest little ordinal separation.
How to Use This Calculator
- Enter labels for the two independent groups.
- Paste numeric observations for each group into the text areas.
- Choose an alpha level for the bootstrap confidence interval.
- Set bootstrap resamples and display precision.
- Press Calculate Correlation to show results below the header and above the form.
- Review the rank-biserial value, U statistics, dominance counts, p-value, and confidence interval.
- Inspect the Plotly graphs to compare distributions and pairwise outcomes.
- Download the current analysis as CSV or PDF.
Frequently Asked Questions
1) What does rank-biserial correlation measure?
It measures how strongly one independent group tends to rank above another. A positive value favors Group A, a negative value favors Group B, and values near zero indicate weak separation.
2) When should I use this instead of Pearson correlation?
Use it for two independent groups when the outcome is ordinal, skewed, or non-normal. It is especially useful when comparing ranked or continuous values without assuming equal variances or normal distributions.
3) Is rank-biserial correlation related to Mann-Whitney U?
Yes. It can be derived directly from the Mann-Whitney U statistic. That link makes it a practical effect size for nonparametric two-group comparisons.
4) How should I interpret a value of 0.60?
A value of 0.60 suggests strong dominance for Group A. Many pairwise comparisons place Group A above Group B, so the ordinal separation is substantial in practical terms.
5) What do ties do to the calculation?
Ties reduce directional dominance because equal comparisons contribute half weight in the common language estimate. The p-value calculation also applies a tie correction to the variance of U.
6) Can I use unequal sample sizes?
Yes. The two groups do not need equal sizes. The calculator handles different sample counts as long as both groups are independent and contain at least two observations.
7) Why include a bootstrap confidence interval?
The bootstrap interval gives a practical uncertainty range for the effect size. It is helpful when sample sizes are modest or when distributional assumptions are hard to justify.
8) Does a significant p-value guarantee a large effect?
No. Statistical significance and effect size answer different questions. A result can be significant yet small, or practically meaningful without crossing a chosen significance threshold.