What the numbers mean
- Cohen’s U3 — the share of the treatment group above the control group’s mean: Φ(d).
- Overlap — the area the two distributions share: 2Φ(−|d|/2).
- Probability of superiority — the chance a random treated person scores higher than a random control person: Φ(d/√2). Also called the common-language effect size.
- Number needed to treat (NNT) — how many people you would treat for one more to do better than without treatment, using Kraemer & Kupfer’s conversion NNT = 1 / (2·P(superiority) − 1).
- Correlation r — the equivalent point-biserial correlation for equal groups: d / √(d² + 4).
| Cohen’s d | U3 | Overlap | P(superiority) | NNT | r |
|---|---|---|---|---|---|
| 0.2 “small” | 57.9% | 92.0% | 55.6% | 8.9 | 0.10 |
| 0.5 “medium” | 69.1% | 80.3% | 63.8% | 3.6 | 0.24 |
| 0.8 “large” | 78.8% | 68.9% | 71.4% | 2.3 | 0.37 |
All values assume two normal distributions with equal standard deviations and equal group sizes, and were checked against SciPy.
Why the “small / medium / large” labels mislead
Cohen proposed 0.2, 0.5 and 0.8 as rough defaults for when nothing better is known. Whether an effect matters depends on context: a d of 0.2 on mortality is enormous, while a d of 0.8 on a lab task may be trivial. Report d with its confidence interval and one of the translations above, and let readers judge. Compute d from your data with the effect size calculator, or read the guide to interpreting effect size.
Frequently asked questions
The two group means are half a standard deviation apart. About 69% of the treatment group scores above the control mean, the two distributions overlap by about 80%, and a randomly chosen treated person beats a randomly chosen control person about 64% of the time.
The chance that a randomly picked member of one group scores higher than a randomly picked member of the other. For two normal groups it equals Phi of d divided by the square root of 2, so d = 0 gives 50%.
The percentage of the treatment group that scores above the mean of the control group. For two normal groups it equals Phi of d.
Using Kraemer and Kupfer's conversion, NNT = 1 / (2 x probability of superiority - 1). It answers how many people you would need to treat for one more to do better than they would have without treatment.