Effect Size Calculator
Cohen’s d for two independent groups, using the pooled standard deviation.
Small effect. Pooled SD = 15; the group means differ by 5 on that scale.
Cohen’s d is (M₁ − M₂) ÷ pooled SD. With means 105 and 100, both SDs 15, and 30 per group, the pooled SD is 15, so d = 5 ÷ 15 = 0.33 — a small-to-medium effect. It scales the difference between two group means into standard-deviation units, so magnitude is comparable across studies.
What effect size measures
A p-value tells you whether a difference is likely real; an effect size tells you how big it is. Cohen’s d expresses the gap between two group means in standard-deviation units, so a d of 0.5 means the groups differ by half a standard deviation regardless of the original scale. Because it strips out the units, d lets you compare and combine results across studies — the reason it anchors meta-analysis and power analysis.
s_pooled = √( ((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2) )
Worked example
Two groups: means 105 and 100, both SDs 15, 30 participants each.
- 1 Pool the two standard deviations. s_pooled = √( ((30−1)·15² + (30−1)·15²) ÷ (30+30−2) ) = √( (6525 + 6525) ÷ 58 ) = √225 = 15.
- 2 Subtract the means. M₁ − M₂ = 105 − 100 = 5.
- 3 Divide by the pooled SD. d = 5 ÷ 15 = 0.333.
- 4 Interpret the magnitude. |d| = 0.33 sits between the 0.2 (small) and 0.5 (medium) benchmarks, so it is a small-to-medium effect.
Cohen’s benchmarks for d
Rules of thumb, not hard cutoffs — interpret in the context of your field.
| |d| | Label | Rough meaning |
|---|---|---|
| 0.2 | Small | means differ by one-fifth of an SD |
| 0.5 | Medium | visible to the naked eye |
| 0.8 | Large | means differ by four-fifths of an SD |
| ≥ 1.2 | Very large | groups barely overlap |
Effect size versus p-value
Significance is not magnitude. A tiny, unimportant difference can be highly significant with a large enough sample, while a big, meaningful difference can miss significance in a small study. Report both: the p-value for whether an effect exists, and d for how large it is.
Sign and direction. The sign of d follows M₁ − M₂; only its absolute value determines the magnitude label. Swap the groups and the sign flips but the size does not.
Assumptions. The pooled-SD version assumes the two groups have similar variances. When spreads differ sharply, prefer Glass’s Δ (using one group’s SD) or Hedges’ g, which corrects the slight upward bias of d in small samples.