Effect Size Calculator (Cohen’s d, Hedges’ g)

Understanding the Cohen's d and Hedges' g effect size — concept and principle

An effect size tells you how large a difference between two group means is, in units that do not depend on the measurement scale. Cohen's d expresses the gap between two means in standard deviations, and Hedges' g is the same quantity with a small-sample bias correction. Researchers report them next to p-values, use them in meta-analyses and plug them into power calculations. Enter each group's mean, standard deviation and sample size.

How it is calculated

1. Pooled SD: s_p = √[((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)].
2. Cohen's d = (M₁ − M₂) ÷ s_p. The sign shows which group is higher.
3. Hedges' g = d × (1 − 3 ÷ (4(n₁ + n₂) − 9)), a standard approximation of the exact correction factor.
4. The size label uses Cohen's benchmarks on |d|: below 0.2 negligible, 0.2 to under 0.5 small, 0.5 to under 0.8 medium, 0.8 or more large.

Means and SDs must be in the same unit; the result itself is unitless.

The principle

A raw difference of 5 points means little on its own: it is huge if scores barely vary and trivial if they vary by 100. Dividing by the pooled SD, a weighted average of the two group variances, puts every study on a common scale. Because the sample SD slightly underestimates the population SD in small samples, d is biased upward; Hedges (1981) derived the correction factor that removes most of this bias. With 40 per group the correction is under 1%, but it matters with fewer than about 20 per group.

Worked example

1. Group 1: M = 105, SD = 15, n = 40. Group 2: M = 100, SD = 15, n = 40.
2. Pooled SD = √[(39 × 225 + 39 × 225) ÷ 78] = 15.
3. d = (105 − 100) ÷ 15 ≈ 0.3333.
4. Correction factor 1 − 3 ÷ (320 − 9) = 1 − 3 ÷ 311 ≈ 0.99035.
5. g ≈ 0.3333 × 0.99035 ≈ 0.3301.
6. |d| lies between 0.2 and 0.5, so the label is small.

How to read the result

If both groups are roughly normal, d = 0.33 means about 63% of group 1 scores lie above the mean of group 2 (Φ(0.33) ≈ 0.63). Cohen himself described 0.2, 0.5 and 0.8 as rough conventions for when no better guide exists; in education a d of 0.3 can be meaningful, while in other fields it may be negligible. Compare with effects typically found in your own field. Also remember the uncertainty: with 40 per group the 95% interval for d here runs roughly from −0.11 to 0.77.

Common mistakes

• Using standard errors instead of standard deviations.
• Applying this formula to paired or repeated measures data, which needs a different standardiser.
• Treating the small, medium and large labels as fixed rules.

Frequently asked questions

Should I report d or g?

Hedges' g is preferred for small samples and in meta-analysis; with large samples the two are practically identical.

Can d be negative?

Yes. A negative value means mean 2 is higher than mean 1; the size depends on the absolute value.

How does d relate to the t statistic?

For two independent groups, d = t × √(1/n₁ + 1/n₂).

Sources: Cohen, J. Statistical Power Analysis for the Behavioral Sciences, 2nd ed. (1988). · Hedges, L. V. (1981). Journal of Educational Statistics, 6(2), 107–128.