Effect size calculator

What is Cohen’s d?

The difference between two group means divided by the pooled standard deviation: d = (M₁ − M₂) ÷ s_pooled. It says how many standard deviations apart the groups are. Because it is unitless, effects measured on different scales can be compared, and it does not grow just because the samples are large.

How is the pooled SD calculated?

s_pooled = √{[(n₁ − 1)s₁² + (n₂ − 1)s₂²] ÷ (n₁ + n₂ − 2)} — a weighted average of the two variances, so the larger group counts for more. Enter standard deviations, not standard errors: a standard error is smaller by a factor of √n and would inflate d.

What counts as a small or large effect?

Cohen suggested 0.2 as small, 0.5 as medium and 0.8 as large, as rough guides when nothing better is known for the field. Lakens (2013) points out that these cut-offs are arbitrary and should not be applied rigidly: a small effect can matter a lot (a d of 0.1 in suicide prevention, for example), and the better habit is to compare your d with effects in the same research area and to explain what it means in practice.

What is Hedges’ g?

Cohen’s d with a small-sample correction: g = d × [1 − 3 ÷ (4(n₁ + n₂) − 9)]. It matters most for small groups. According to Lakens, d slightly overestimates the population effect, especially when there are fewer than about 20 observations per group, and g removes that bias; above that size the two are almost the same.

Worked example, step by step

Group 1 has mean 20, SD 4 and n = 30; group 2 has mean 18, SD 5 and n = 35. The pooled variance is (29 × 16 + 34 × 25) ÷ 63 = 20.86, so the pooled SD is 4.57. The mean difference is 2, so d = 2 ÷ 4.57 = 0.44. The correction factor is 1 − 3 ÷ (4 × 65 − 9) = 0.9880, which gives g = 0.43. Read against Cohen’s guides, the groups sit a little under half a standard deviation apart: between small and medium.

How is d related to the t-test?

For two independent groups with the pooled variance, t = d × √[n₁n₂ ÷ (n₁ + n₂)]. In the example, t = 0.44 × 4.02 = 1.76 with 63 degrees of freedom, p = 0.083. The p-value depends on the sample size and d does not: with 6,000 people per group, means of 20 and 19.8 and SDs of 4 and 5 give d = 0.04, a trivial effect, yet t = 2.42 and p = 0.016. Report both the p-value and the effect size.

Common mistakes with effect sizes

First, using the benchmarks as a verdict. Second, mixing up standard deviation and standard error. Third, pooling when the standard deviations differ a lot between groups, for example after a treatment that changes the spread; Lakens recommends Glass’s Δ, which divides by the control group’s SD, whenever the SDs differ substantially. Fourth, using this calculator for before-and-after data on the same people: it assumes two independent groups, and versions of d for paired designs use a different denominator. Fifth, forgetting the sign: a negative d only means group 2 is higher.

What sample size do I need for a given d?

It depends on the power you want. For a two-sided test at α = 0.05 and 80% power, a medium effect (d = 0.5) needs 64 people per group. Smaller effects need many more, since the sample size grows roughly with 1 ÷ d². Use the statistical power calculator to plan a study before collecting data.

Sources: Lakens, D. (2013), “Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs”, Frontiers in Psychology 4:863; Cohen, J. (1988), Statistical Power Analysis for the Behavioral Sciences, 2nd ed. (benchmarks 0.2, 0.5, 0.8, as cited by Lakens). Related: t-test calculator, statistical power calculator, standard deviation calculator, standard error calculator and paired t-test calculator.