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Effect Size Calculator

Calculate effect sizes: Cohen's d and Hedges' g from two means, SDs and sample sizes, or the odds ratio, risk ratio and risk difference from a 2 × 2 table, with intervals.

Formula

d=xˉ1−xˉ2sp;g=d(1−34(n1+n2)−9)d = \dfrac{\bar{x}_1 - \bar{x}_2}{s_p};\quad g = d\left(1 - \dfrac{3}{4(n_1 + n_2) - 9}\right)
OR=adbc;RR=a/(a+b)c/(c+d)\mathrm{OR} = \dfrac{ad}{bc};\quad \mathrm{RR} = \dfrac{a/(a + b)}{c/(c + d)}
interval:ln⁡(ratio)±1.96 SE\text{interval:}\quad \ln(\text{ratio}) \pm 1.96\,\mathrm{SE}
d, gd,\ g
standardised mean difference, without and with the small-sample correction
a, b, c, da,\ b,\ c,\ d
counts in the 2 × 2 table: exposed with and without the outcome, unexposed with and without

How it works

A p-value depends on the sample size, so it cannot say how big an effect is. An effect size can. For a difference between two means, Cohen's d divides the difference by the pooled standard deviation, so it measures the difference in units of the variability; Hedges' g corrects it for small samples. For binary outcomes, the risk ratio and the odds ratio compare how often the outcome occurs in the two groups.

The page also gives the probability of superiority, the chance that a random member of the higher group exceeds one of the other. The odds ratio and risk ratio are not interchangeable: they are close only when the outcome is rare. Cohen's benchmarks (0.2, 0.5, 0.8) are conventions that he himself cautioned against using mechanically.

Worked example

Means 5.4 (SD 0.8, n = 7) and 6.6 (SD 0.7, n = 8); and a table of 30/70 exposed against 15/85 unexposed.

  1. Pooled SD = 0.748, d = −1.605, g = −1.51 (95% CI −2.61 to −0.41); probability of superiority 87.2%.
  2. Risk 30% against 15%: risk ratio 2.00 (1.15 to 3.48); odds ratio 2.43 (1.21 to 4.87).

A large standardised difference (g = −1.51); and an outcome twice as common in the exposed group, with an odds ratio of 2.4.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

  • For d and g, roughly normal values with similar variances; the interval is approximate.
  • For the ratios, independent subjects and no zero cells.
  • An effect size describes association in these data, not causation.

Common mistakes

  • Interpreting an odds ratio as a risk ratio when the outcome is common.
  • Using Cohen's benchmarks without regard to the field.
  • Reporting an effect size without an interval.

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