An observed Cohen's d of 0.50 from a study with 64 people in each group has an exact 95% confidence interval of 0.15 to 0.85, on our computation. Sixty-four per group is the size Cohen gave in 1992 for 80% power to detect a medium effect at a two-sided α of .05. So a study of exactly that size, returning exactly his "medium", is still compatible with anything from 0.15, below his "small", to 0.85, above his "large". The label describes a point; the interval describes how loosely the data hold it.
All three common effect sizes divide the gap between two group means by a standard deviation, a measure of spread. Cohen's d uses the pooled SD, the two groups' spreads combined with each weighted by its size minus one, as the NIST/SEMATECH e-Handbook does for the two-sample t test. Glass's Δ divides by the control group's SD alone. Hedges' g is d multiplied by a factor that offsets d's tendency to overstate in small samples. A d of 0.5 means the difference between the means equals half a standard deviation.
An exact interval tempers a medium d even at 64 per group
We computed exact 95% intervals for an observed d of 0.50 by inverting the noncentral t distribution, assuming normal data and equal population SDs. Our guide to confidence intervals covers what the 95% does and does not promise.
| People per group | Exact 95% interval for d = 0.50 | Cohen's labels the interval spans |
|---|---|---|
| 20 | −0.13 to 1.13 | below zero through large |
| 64 | 0.15 to 0.85 | below small through large |
| 100 | 0.22 to 0.78 | small through medium |
| 400 | 0.36 to 0.64 | small through medium |
Thirty-two per group is the first size at which the interval for an observed 0.50 lies wholly above zero, and only just. For the whole interval to stay inside one of Cohen's bands, the required sample size is much larger, even in the best case where the estimate lands mid-band: 348 per group for an observed 0.35, the middle of "small", and 361 for an observed 0.65, the middle of "medium". An estimate nearer a boundary takes more, and "large" has no upper edge to fit inside.
Hedges' factor trims a bias that roughly halves as the groups double
In a 1981 paper in the Journal of Educational Statistics, L. V. Hedges showed that the expected value of d is the true effect divided by a factor c(m), slightly below one, where m is the degrees of freedom behind the SD (for the pooled SD, the two group sizes added, minus two). The result is a systematic upward bias rather than random scatter. Our figures, for normal data with equal SDs and equal groups:
| People per group | d overstates the true effect by | Glass's Δ overstates it by |
|---|---|---|
| 5 | 10.8% | 25.3% |
| 10 | 4.4% | 9.4% |
| 20 | 2.0% | 4.2% |
| 30 | 1.3% | 2.7% |
No single size marks a break: going from 10 to 20 to 40 per group takes d's bias from 4.4% to 2.0% to just under 1%. Hedges' own example, five per group with the SD taken from one group, overstated by "approximately 25 percent"; we get 25.33%. The familiar shortcut, c ≈ 1 − 3/(4m − 1), is his approximation, and in effect terms it overshoots the exact correction by 0.058% at 5 per group.
Later usage amended Hedges' own meaning of g
In the 1981 paper, "g" is Glass's estimator, the biased one, and g′ the pooled version; the corrected estimator is written gᵁ. What is now called Hedges' g is that gᵁ. The Cochrane Handbook, chapter 6, defines the standardised mean difference (SMD) in its reviews as "the effect size known in social science as Hedges' (adjusted) g". Lakens' 2013 tutorial adds that d "is sometimes referred to as Cohen's g, which can be confusing". Cohen's 1992 table uses g for a third, unrelated index built on a sign test.
"Unbiased" carries conditions too: the exact c(m), normal data and equal population SDs. Lakens notes that such corrections "do not always lead to a completely unbiased effect size estimate".
Pooled and control-group SDs rescale the same gap into different labels
Glass's reason for the control SD, as Hedges reports it, was that "pooling two variances could lead to different standardized values of the identical mean difference" when several treatments share one control. Cochrane gives another: if an intervention changes between-person variation, "such an impact of the intervention should not influence the effect estimate."
With equal groups, Δ divided by d equals the pooled SD over the control SD. Where the treatment group's SD is half the control's, Δ comes out at 0.79 times d; where it is double, 1.58 times. A gain of 0.8 control-group SDs with a treatment SD twice the control's, 30 per group, gives Δ = 0.80, "large", and d = 0.51, "medium", from one mean difference. Δ is also the more biased, because its SD rests on one group: about twice d's bias from about ten per group, as the table above shows.
The quick correlation shortcut comes calibrated for equal groups
Converting d to a correlation with r = d/√(d² + 4) assumes groups of the same size. Cohen's statement that a d of .50 "is equivalent to a point-biserial correlation of .243" is that equal-groups case. The point-biserial correlation, between the outcome and group membership, exists for any split, and Lakens gives the general formula. At d = 0.5 it is 0.246 with 30 people against 30, close to the shortcut's 0.243. With 10 against 90 it falls to 0.150, and with 5 against 95 to 0.109, where the shortcut is 2.22 times too high.
Until 12 October 2026 our effect size calculator showed the shortcut r for every input, 1.62 times the point-biserial for that same d at 10 against 90. It labelled d rather than g and gave no interval. Otherwise its arithmetic was exact: across 420 input combinations run against an independent implementation, it matched every one. The revision of that date computes the point-biserial for any split, prints the exact interval for d and labels g against Cohen's three conventions only. For an observed 0.50 at 64 per group, the correction alone moves the revised label to "small", with g at 0.497.
Cohen qualified his own benchmarks as made subjectively
In "A power primer" (1992), Cohen wrote: "My intent was that medium ES represent an effect likely to be visible to the naked eye of a careful observer." He placed "small" and "large" an equal distance to either side of it. He added: "Although the definitions were made subjectively, with some early minor adjustments, these conventions have been fixed since the 1977 edition" of his textbook. Lakens calls the values "arbitrary".
In the two fields those reviews covered, measured effects sat lower on the ladder. Lovakov and Agadullina, in the abstract of their 2021 paper, report 6,447 values of d from studies in 134 social-psychology meta-analyses. The median was 0.36, the middle half ran from 0.15 to 0.65, and the authors conclude that Cohen's guidelines "tend to overestimate medium and large effect sizes". For group differences in gerontology, Brydges' 2019 abstract gives a median Hedges' g of 0.38, with the middle half from 0.16 to 0.76.
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