Effect Size Calculator

A p-value cannot tell you how big something is

This is the single most consequential misreading in applied statistics, and the tool above demonstrates it in one click.

Take a five-point difference between two groups with a standard deviation of 15. Cohen’s d is 0.333, whatever the sample size.

At thirty per group, t = 1.29 and p = 0.20. Not significant; the conventional reading is “no effect found”.

At ten thousand per group, t = 23.57 and p is around 10⁻¹²¹. Overwhelmingly significant. Identical difference, identical spread, opposite conclusions.

The formula says why. t = d × √(n₁n₂/(n₁+n₂)). The effect and the sample size enter as separate factors, so t is a product of “how big” and “how much data”. A p-value is a statement about both at once, and cannot be decomposed back into either.

Two consequences follow, and they cut in opposite directions. A non-significant result from a small study is not evidence of no effect — it is frequently a real effect the study was too small to resolve. And a highly significant result from a large study can be an effect far too small to care about: with a million observations, a d of 0.005 is significant and meaningless.

So report both, always. The p-value says whether chance is a plausible explanation; the effect size says whether the answer matters. Most journals now require it, and the requirement exists because decades of literature reported only the first.

Small, medium and large were provisional

The 0.2 / 0.5 / 0.8 benchmarks are quoted as though they were derived. They were not.

Cohen proposed them for fields with no existing benchmarks, explicitly as a stopgap, and warned against treating them as standards. The wording in his own book is unambiguous about how provisional they were, and it is routinely ignored.

The right comparison is to your own field. In personnel selection a d of 0.2 on a hiring criterion can be worth a great deal of money at scale. In a pharmaceutical trial a d of 0.8 on a surrogate endpoint may mean nothing clinically. The labels have no way of expressing that, because they are labels for numbers rather than for consequences.

The overlap figure is a far better translation, and the tool prints it. At d = 0.2 the distributions overlap by 92%. At the “medium” d = 0.5 they still overlap by 80%. At the “large” d = 0.8, by 69%. Even a large effect describes two groups that are mostly the same people, which is a fact the word “large” actively conceals.

The probability of superiority is the other useful translation. At d = 0.5, a randomly chosen member of the higher group beats a randomly chosen member of the lower one 64% of the time — against 50% for no effect. That is a sentence a non-specialist can weigh, and it is the same number as the effect size.

Report the raw difference too. “Five points on a 100-point scale” is interpretable to anyone who knows the scale. Standardising is what makes effects comparable across studies; it is not what makes them meaningful within one.

Which measure, and when the correction matters

Three standardised mean differences are in common use, and they differ in what they divide by.

Cohen’s d divides by the pooled standard deviation of both groups. The default, and the right choice when both groups are samples of comparable things.

Hedges’ g is d with the small-sample bias removed. d systematically overestimates the population effect when n is small; the correction factor is 1 − 3/(4df − 1). At thirty per group it is 0.987, so d runs about 1.3% high. Below about twenty per group the correction is worth applying; above fifty it is cosmetic. There is no reason not to use g always — it converges to d.

Glass’s delta divides by the control group’s SD only. Use it when the treatment might have changed the variance as well as the mean, which pooling would then contaminate. It is noisier, because it uses half the data to estimate the denominator.

A caution about the pooled SD. Pooling assumes the two groups have similar variances. If they do not — and a treatment that helps some people and harms others produces exactly that — then d is standardising by a quantity that describes neither group. Check the two SDs before pooling them; the tool shows both.

For correlations rather than group differences, r is itself an effect size and the correlation calculator covers it. The tool converts between d and r, which is what meta-analyses need when studies report different designs.

Where the effect size is the input rather than the output

Two applications where you need an effect size before you have any data, which is a genuinely awkward position and worth being honest about.

Power analysis. Deciding how many participants a study needs requires assuming the effect size you are trying to detect. There is no way round it — a study can only be powered for a specific effect, and “powered to detect any effect” is not a thing. Use a published effect from similar work, or the smallest effect that would change a decision, and say which. The sample size calculator takes it as input.

Do not use the effect you observed to justify the sample size you used. Post-hoc power computed from the observed effect is a deterministic function of the p-value and adds no information at all. It is a common and empty ritual.

Meta-analysis. Combining studies requires a common scale, which is precisely what standardising provides. Study-level effect sizes are pooled with weights reflecting their precision, so a large study contributes more — and each study’s d is unaffected by its own size, which is what makes the pooling valid.

Report the confidence interval on the effect size, not just the point. A d of 0.5 with an interval from 0.1 to 0.9 and a d of 0.5 with an interval from 0.45 to 0.55 are different findings, and the point estimate alone hides which you have. This is the same argument the confidence interval calculator makes for means.

Effect sizes for designs that are not two independent groups

Cohen's d answers one shape of question. Using it outside that shape is a common way to report a number that does not mean what it says.

Paired data has its own d, and the two are not interchangeable. For before-and-after measurements on the same subjects, dividing the mean difference by the standard deviation of the differences gives a much larger value than dividing by the pooled between-subject standard deviation — because the pairing removed the between-subject variation. Both are used, both are called Cohen's d, and a meta-analysis mixing them is combining incompatible quantities. Say which you computed.

For a relationship between two continuous variables, r is already an effect size. No conversion needed, and the correlation calculator reports it. Square it before judging strength, for the same reason this page insists on the overlap figure.

For more than two groups, eta squared and omega squared play the role d plays here — the share of variance the grouping accounts for. The ANOVA calculator reports both, and prefers omega squared because eta squared is biased upward.

For two categorical variables, use Cramer's V or an odds ratio. A chi-square statistic is not an effect size: like a p-value it grows with the sample, so a large chi-square on a large table says nothing about how strong the association is.

The common thread is that every one of these is unaffected by sample size, which is what makes them effect sizes. Any quantity that grows as you collect more data — t, F, chi-square, a z-score, a p-value — is measuring certainty, not magnitude.

Sources and methodology

References for the measures and the benchmarks.

• NIST/SEMATECH e-Handbook — two-sample comparisons and the pooled standard deviation
• Hedges, “Distribution Theory for Glass’s Estimator of Effect Size”, Journal of Educational Statistics 6(2), 1981 — the small-sample correction
• Wasserstein & Lazar, “The ASA Statement on p-Values”, The American Statistician 70(2), 2016 — on why a p-value is not a measure of effect size
• P-Value Calculator — the other half of the report, and what it does not say
• Sample Size Calculator — which takes an assumed effect size as its input
• How we calculate — methodology

Method. Cohen’s d is computed from the pooled standard deviation with the n − 1 weighting, and Hedges’ g applies the exact 1 − 3/(4df − 1) correction rather than the √ approximation sometimes quoted. The t statistic is derived from d rather than recomputed from the raw data, so the two can never disagree — which is what makes the sample-size demonstration honest rather than a coincidence of rounding. Overlap and probability of superiority assume normality and equal variances, and the tool shows both group SDs so that assumption can be checked rather than presumed. The suite asserts that d is invariant to sample size while t is not — the same d of 1/3 giving t = 1.2910 at thirty per group and 23.5702 at ten thousand — that Hedges’ g converges to d as n grows, that d and r convert in both directions, and that the overlap at d = 0.5 is 80.26%. That engine is verified on every change against 49 assertions. The count and the per-case breakdown are published on the formula verification page.

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