Cohen’s d and Hedges’ g calculator with confidence intervals

A p value tells you whether a difference is detectable; an effect size tells you whether it is large. With huge samples, trivial differences come out significant, and with small samples an important effect can be missed. That is why journals ask for an effect size every time and, increasingly, with its confidence interval.

Paste the data of two groups (or of the same people before and after), or type the means, SDs and sample sizes from a paper. You get Cohen’s d, Hedges’ g with its interval, the easier-to-grasp equivalents and an APA-style sentence for your write-up. Everything is calculated in your browser.

The difference in means is divided by a standard deviation: the pooled SD of the two groups (independent design) or the SD of the differences (before and after). The result is in standard-deviation units, so it is comparable across studies that use different scales.

Hedges’ g corrects the small-sample bias of d. The confidence interval comes from inverting the noncentral t distribution, which is the exact method and the one style guides recommend; it is not the rough “±1.96 standard errors” some spreadsheets use.

What an effect size is

A result can be statistically significant and practically irrelevant. The p value depends on sample size; the effect size does not. It measures how far apart two groups are in standard-deviation units, which is why it can be compared across studies and pooled in a meta-analysis.

Cohen’s d, Hedges’ g, Glass’s Δ, d_z and d_av

• **Cohen’s d**: the difference in means divided by the pooled SD. The standard measure for two independent groups.
• **Hedges’ g**: the same d with a small correction that removes its small-sample bias. With more than about 20 cases per group they barely differ.
• **Glass’s Δ**: uses only the control group’s SD. Preferable if the treatment changes the spread.
• **d_z**: in a paired design, the mean of the differences divided by their SD. It grows when the two measures are highly correlated.
• **d_av**: in a paired design, the difference divided by the average of the two SDs. More comparable with d from independent groups.

Why the confidence interval matters

An effect size without an interval conveys false precision. With 10 cases per group, a d of 0.8 can, with the same confidence, be anything from almost nothing to enormous. The interval shows it, and reporting standards (APA 7, CONSORT) recommend including it.

How to interpret the magnitude

Cohen’s benchmarks (0.2 small, 0.5 medium, 0.8 large) are a last-resort guide: Cohen himself asked for them to be used only in the absence of anything better. Read the effect in context: in education, 0.4 is considered notable; in medicine, a “small” effect on mortality can save many lives. The probability of superiority helps explain it to someone who does not know d.

Common mistakes

• Reporting only the p value, with no effect size.
• Presenting the d of a paired design as if it were comparable with that of independent groups.
• Using Cohen’s benchmarks as an automatic rule, with no context.
• Forgetting the sign: it shows which group has the higher mean.
• Overlooking that d is biased with small samples: use g.
• Computing the interval with an approximate formula when there are few cases.

Limits of this calculator

• It assumes roughly normal data and, in the independent design, similar spread; if not, Glass’s Δ or a nonparametric test may be more suitable.
• The probability of superiority and the overlap are equivalents that assume normality.
• For three or more groups, use the ANOVA calculator (η² and ω²).
• It accepts up to 5,000 values per group.

Notice

This is an estimate. For publication, check the figures with your usual statistics package and cite the method.