Effect Size Calculator (Cohen's d + Hedges' g)
Effect size calculator. Cohen's d with its exact 95% confidence interval, Hedges' g, Glass's Δ and the point-biserial r, labelled by Cohen's conventions read from g.
Effect Size
| Hedges' g, bias-corrected: c(m) × dHedges (1981) writes this corrected value gᵁ and uses g for the uncorrected one; Cochrane calls it "Hedges' (adjusted) g" | — |
| 95% CI for gHedges (1981) normal approximation, variance 1/ñ + g²/(2m) | — |
| Correction factor c(m)Exact, from the gamma function; m = n₁ + n₂ − 2 | — |
| Glass's ΔGroup 2 (control) SD only, uncorrected | — |
| Glass's Δ, bias-correctedc(n₂ − 1) × Δ | — |
| r, point-biseriald / √(d² + N(N − 2)/(n₁n₂)), valid for unequal groups | — |
| Pooled SD | — |
Labels use Cohen's conventions, 0.2 small, 0.5 medium and 0.8 large, read from g. Cohen wrote in 1992 that "the definitions were made subjectively". Both intervals assume normally distributed data with equal population SDs.
How to use the effect size calculator
Enter the mean, SD and size of each group
You need the mean, standard deviation and number of people for both groups. Group 1 is usually the treatment and Group 2 the control. Order only sets the sign: a positive d means Group 1's mean is higher. Every field is required. A blank field, a negative SD, or a group size that is not a whole number of at least 2 is refused rather than guessed.
Read Cohen's d and its interval
d is (M₁ − M₂) divided by the pooled SD, so d = 0.5 means the difference between the means equals half a standard deviation. The 95% confidence interval under it is exact: it inverts the noncentral t distribution, assuming normal data and equal population SDs. At the default inputs d is 0.333 and the interval runs from −0.18 to 0.84.
Read Hedges' g in small samples
d runs high in small samples. With normal data and equal SDs it overstates the population effect by 10.8% at 5 per group, 4.4% at 10 and 1.3% at 30, which follows from Hedges' (1981) result that its expected value is δ / c(m). Hedges' g multiplies d by c(m) and removes that bias. The calculator uses the exact c(m); the familiar shortcut 1 − 3 / (4N − 9) differs from it by at most 0.06% from 5 per group up. The size label is read from g.
Glass's Δ divides by the control SD
Glass's Δ = (M₁ − M₂) / s₂. Hedges (1981) reports Glass's reason: pooling "could lead to different standardized values of the identical mean difference" when several treatments are compared with one control. The Cochrane Handbook gives another: if an intervention changes between-person variation, that should not change the effect estimate. Δ is biased too, and more than d, because its SD rests on n₂ − 1 degrees of freedom: 9.4% high at 10 per group, against 4.4% for d. The calculator shows it corrected by c(n₂ − 1) as well.
Report the size alongside any p-value
The American Statistical Association's 2016 statement says: "A p-value, or statistical significance, does not measure the size of an effect or the importance of a result." An effect size with its interval answers the question a p-value cannot.
Effect size — magnitude vs significance, and why both matter
A p-value measures how incompatible the data are with a model of no difference. The American Statistical Association's 2016 statement adds that p-values "do not measure the probability that the studied hypothesis is true", and that they do not measure the size of an effect either. Size is a separate question. Take a blood-pressure trial with 25,000 people per arm, an outcome SD of 15 mmHg and a 0.5 mmHg difference between arms. There d is 0.033, and its 95% interval of 0.02 to 0.05 sits clear of zero yet far below Cohen's "small". Turn it round: a pilot with 15 per arm, a 10-point IQ difference and an SD of 15 has d = 0.667, with an interval from −0.08 to 1.40. That is a sizeable estimate, but the data have not ruled out zero.
The Cohen's d family
Cohen's d divides the difference in means by the pooled SD, weighting each group's variance by n − 1 as the NIST/SEMATECH e-Handbook does for the two-sample t test. Hedges' g multiplies d by the factor c(m) from Hedges (1981), which removes d's upward bias in small samples. The naming needs care: Hedges' own paper calls the uncorrected estimator g and the corrected one gᵁ, and Lakens (2013) notes that d "is sometimes referred to as Cohen's g". The Cochrane Handbook settles on "Hedges' (adjusted) g" for the corrected value, which is what this calculator labels g. Glass's Δ divides by the control group's SD alone. The correlation form is the point-biserial r = d / √(d² + N(N − 2)/(n₁n₂)) (Lakens 2013, eq. 3). With large equal groups this tends to d / √(d² + 4), and Cohen (1992) gives d = .50 as r = .243. With unequal groups the shortcut runs high: at 10 against 90 people, d = 0.5 is r = 0.150, not 0.243.
"My intent was that medium ES represent an effect likely to be visible to the naked eye of a careful observer." — Cohen J. (1992) "A power primer", Psychological Bulletin 112(1), 155–159.
Cohen's benchmarks — useful, but contextual
Cohen set 0.2 (small), 0.5 (medium) and 0.8 (large) in his Statistical Power Analysis for the Behavioral Sciences, first published in 1969 and in its second edition in 1988. In 1992 he wrote that "the definitions were made subjectively" and had been fixed since the 1977 edition, with small set noticeably below medium and large the same distance above it. Measured effects are often smaller than his ladder suggests. Lovakov and Agadullina (2021) collected 6,447 values of d from studies in 134 social-psychology meta-analyses and report, in their abstract, 25th, 50th and 75th percentiles of 0.15, 0.36 and 0.65. Brydges (2019) reports 0.16, 0.38 and 0.76 for Hedges' g in gerontology. Both conclude that Cohen's guidelines overestimate the effects typical of their field.
Cohen's d, Hedges' g and Glass's Δ — and what a p-value cannot tell you
Cohen's d = (M₁ − M₂) / pooled SD, with each group's variance weighted by n − 1 (NIST/SEMATECH §7.3.1).
Cohen's labels, 0.2 small · 0.5 medium · 0.8 large, "were made subjectively", he wrote in 1992.
Uncorrected d runs 4.4% high at 10 per group and 1.3% high at 30, with normal data and equal SDs.
Hedges' 1981 paper calls the biased estimator g and the corrected one gᵁ, the reverse of today's usage. Cochrane writes "Hedges' (adjusted) g".
Glass's Δ runs 9.4% high at 10 per group, more than d, because its SD has only n₂ − 1 degrees of freedom.
At 64 per group, Cohen's 1992 figure for 80% power at a medium d with a two-sided α of .05, an observed d of 0.50 has an exact 95% interval of 0.15 to 0.85.
d = .50 is r = .243 for large equal groups (Cohen 1992). With 10 against 90 people the same d is r = 0.150.
A p-value measures neither the probability that a hypothesis is true nor the size of an effect (ASA 2016, principles 2 and 5).
Across 6,447 values of d from 134 social-psychology meta-analyses, the median was 0.36 (Lovakov & Agadullina 2021).
For group differences in gerontology research the median Hedges' g was 0.38 (Brydges 2019).
Frequently asked questions
Both divide the mean difference by the pooled SD. g then multiplies d by Hedges' correction c(m), which removes d's upward bias when the data are normal with equal SDs. That bias is 10.8% at 5 per group, 4.4% at 10, 1.3% at 30 and 0.4% at 100. Cochrane reviews use the corrected form, which the Handbook calls "Hedges' (adjusted) g". The calculator shows both and reads the size label from g.
A weighted average of the two groups' variances: SD_pooled = √[((n₁−1)·s₁² + (n₂−1)·s₂²) / (n₁ + n₂ − 2)]. It is the quantity the NIST/SEMATECH e-Handbook uses for the two-sample t test, which assumes the two populations have equal variance.
The Cochrane Handbook says Glass's Δ uses only the comparator group's SD "on the basis that if the experimental intervention affects between-person variation, then such an impact of the intervention should not influence the effect estimate". Glass's original reason, as Hedges (1981) reports it, was that several treatments compared with one control should not get different standardised values for the same mean difference. Δ needs its own small-sample correction, which the calculator applies with c(n₂ − 1).
No, and Cohen did not present them that way. In 1992 he wrote that medium was meant as an effect "visible to the naked eye of a careful observer", that small and large were set either side of it, and that "the definitions were made subjectively". Measured distributions run lower: the abstracts of Lovakov and Agadullina (2021) and Brydges (2019) report medians of d = 0.36 in social psychology and g = 0.38 in gerontology.
Because one study's d is an estimate. For an observed d of 0.50, the exact 95% interval is −0.40 to 1.38 with 10 per group, 0.15 to 0.85 with 64 per group and 0.36 to 0.64 with 400 per group. At 64 per group, the sample Cohen (1992) gives for 80% power at a medium effect with a two-sided α of .05, the interval still runs from below "small" to "large".
For two normal distributions with equal SDs, d = 0.5 leaves about 33% of the combined area non-overlapping, d = 0.8 about 47% and d = 2.0 about 81%. Put another way, at d = 0.5 the average member of Group 1 scores above 69% of Group 2.
d is a measure for continuous outcomes, and both intervals here assume normal data with equal population SDs. For yes/no outcomes, chapter 6 of the Cochrane Handbook covers ratio and difference measures instead.
Yes. d = t · √(1/n₁ + 1/n₂) for independent samples (Lakens 2013, eq. 2). Useful when a paper reports only t and the group sizes.
No. All inputs stay in your browser.
Cohen J. (1992) "A power primer", Psychological Bulletin 112(1). Hedges L.V. (1981) "Distribution theory for Glass's estimator of effect size and related estimators", Journal of Educational Statistics 6(2). Lakens D. (2013) "Calculating and reporting effect sizes to facilitate cumulative science", Frontiers in Psychology 4:863. Cochrane Handbook, chapter 6. ASA Statement on Statistical Significance and P-Values (2016).
Method & sources
How it computes
d = (M₁ − M₂) ÷ pooled SD (n − 1 weights). g = c(m)·d with Hedges' (1981) exact c(m), m = n₁ + n₂ − 2; Glass's Δ = (M₁ − M₂) ÷ s₂, also shown as c(n₂ − 1)·Δ. 95% CI for d: exact, by inverting the noncentral t; for g: Hedges' normal approximation. r = d ÷ √(d² + N(N − 2)/(n₁n₂)). Label read from g against Cohen's 0.2/0.5/0.8. Blank, negative-SD, non-whole or under-2 inputs and two zero SDs are refused.
What this tool implements
- Pooled SD with n − 1 weights, as NIST/SEMATECH §7.3.1 gives it for the two-sample t test and Lakens (2013) eq. 1 for d.
- Hedges' g uses the exact factor c(m) = Γ(m/2) / (√(m/2)·Γ((m − 1)/2)), which reproduces Hedges' (1981) Table I. Hedges' paper calls the uncorrected estimator g and the corrected one gᵁ; Cochrane's Handbook calls the corrected one "Hedges' (adjusted) g", which is what the page labels g. The shortcut 1 − 3/(4m − 1) differs by at most 0.06% from 5 per group.
- The 95% interval for d inverts the noncentral t distribution of d·√(n₁n₂/(n₁ + n₂)) on n₁ + n₂ − 2 df; the interval for g is Hedges' (1981) normal approximation, variance 1/ñ + g²/(2m). Both assume normal data with equal population SDs. At 64 per group an observed d of 0.50 gives 0.15 to 0.85.
- r is the point-biserial r = d / √(d² + N(N − 2)/(n₁n₂)) (Lakens 2013 eq. 3). Corrected 2026-10-12: the page used d / √(d² + 4), the equal-groups form, for every input, 1.62 times too high at 10 against 90.
- Labels are Cohen's 0.2 / 0.5 / 0.8, read from g (until 2026-10-12, from d). Cohen (1992) wrote that "the definitions were made subjectively" and had been fixed since the 1977 edition of his book (first edition 1969, second 1988).
- Corrected 2026-10-12: blank fields were replaced by defaults, a negative SD by 0 and n = 1 or 2.9 by 2, and two zero SDs printed "Very large". All are now refused.
Sources
- Hedges LV. Distribution theory for Glass's estimator of effect size and related estimators. J Educ Stat 1981;6(2):107-128. https://doi.org/10.3102/10769986006002107
- Cohen J. A power primer. Psychol Bull 1992;112(1):155-159. https://doi.org/10.1037/0033-2909.112.1.155
- Lakens D. Calculating and reporting effect sizes to facilitate cumulative science. Front Psychol 2013;4:863. https://doi.org/10.3389/fpsyg.2013.00863
- Cochrane Handbook for Systematic Reviews of Interventions, v6.5 (2024), chapter 6: Choosing effect measures and computing estimates of effect. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/c…
- American Statistical Association. Statement on Statistical Significance and P-Values (2016). https://www.amstat.org/asa/files/pdfs/P-ValueStatement.pdf
- NIST/SEMATECH. e-Handbook of Statistical Methods, §7.3.1 Do two processes have the same mean? https://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm
- Lovakov A, Agadullina ER. Empirically derived guidelines for effect size interpretation in social psychology. Eur J Soc Psychol 2021;51:485-504. https://doi.org/10.1002/ejsp.2752
- Brydges CR. Effect size guidelines, sample size calculations, and statistical power in gerontology. Innov Aging 2019;3(4). https://doi.org/10.1093/geroni/igz036
Abridged — the full review record for this tool runs longer than the list above.
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