Cohen's d Effect Size Calculator (with Hedges' g)

An effect size answers the question a p-value cannot: how big is the difference? Cohen's d expresses the gap between two means in standard-deviation units, so a result is comparable across studies, scales, and sample sizes. Journals and supervisors now expect it reported alongside the t-test p-value.

1. Pooled SD (independent design), per Cohen (1988). s_pooled = √[((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2)] weights each group's variance by its degrees of freedom.
2. Cohen's d. Divide the mean difference by the standardiser: d = (M₁ − M₂) / s_pooled. A one-sample design uses the sample SD against a reference μ₀; a paired design uses the SD of the difference scores (the d_z of Lakens 2013).
3. Hedges' g (bias correction), per Hedges (1981). Small samples make d run high, so g multiplies it by J = 1 − 3/(4·df − 1), with df = n₁+n₂−2 (independent) or n−1 (one-sample and paired).
4. Confidence interval (Lakens 2013). The standard error uses the large-sample normal approximation SE(d) ≈ √((n₁+n₂)/(n₁n₂) + d²/(2(n₁+n₂))), and the interval is d ± 1.96·SE. This is an approximation, labelled as such — it is close for moderate samples and slightly narrow for tiny ones.
5. Overlap (Cohen 1988; McGraw & Wong 1992).Cohen's U₃ = Φ(d) is the share of one group below the other's mean, and the overlap coefficient 2·Φ(−|d|/2) is how much the two equal-variance normal curves share. Both use the standard-normal CDF.
6. Magnitude.The verdict reads |d| against Cohen's 0.2 / 0.5 / 0.8 cut-offs. These are deliberately blunt benchmarks; the page flags them as rules of thumb rather than field-specific truth.

As a credibility check, the tool also recovers d from the t statistic — d = t·√(1/n₁ + 1/n₂) (Lakens 2013) — and confirms it equals the value computed directly from the means and SDs, to machine precision. If you are sizing a future study instead, the sample size calculator turns a target d back into a required n.

• Lakens, D. (2013). Calculating and reporting effect sizes — Frontiers in Psychology 4:863
• Hedges, L. V. (1981). Distribution theory for Glass's estimator — J. Educational Statistics 6(2)
• McGraw & Wong (1992). A common language effect size statistic — Psychological Bulletin 111(2)
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.)

The formulas and benchmarks on this page were last cross-checked against these sources on 2026-06-12, and the computed examples reconcile to the cited values to within ±0.001.