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Cohen's d Effect Size Calculator

Cohen's d is a standardized mean difference: it divides the gap between two group means by their pooled standard deviation, letting you compare effect sizes across studies and units. This tool computes d, Hedges' g (small-sample bias correction), pooled SD, a 95 % confidence interval, and the common-language effect size (CLES) from McGraw & Wong (1992) — and classifies the result against Cohen 1988 with Sawilowsky 2009's very-large extension.

Group 1 (M1 / SD1 / N1)
Group 2 (M2 / SD2 / N2)

Example: experimental group M = 105 vs control M = 100, both SD = 15, N = 30 — d ≈ 0.33 (small).

Cohen's d

0.33

Effect-size benchmark (Cohen 1988)

0 0.2 0.5 0.8 ≥ 1.2

Marker tracks |d|; the sign of d is shown by the headline number above-left.

Hedges' g (small-sample correction)

Prefer g over d when total N < 50 — it shrinks toward 0 and is less biased.

Probability of superiority

Chance a random Group 1 score exceeds a random Group 2 score (CLES).

Pooled SD (sp)

95% confidence interval for d

Pooled-variance formula for two independent samples (Cohen 1988). Hedges' correction uses J = 1 − 3/(4·df − 1); CI uses the large-sample SE from Hedges & Olkin 1985.

Formula

Cohen's d = (M₁ − M₂) ÷ sp sp = √[((N₁ − 1)·SD₁² + (N₂ − 1)·SD₂²) ÷ (N₁ + N₂ − 2)] Hedges' g = d · (1 − 3 ÷ (4·df − 1)), df = N₁ + N₂ − 2 CLES = Φ(d ÷ √2)

Frequently asked

How is Cohen's d different from a t-test?

A t-test answers "is this difference statistically significant?" and is highly sample-size dependent — with enough data, even a trivial gap becomes "significant". Cohen's d answers "how big is the difference?" by standardising it with the pooled SD. The two are complementary: modern reporting standards (APA, PSPB, AMA) require both p and d (with 95 % CI) so the reader sees significance and magnitude at once.

When should I report Hedges' g instead of d?

Hedges' g is the small-sample unbiased estimator. Once total N is below ~50 (df ≈ 48), d slightly overestimates the population effect, and g shrinks it back using J ≈ 1 − 3/(4·df − 1). Meta-analytic conventions (Cochrane, R's metafor and effsize packages) default to g. For large samples J → 1 and the two values converge. Use g for clinical trials, educational research, or any meta-analysis; d is fine otherwise.

Can I still use Cohen's d if the two SDs differ a lot (say one is twice the other)?

You can, but interpret with care. The pooled SD assumes roughly equal variance. If SD₁ ≈ 2·SD₂, sp averages the two and pulls d toward the smaller side. Common alternatives: (1) Glass' Δ uses only the control SD as the denominator (popular in intervention studies); (2) report each group's SD alongside d and its 95 % CI; (3) pair Welch's t with Welch's d. Most psychology/education papers use pooled d when the variance ratio is under ~4, switching to Glass beyond that.

What is the "probability of superiority" (CLES) and why is it good for lay audiences?

CLES = Φ(d/√2). Assuming both groups are normal with equal variance, it is the probability that a randomly sampled Group 1 score exceeds a randomly sampled Group 2 score. For example d = 0.5 → CLES ≈ 64 %, d = 0.8 → 71 %. This "probability of being higher" framing is much easier for students, patients, or journalists to grasp than an abstract d value — "Group 1 wins 64 % of head-to-head matchups" lands better than "d = 0.5, a medium effect". McGraw & Wong (1992) introduced this metric for exactly this audience.

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