Effect Size Calculator (Cohen's d)

A raw mean difference is hard to judge on its own: is a 7-point gap between two classes large? It depends entirely on how much scores vary in the first place. Cohen's d resolves this by expressing the gap in units of standard deviation — a d of 0.7 means the group means sit 0.7 pooled standard deviations apart, whatever the original measurement scale was.

Because it is scale-free, d serves two purposes raw differences cannot. It lets you compare effects measured on different instruments (a reading intervention scored in one test against a math intervention scored in another), and it is the common currency of meta-analysis, where results from many studies are pooled into a single estimate.

The calculator runs three computations in sequence:

The pooled standard deviation is a weighted average of the two group variances, with each group weighted by its degrees of freedom (n − 1). This gives larger groups proportionally more influence over the yardstick used to standardize the difference.

The interpretation labels follow the benchmarks Jacob Cohen proposed in his 1988 book Statistical Power Analysis for the Behavioral Sciences: |d| below 0.2 is negligible, 0.2 to 0.5 small, 0.5 to 0.8 medium, and 0.8 or above large. A helpful intuition: at d = 0.8, the average member of the higher group scores above roughly 79% of the lower group; at d = 0.2, above only about 58%.

Cohen himself warned that these are conventions of last resort, not laws. In fields where interventions are cheap and outcomes matter at scale — education, public health — a “small” d of 0.2 can be practically important, while in tightly controlled laboratory work a d of 0.5 might be unremarkable. Always interpret the number against effects typical of your own field.

Cohen's d has a known flaw: with small samples it systematically overestimates the population effect size, because the pooled standard deviation is itself a noisy estimate. Hedges' g applies a correction factor — always slightly below 1 — that removes most of this bias. The two converge as samples grow: with 60 total participants the correction is about 1.3%, and with 300 it drops below 0.3%.

A sensible reporting rule: quote g when either group has fewer than about 20 participants or when contributing to a meta-analysis, and note that many journals now request g by default. Since g is never larger in magnitude than d, reporting g is also the conservative choice.

A t-test answers a yes-or-no question — is the difference distinguishable from zero? — and its p-value depends heavily on sample size. With thousands of participants, a trivial d of 0.05 becomes “highly significant”; with ten participants, a substantial d of 0.7 may fail to reach significance. Neither p-value tells you what you usually want to know: how big the effect is.

Effect size and significance therefore answer complementary questions, and modern reporting standards (including APA style) expect both. The pairing also powers study planning: a power analysis takes the effect size you hope to detect and returns the sample size required to detect it reliably.

An education researcher compares final exam scores. The new method group has m₁ = 85, s₁ = 10, n₁ = 30; the traditional group has m₂ = 78, s₂ = 12, n₂ = 30. The calculator proceeds step by step:

1. Weight each variance: (30 − 1) × 10² = 29 × 100 = 2900 and (30 − 1) × 12² = 29 × 144 = 4176.
2. Pooled variance: (2900 + 4176) / (30 + 30 − 2) = 7076 / 58 = 122.
3. Pooled SD: √122 ≈ 11.0454.
4. Cohen's d: (85 − 78) / √122 = 7 / 11.045361 = 0.63375, displayed as 0.6338 — a medium effect.
5. Correction factor: 1 − 3 / (4 × 60 − 9) = 1 − 3/231 ≈ 0.987013.
6. Hedges' g: 0.63375 × 0.987013 ≈ 0.6255.

Interpretation: the new method's average sits about 0.63 pooled standard deviations above the traditional method's — a medium effect by Cohen's benchmarks, meaning the typical student under the new method outscored roughly 74% of students under the old one. The bias-corrected g of 0.6255 is the number to carry into a meta-analysis.