A p-value tells you whether an effect could be noise; an effect size tells you whether it matters. Cohen’s d — the difference between two means measured in pooled standard deviations — is the standard currency, the number meta-analyses trade in and power analyses are built from. This effect size calculator computes it from the two groups’ means, standard deviations and sample sizes, showing the pooled SD so the standardisation is auditable, and reads the result against Cohen’s benchmarks.
The Formula
d = (mean₁ − mean₂) ÷ pooled SD, where the pooled SD combines both groups’ spreads weighted by their degrees of freedom: √[ ((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2) ]. The page’s default — group means 80 and 74, SDs 10 and 12, thirty per group — pools to √[(29×100 + 29×144) ÷ 58] = √122 ≈ 11.05, so d = 6 ÷ 11.05 ≈ 0.54: a medium effect, six raw points that amount to just over half a standard deviation. The sign carries the direction (group 1 above group 2); the magnitude carries the importance, and reports usually quote the absolute value with the direction stated in words.
Reading d Without Superstition
Cohen offered 0.2 / 0.5 / 0.8 as rough landmarks — small, medium, large — and spent pages warning they are conventions of last resort, not laws. Translate d into overlap language to feel it: at d = 0.54 the average group-1 member outscores about 70 percent of group 2, and the two distributions still overlap by roughly 79 percent — a real, visible-on-average difference that predicts individuals poorly. In medicine, d ≈ 0.2 for a cheap, safe intervention can be a triumph; in education research a d of 0.4 is the hinge point Hattie made famous; in particle physics nothing under 5σ-equivalents gets announced at all. The number is a measurement; whether it is big is a field-specific judgement the calculator deliberately leaves to you.
d, g, and the Small-Sample Correction
Cohen’s d computed from samples slightly overestimates the population effect, especially with few observations. Hedges’ g applies the correction factor — approximately 1 − 3 ÷ (4(n₁+n₂) − 9) — shrinking d toward honesty: at thirty per group g ≈ 0.987 × d (our 0.543 becomes about 0.536), while at eight per group the shrink is a serious 5 percent. Journals in psychology and medicine increasingly ask for g by name. The calculator reports d, the canonical headline; multiply by the factor above when your venue demands g, and note that with unequal variances, Glass’s Δ (using only the control group’s SD) is the third sibling in the family.
Why Effect Size, Not Just Significance
With enough participants, a difference of no practical consequence becomes “highly significant”; with too few, a large real effect stays “not significant.” Effect size is immune to that sample-size theatre because it divides the difference by spread rather than by standard error — d does not grow just because n did. That is why meta-analysis can average d across studies of wildly different sizes, why power analysis starts from the d you refuse to miss, and why reporting standards (APA among them) require effect sizes beside every p. A result reported as “p = .03” is a claim about data volume; “d = 0.54” is a claim about the world.
From d to Required Sample Size
Turn the number around and it designs the next study: detecting d = 0.5 at 80 percent power and α = .05 (two-tailed, two groups) takes about 64 participants per group; d = 0.2 takes about 394 per group; d = 0.8 about 26. The arithmetic punishes small-effect hunting brutally, which is precisely its pedagogical value — most underpowered studies in the literature were, on paper, only ever capable of finding effects that did not exist. Compute your plausible d from pilot data with this page, then plan n from it, in that order.
Common Mistakes
Pooling wrongly — averaging the two SDs instead of pooling variances with degree-of-freedom weights. Quoting d from a paired design as if it were independent-groups d (the standardisations differ). Comparing d across measures with different reliabilities — noise inflates the SD and deflates d, so an unreliable instrument manufactures “small effects.” And the benchmark superstition: a d of 0.19 is not “nothing” and 0.81 is not automatically “huge”; context, cost and base rates finish the interpretation.
Related Tools
The inference bench continues in the Statistics Calculators hub — the Class Width Calculator organises raw data for display, and the Conditional Probability Calculator covers the probability side.
Frequently Asked Questions
How do you calculate Cohen’s d?
Divide the difference of the two means by the pooled standard deviation: the square root of the degree-of-freedom-weighted average of the two variances. Means 80 and 74 with SDs 10 and 12 (n = 30 each) pool to 11.05, giving d ≈ 0.54.
What is a good effect size?
Cohen’s landmarks call 0.2 small, 0.5 medium and 0.8 large, but “good” is field-specific: 0.2 can matter enormously for a cheap public-health intervention, while product experiments often ignore anything under 0.5. Judge against effects your field already respects.
What is the difference between Cohen’s d and Hedges’ g?
g is d with a small-sample correction factor (about 1 − 3/(4N − 9)) that removes d’s slight upward bias. At 30 per group the difference is about one percent; at 8 per group it reaches five — report g when samples are small or your journal asks.
Can effect size be negative?
Yes — the sign encodes direction: a negative d means group 2’s mean exceeded group 1’s. Magnitude is interpreted from the absolute value; always state which group scored higher in words.
How does effect size relate to statistical significance?
They measure different things: significance asks whether the data distinguish the effect from zero, and depends heavily on sample size; effect size measures the standardised magnitude and does not grow with n. Report both — one without the other misleads.
How many participants do I need to detect d = 0.5?
About 64 per group for 80 percent power at α = .05 in a two-group comparison. Halving the target effect roughly quadruples the requirement — d = 0.2 needs about 394 per group.