Effect Size (Cohen's d) Calculator
Find Cohen's d and Hedges' g from raw data, summary statistics, or a t-statistic — for independent-samples, paired/repeated-measures, and one-sample designs. See exactly how much two distributions overlap, get a step-by-step solution, and translate the number into plain English.
Background
Cohen's d is a standardized measure of how far apart two means are, expressed in standard-deviation units rather than the original scale. Unlike a p-value, effect size doesn't depend on sample size — it answers "how big is the difference?" rather than "is the difference statistically detectable?"
How to use this calculator
- Pick the design that matches your data: two separate groups, the same people measured twice, one group against a known value, or a t-statistic you already have.
- Enter either summary statistics (mean, SD, n) or raw data — the calculator computes the mean and SD for you if you paste in raw values.
- Click Calculate to get Cohen's d, Hedges' g, an overlapping-distributions chart, and a plain-English translation of what the effect size means.
How this calculator works
- Independent groups: d = (M₁ − M₂) / SD_pooled, where the pooled SD is a sample-size-weighted combination of both groups' variances — not a simple average of the two SDs.
- Paired / repeated measures: d = mean(difference) / SD(difference), using the spread of the within-person change scores.
- One sample: d = (M − μ₀) / SD, comparing a sample mean to a known or hypothesized population value.
- From a t-statistic: d = t·√(1/n₁ + 1/n₂) for independent samples, or d = t/√n for paired/one-sample designs.
- Every mode also reports Hedges' g, which multiplies d by a small correction factor that removes the slight upward bias d has in small samples.
Formula & Equations Used
Cohen's d (independent): d = (M₁ − M₂) / SD_pooled
Pooled SD: SD_pooled = √[ ((n₁−1)SD₁² + (n₂−1)SD₂²) / (n₁+n₂−2) ]
Hedges' g: g = d × [1 − 3/(4·df − 1)]
Standard error of d: SE(d) = √[ (n₁+n₂)/(n₁n₂) + d²/(2(n₁+n₂)) ]
Probability of superiority: CL = Φ(d/√2)
Distribution overlap: OVL = 2·Φ(−|d|/2)
Example Problems & Step-by-Step Solutions
Example 1 — Independent groups
Method A: M=85, SD=10, n=30. Method B: M=78, SD=12, n=28. Find Cohen's d.
- Pooled SD = √[ (29×10² + 27×12²) / 56 ] ≈ 11.01.
- d = (85 − 78) / 11.01 ≈ 0.64 — a medium-to-large effect.
- Hedges' g (df=56) ≈ 0.63 — nearly identical to d since the sample isn't tiny.
Example 2 — Paired samples
A weight-loss program: mean change = −4.2 lb, SD of change = 5.1 lb, n = 25.
- d = −4.2 / 5.1 ≈ −0.82 — a large effect (weight decreased).
- This uses the SD of the individual change scores, which is usually smaller than the SD of the raw before/after scores — so paired d values often run larger than independent-groups d for a similarly-sized effect.
Example 3 — From a t-statistic
A published independent-samples t-test reports t = 2.86, n₁ = 20, n₂ = 22.
- d = 2.86 × √(1/20 + 1/22) ≈ 2.86 × 0.309 ≈ 0.88 — a large effect.
- Handy when a paper reports significance testing but not effect size directly.
Example 4 — One sample vs. reference
A class's average IQ score is 112 (SD=15, n=20) vs. the population mean of 100.
- d = (112 − 100) / 15 = 0.80 — a large effect.
- Hedges' g (df=19) ≈ 0.77 — the small-sample correction pulls it in slightly.
Frequently Asked Questions
What's the difference between Cohen's d and Hedges' g?
They use the same basic formula, but Hedges' g applies a small correction that removes the slight tendency of d to overestimate the true effect size in small samples. For large samples, d and g are nearly identical.
What counts as a "large" effect size?
Cohen's original (1988) rules of thumb are |d| ≈ 0.2 (small), 0.5 (medium), and 0.8 (large) — but these are general guidelines, not fixed rules, and what counts as meaningful varies by field.
Can Cohen's d be negative?
Yes — the sign just reflects which mean was subtracted from which, or which direction scores changed. The magnitude (absolute value) is what determines the size of the effect.
Why does this calculator ask for raw data as an alternative to summary stats?
If you have the actual data points rather than pre-computed statistics, entering them directly avoids rounding errors and lets the calculator compute the mean and SD for you.
Is effect size the same as statistical significance?
No. A p-value tells you whether an effect is unlikely to be due to chance; effect size tells you how big that effect actually is. A tiny, practically meaningless difference can still be statistically significant in a large enough sample.
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