Cohen's d Effect Size Calculator

Q: Should I provide sample sizes?

Yes, when the groups have different sizes. The weighted pooled SD is more accurate than the simple average when n₁ ≠ n₂. If group sizes are equal, both formulas give the same result.

Q: What is Hedges' g?

Hedges' g corrects Cohen's d for small-sample bias using the factor J = 1 − 3/(4df−1). For n ≥ 20 per group the correction is small (< 5%), but for very small samples g is preferred for meta-analyses.

Q: How do I convert r to d?

The formula is d = 2r ÷ √(1−r²). This is an exact algebraic conversion. For r = 0.4: d = 0.8 ÷ √(1−0.16) = 0.8 ÷ √0.84 ≈ 0.873 - a large effect by Cohen's benchmarks.

Frequently Asked Questions

Why report effect size at all?

Statistical significance (p < 0.05) only says an effect is probably real given the sample size. Effect size says whether it is large enough to matter. A p-value of 0.001 with d = 0.05 is real but trivial; a p-value of 0.06 with d = 0.9 is practically important but narrowly misses the conventional threshold.

Should I provide sample sizes?

Yes, when the groups have different sizes. The weighted pooled SD is more accurate than the simple average when n₁ ≠ n₂. If group sizes are equal, both formulas give the same result.

What is Hedges' g?

Hedges' g corrects Cohen's d for small-sample bias using the factor J = 1 − 3/(4df−1). For n ≥ 20 per group the correction is small (< 5%), but for very small samples g is preferred for meta-analyses.

How do I convert r to d?

The formula is d = 2r ÷ √(1−r²). This is an exact algebraic conversion. For r = 0.4: d = 0.8 ÷ √(1−0.16) = 0.8 ÷ √0.84 ≈ 0.873 - a large effect by Cohen's benchmarks.

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How This Calculator Works

Effect size answers the question a p-value cannot: not "is there a difference?" but "how big is it?" Cohen's d is the most common standardized effect-size measure for comparing two means. You can calculate it two ways:

From group means: d = (m₁ − m₂) ÷ pooled SD. When sample sizes are provided, the pooled SD uses the weighted formula √(((n₁−1)·s₁² + (n₂−1)·s₂²) ÷ (n₁+n₂−2)). Without sample sizes, it falls back to the simple average √((s₁²+s₂²)÷2).

From a correlation: d = 2r ÷ √(1−r²), which converts a Pearson r to an equivalent d.

Because d is expressed in pooled standard-deviation units, it is comparable across studies and disciplines regardless of the original measurement scale.

Interpreting Cohen's d

Cohen's conventional benchmarks for |d|:

≥ 0.8: Large effect

0.5 to 0.8: Medium effect

0.2 to 0.5: Small effect

< 0.2: Negligible

Rough distribution-overlap interpretation: d = 0.2 gives about 85% overlap between the two distributions, d = 0.5 gives about 69%, and d = 0.8 gives about 53%. A d of 0.8 means the average person in group 1 scores higher than about 79% of people in group 2.

Worked Example

Group 1: mean = 50, SD = 10. Group 2: mean = 45, SD = 12. No sample sizes entered. Pooled SD = √((10² + 12²) ÷ 2) = √(244 ÷ 2) = √122 ≈ 11.05. Then d = (50 − 45) ÷ 11.05 ≈ 0.453 - a medium-leaning small effect. The two distributions overlap substantially. With sample sizes n₁ = n₂ = 30 provided, the weighted pooled SD changes to √((29×100 + 29×144) ÷ 58) ≈ 11.05 (identical when n are equal), and d stays the same.

Common Mistakes

Confusing d with a percentage. d = 0.5 means the means differ by half a pooled SD, not 50%. The scale depends on the original measurement.

Treating significance as large effect. With a large sample, a tiny d can produce p < 0.001. Always report d alongside the p-value.

Using Cohen's thresholds rigidly. A "small" medical effect that prevents 1 heart attack per 100 treated patients is clinically important at population scale. Context matters.

Negative SDs. Standard deviations must be positive. The calculator requires s > 0 for the pooled term to be defined.

Alternative Effect Size Measures

Cohen's d is one of many effect size indices. Others include:

r and R²: for correlations - r = 0.1 small, 0.3 medium, 0.5 large (Cohen's conventions for r).

η² and partial η²: for ANOVA - proportion of variance explained by the factor.

Number Needed to Treat (NNT): for medical trials - the most clinically intuitive measure, expressing how many patients must be treated for one to benefit.

Odds ratio and relative risk: for binary outcomes in epidemiology.

Always choose the effect size measure most natural to your outcome type and audience.

Cohen's d Effect Size Calculator

Frequently Asked Questions

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