Cohen's d Calculator
Print PageA Cohen’s d Calculator (also known as a Standardized Mean Difference Effect Size Calculator, Pooled Standard Deviation [spooled] Utility, Hedges’ g Unbiased Small-Sample Generator, or Glass’s Δ Analyzer) computes exact standardized effect sizes (d = ( X̄1 - X̄2 ) ÷ spooled), pooled sample standard deviations (spooled = √[ ( (n1 - 1)s12 + (n2 - 1)s22 ) ÷ (n1 + n2 - 2) ]), Hedges’ g unbiased corrections (g = d · [ 1 - (3 ÷ (4[n1+n2]-9)) ]), Glass’s Δ for unequal variances, paired samples dz, and Common Language Effect Size (CLES) probabilities.
In biomedical clinical trials, psychological experiments, educational interventions, and meta-analytic evidence reviews, Cohen’s d measures the practical magnitude of a treatment effect independent of sample size, preventing large sample sizes from making trivial differences appear misleadingly important.
Our free online Cohen’s d Calculator provides instant calculations across all standardized effect size parameters:
- Independent Samples Cohen’s d Formula (d):
d = ( X̄1 - X̄2 ) ÷ spooled(whereX̄1, X̄2are group means). - Pooled Standard Deviation Formula (spooled):
spooled = √[ ( ( n1 - 1 ) s12 + ( n2 - 1 ) s22 ) ÷ ( n1 + n2 - 2 ) ]. - Hedges’ g Small-Sample Unbiased Correction (g for n < 50):
g = d · J(df) = d · [ 1 - ( 3 ÷ ( 4 · [n1 + n2] - 9 ) ) ]. - Glass’s Delta (Δ for Unequal Variances):
Δ = ( X̄1 - X̄2 ) ÷ scontrol(Uses control group standard deviationscontrol). - Paired / Repeated Measures Cohen’s d (dz):
dz = d̄ ÷ sd = ( X̄post - X̄pre ) ÷ sdiff. - Common Language Effect Size (CLES / Probability of Superiority):
CLES = Φ( d ÷ √2 )(Probability a randomly selected unit from Group 1 exceeds Group 2). - Effect Size Magnitude Classification Rules:
|d| < 0.20: Negligible Effect.|d| ≈ 0.20: Small Effect.|d| ≈ 0.50: Medium Effect.|d| ≈ 0.80: Large Effect.|d| ≈ 1.20: Very Large Effect.|d| ≥ 2.00: Huge / Transformation Effect.
Master Cohen’s d Reference Table (Antidepressant Clinical Trial: n1 = 30 Treatment, n2 = 30 Placebo)
The table below displays sample means (X̄), sample standard deviations (s), calculated pooled standard deviations spooled, Cohen’s d, Hedges’ g, and CLES probabilities for a 60-patient antidepressant drug trial (n1 = 30 Treatment Patients, n2 = 30 Placebo Patients):
| Trial Subgroup / Metric | Sample Size (n) | Group Mean (X̄) | Standard Deviation (s) | Calculated Effect Size Metric | Clinical Magnitude Interpretation |
|---|---|---|---|---|---|
| Group 1: New Antidepressant Drug | n1 = 30 patients | X̄1 = 18.50 points | s1 = 4.20 points | Mean Difference = +4.50 pts | Superior Clinical Improvement |
| Group 2: Placebo Control | n2 = 30 patients | X̄2 = 14.00 points | s2 = 4.80 points | scontrol = 4.80 points | Baseline Control Benchmark |
| Pooled Standard Deviation (spooled) | df = 58 degrees | spooled2 = 20.340 | spooled = 4.510 pts | Combined Within-Group Variation | Standardized Denominator Scale |
| Cohen’s d Effect Size (d) | N = 60 total | 4.50 ÷ 4.510 | d = 0.9978 | d = 0.998 Standardized SDs | LARGE to VERY LARGE Effect (|d| ≈ 1.0) |
| Hedges’ g Unbiased Correction (g) | J(58) = 0.98701 | 0.9978 · 0.98701 | g = 0.9848 | g = 0.985 Unbiased Correction | Small-Sample Unbiased Estimate |
| Common Language Effect Size (CLES) | Φ(0.9978 / √2) | Φ(0.7056) | 75.9800% (0.7598) | CLES = 75.98% Superiority Chance | 76% chance drug patient beats placebo |
Step-by-Step Antidepressant Clinical Trial Cohen’s d Calculation
To calculate Cohen’s d effect size for a clinical trial comparing Treatment (X̄1 = 18.50, s1 = 4.20, n1 = 30) against Placebo (X̄2 = 14.00, s2 = 4.80, n2 = 30):
Step 1 (Calculate Mean Difference): X̄_1 - X̄_2 = 18.50 - 14.00 = 4.50 points
Step 2 (Calculate Pooled Variance s_pooled^2): s_pooled^2 = [ (29 · 4.20^2) + (29 · 4.80^2) ] ÷ (30 + 30 - 2) = [ (29 · 17.64) + (29 · 23.04) ] ÷ 58 = [ 511.56 + 668.16 ] ÷ 58 = 1,179.72 ÷ 58 = 20.340
Step 3 (Calculate Pooled Standard Deviation s_pooled): s_pooled = √20.340 = 4.510 points
Step 4 (Calculate Cohen's d): d = 4.50 ÷ 4.510 = 0.9978 ≈ 0.998 (Large to Very Large Effect Size)
Step 5 (Calculate Hedges' g Correction Factor J): J = 1 - ( 3 ÷ [ (4 · 60) - 9 ] ) = 1 - (3 ÷ 231) = 1 - 0.012987 = 0.98701 &implies; g = 0.9978 · 0.98701 = 0.9848 ≈ 0.985
Step 6 (Calculate Common Language CLES): CLES = Φ(0.9978 ÷ 1.4142) = Φ(0.7056) = 0.7598 ≈ 75.98%
Thus, the treatment demonstrates a large effect size of d = 0.998, showing that a randomly selected treatment patient has a 75.98% probability of outperforming a placebo patient.
Effect Size Metrics Comparison: Cohen’s d vs. Hedges’ g vs. Glass’s Delta vs. Eta-Squared (η2)
Below is a comparative reference chart detailing when to use Cohen’s d versus alternative effect size measures:
| Effect Size Metric | Denominator Standard Deviation Used | Small Sample Bias Correction | Primary Practical Application |
|---|---|---|---|
| Cohen’s d | Pooled Standard Deviation spooled | Unadjusted (Overestimates for n < 20) | Standard t-test reporting (n ≥ 30 per group). |
| Hedges’ g | Pooled Standard Deviation spooled | UNBIASED (Multiplies d by J correction factor) | Meta-analysis & small sample sizes (n < 30). |
| Glass’s Delta (Δ) | Control Group Standard Deviation scontrol | Unadjusted | Trials where treatment alters variance. |
| Eta-Squared (η2 / Partial η2) | Total Variance SStotal | Proportion of Variance Explained | ANOVA multi-group factorial experiments. |
History & Mathematics: 1969 Jacob Cohen to 1981 Larry Hedges
1969 & 1988 Jacob Cohen & Behavioral Power Analysis
In 1969, American psychologist and statistician Jacob Cohen published Statistical Power Analysis for the Behavioral Sciences, creating d to provide a standardized measure of effect size that remains comparable regardless of measurement scale units or sample sizes.
1981 Larry Hedges & Meta-Analysis Unbiased Correction
In 1981, University of Chicago statistician Larry Hedges derived the exact small-sample correction factor J(df), creating Hedges’ g to remove sample size overestimation bias in quantitative meta-analysis syntheses.
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Frequently Asked Questions (FAQ)
What is the formula for Cohen’s d?
The formula is d = ( X̄1 - X̄2 ) ÷ spooled, where spooled = √[ ( (n1-1)s12 + (n2-1)s22 ) ÷ (n1+n2-2) ].
What are Cohen’s benchmark thresholds for small, medium, and large effect sizes?
Jacob Cohen established standard benchmarks: |d| = 0.20 is Small, |d| = 0.50 is Medium, |d| = 0.80 is Large, and |d| = 1.20 is Very Large.
What is the difference between Cohen’s d and Hedges’ g?
Cohen’s d uses the sample pooled standard deviation directly, which slightly overestimates effect size in small samples. Hedges’ g applies an unbiased correction factor J = 1 - (3 ÷ [4(n1+n2)-9]) to remove small-sample bias.