Effect Size Calculator
Print PageAn Effect Size Calculator (also known as a Cohen’s d Calculator, Hedges’ g Utility, Glass’s Delta Analyzer, or Meta-Analysis Magnitude Calculator) quantifies the practical magnitude or strength of an experimental treatment effect or difference between two group means (&x̄;1 - &x̄;2) independent of sample size n.
While p-values indicate whether an observed difference is statistically significant, effect size measures how large and meaningful that difference is in the real world. In clinical medicine, psychology, and A/B testing, computing Cohen’s d, Hedges’ g, or Glass’s Delta is required to evaluate practical importance and perform sample size power calculations.
Our free online Effect Size Calculator provides instant calculations across all standard effect size metrics:
- Cohen’s d (Pooled Standard Deviation):
d = ( &x̄;1 - &x̄;2 ) ÷ spooled. - Pooled Standard Deviation (spooled):
spooled = √[ ( (n1 - 1)s12 + (n2 - 1)s22 ) ÷ (n1 + n2 - 2) ]. - Hedges’ g (Small-Sample Bias Correction):
g = d · [ 1 - ( 3 ÷ [4(n1 + n2) - 9] ) ]. - Glass’s Delta (Δ):
Δ = ( &x̄;treatment - &x̄;control ) ÷ scontrol. - Correlation Coefficient (r from Cohen’s d):
r = d ÷ √[ d2 + 4 ]. - Eta-Squared (η2 for ANOVA):
η2 = SSbetween ÷ SStotal.
Master Cohen’s d & Effect Size Interpretation Reference Table
The table below displays Jacob Cohen’s benchmark thresholds, mathematical definitions, percentiles of non-overlap, and required sample sizes per group for 80% statistical power (α = 0.05):
| Effect Size Classification | Cohen’s d Value | Equivalent Correlation (r) | Group Non-Overlap (%) | Sample Size per Group (80% Power) | Real-World Practical Meaning |
|---|---|---|---|---|---|
| Small Effect Size | d = 0.20 | r = 0.100 | 14.8% Non-Overlap | n = 393 per group | Subtle difference (e.g. 1cm height difference) |
| Medium Effect Size | d = 0.50 | r = 0.243 | 33.0% Non-Overlap | n = 64 per group | Visually noticeable to careful observer |
| Large Effect Size | d = 0.80 | r = 0.371 | 47.4% Non-Overlap | n = 26 per group | Substantial difference easily seen |
| Very Large / Huge Effect Size | d ≥ 1.20 | r ≥ 0.514 | 65.3%+ Non-Overlap | n ≤ 12 per group | Massive clinical treatment breakthrough |
Step-by-Step Clinical Trial Calculation (Treatment vs. Control)
To evaluate a pharmaceutical drug trial comparing a treatment group (n1 = 30 patients, &x̄;1 = 85.0, s1 = 10.0) to a placebo control group (n2 = 30 patients, &x̄;2 = 75.0, s2 = 10.0):
Step 1 (Calculate Mean Difference): Δ&x̄; = 85.0 - 75.0 = 10.0 points
Step 2 (Calculate Pooled Standard Deviation s_pooled): s_pooled = √[ (29 · 100 + 29 · 100) ÷ 58 ] = √[ 5800 ÷ 58 ] = √100 = 10.0
Step 3 (Calculate Cohen's d): d = 10.0 ÷ 10.0 = 1.000 (Large Effect Size!)
Step 4 (Calculate Hedges' g Small Sample Correction): Correction Factor = 1 - ( 3 ÷ [4(60) - 9] ) = 1 - ( 3 ÷ 231 ) = 0.9870 &implies; g = 1.000 · 0.9870 = 0.9870
Step 5 (Convert to Correlation r): r = 1.000 ÷ √[ 1.000^2 + 4 ] = 1.000 ÷ √5 = 1.000 ÷ 2.2361 = 0.4472
Thus, the treatment demonstrates a large effect size of Cohen’s d = 1.000 (Hedges’ g = 0.9870, r = 0.4472), establishing high clinical efficacy.
Effect Size Metrics Comparison: Cohen’s d vs. Hedges’ g vs. Glass’s Delta
Below is a comparative reference chart detailing when to use each major effect size metric:
| Effect Size Metric | Standard Deviation Denominator Used | Best Research Condition | Primary Practical Advantage |
|---|---|---|---|
| Cohen’s d | Pooled Standard Deviation spooled | Equal group sizes & equal variances (n > 30) | Universal standard benchmark for power analysis. |
| Hedges’ g | Pooled Standard Deviation + Bias Correction | Small sample sizes (n < 20 per group) | Removes upward sample size bias in meta-analysis. |
| Glass’s Delta (Δ) | Control Group Standard Deviation scontrol | Treatment alters variance significantly | Unaffected by treatment variance inflation. |
History & Mathematics: 1969 Jacob Cohen to 1981 Larry Hedges
1969 Jacob Cohen & Statistical Power
In 1969, American psychologist and statistician Jacob Cohen published his landmark textbook Statistical Power Analysis for the Behavioral Sciences, introducing Cohen’s d and defining small (0.20), medium (0.50), and large (0.80) effect size rules of thumb.
1981 Larry Hedges & Meta-Analysis Correction
In 1981, statistician Larry Hedges published the exact small-sample correction factor for Cohen’s d (Hedges’ g), establishing modern meta-analytic synthesis standards.
1976 Gene V. Glass & Glass’s Delta
In 1976, researcher Gene V. Glass introduced Glass’s Delta and coined the term “meta-analysis” in his APA presidential address.
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Frequently Asked Questions (FAQ)
What is Cohen’s d formula?
The formula is d = ( &x̄;1 - &x̄;2 ) ÷ spooled.
What is the difference between Cohen’s d and Hedges’ g?
Hedges’ g applies a small-sample correction factor 1 - 3/[4(n1+n2)-9] to Cohen’s d to eliminate upward bias when sample sizes are small (n < 20).
Why is Effect Size better than p-value alone?
A p-value only tells you if an effect is statistically non-zero, whereas effect size tells you how large and practical the effect actually is regardless of sample size.