Hedges' g & Glass's Delta Calculator

Use Hedges' g or Glass's delta when you need alternatives to Cohen's d. Hedges' g corrects for small-sample bias, while Glass's delta uses only the control group's standard deviation—useful when treatments might affect variability.

What Are These Effect Sizes?

All three measure the standardized difference between two group means:

Cohen's d

$d = \frac{\bar{X}_1 - \bar{X}_2}{s_{pooled}}$

Uses pooled standard deviation from both groups.

Hedges' g

$g = d \times J$

Where $J = 1 - \frac{3}{4(n_1 + n_2) - 9}$ is the bias correction factor.

Glass's Delta (Δ)

$\Delta = \frac{\bar{X}_1 - \bar{X}_2}{s_{control}}$

Uses only the control group's standard deviation.

Why Different Measures?

Cohen's d

• Most common and widely understood
• Slightly biased upward in small samples
• Assumes equal variances

Hedges' g

• Corrects Cohen's d for small-sample bias
• Preferred for meta-analysis
• Nearly identical to d for large samples

Glass's Delta

• Appropriate when treatment affects variability
• Uses untreated (control) group as reference
• Avoids contaminating the denominator with treatment effects

The Bias Correction Factor

Cohen's d overestimates the population effect, especially with small samples. Hedges' correction:

$J = 1 - \frac{3}{4(n_1 + n_2 - 2) - 1}$

Or approximately:

$J \approx 1 - \frac{3}{4N - 9}$

For N > 50, the difference is negligible.

Example Calculations

Data:

• Treatment: n₁ = 25, M₁ = 75, SD₁ = 12
• Control: n₂ = 25, M₂ = 68, SD₂ = 10

Pooled SD: $s_{pooled} = \sqrt{\frac{(24)(12^2) + (24)(10^2)}{48}} = \sqrt{\frac{3456 + 2400}{48}} = 11.07$

Cohen's d: $d = \frac{75 - 68}{11.07} = 0.632$

Hedges' g: $J = 1 - \frac{3}{4(50) - 9} = 1 - \frac{3}{191} = 0.984$ $g = 0.632 \times 0.984 = 0.622$

Glass's Delta: $\Delta = \frac{75 - 68}{10} = 0.70$

When to Use Each

Confidence Intervals

For Cohen's d and Hedges' g

Standard error:

$SE_d = \sqrt{\frac{n_1 + n_2}{n_1 n_2} + \frac{d^2}{2(n_1 + n_2)}}$

95% CI: $d \pm 1.96 \times SE_d$

For Hedges' g, multiply SE by J or compute directly using the corrected variance formula.

For Glass's Delta

$SE_\Delta = \sqrt{\frac{n_1 + n_2}{n_1 n_2} + \frac{\Delta^2}{2(n_2 - 1)}}$

Note the denominator uses n₂ - 1 because variance is estimated from control group only.

Interpreting Effect Sizes

Cohen's benchmarks apply to all three:

But context matters:

• In education, d = 0.4 may be practically significant
• In medicine, even d = 0.2 could save lives at scale
• In psychology, most effects are d = 0.2-0.5

Converting Between Measures

If variances are equal, d ≈ Δ.

If variances differ: $d = \Delta \times \frac{s_{control}}{s_{pooled}}$

From d to g: $g = d \times J$

For Paired Data

Hedges' g for paired data:

$g = \frac{\bar{D}}{s_D} \times J_{paired}$

Where $\bar{D}$ is the mean difference, $s_D$ is the SD of differences, and:

$J_{paired} = 1 - \frac{3}{4n - 5}$

Meta-Analysis Considerations

Hedges' g is standard for meta-analysis because:

1. Less biased across different sample sizes
2. Comparable across studies
3. Standard error formulas are well-established
4. Most meta-analysis software expects g

When combining studies: $g_{pooled} = \frac{\sum w_i g_i}{\sum w_i}$

Where weights $w_i = 1/SE_i^2$.

Common Scenarios

Scenario 1: Treatment Changes Variability

Psychotherapy often increases outcome variability (some people improve dramatically, others don't respond). Use Glass's Δ with the control group SD.

Scenario 2: Small Pilot Study

With n = 15 per group, use Hedges' g to avoid overestimating the effect.

Scenario 3: Comparing to Historical Control

If the control group is from historical data with well-established SD, Glass's Δ makes sense.

Scenario 4: Meta-Analysis Contribution

Convert your results to Hedges' g for inclusion in meta-analyses.

Relationship to Other Statistics

From t-statistic: $d = t \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$

From F-statistic (two groups): $d = \sqrt{F \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$

To r (correlation): $r = \frac{d}{\sqrt{d^2 + 4}}$

Assumptions

1. Independent samples (for between-group versions)
2. Normal distributions (for accurate SEs)
3. Equal variances (for Cohen's d; Glass's Δ relaxes this)

Common Mistakes

Using d with Tiny Samples

Cohen's d can be heavily biased with n < 20. Always use Hedges' g for small samples.

Ignoring Variance Differences

If one group has much larger variance, pooled SD obscures this. Consider Glass's Δ.

Forgetting the Direction

Effect sizes can be negative. Report sign or clearly indicate which group was higher.

Treating Benchmarks as Absolutes

"Small/medium/large" are rough guides. A "small" effect can be important in context.

Historical Context

Jacob Cohen developed Cohen's d in the 1960s, establishing effect size reporting in behavioral sciences.

Larry Hedges developed the bias correction in the 1980s, motivated by meta-analysis needs.

Gene Glass proposed using control group SD in the 1970s, particularly for educational research.

Related Calculations

• Cohen's d Calculator — Standard effect size
• Eta-Squared Calculator — ANOVA effect sizes
• Two-Sample T-Test — Hypothesis test
• Power Calculator — Sample size with effect size
• Paired T-Test — Dependent groups

Summary

Hedges' g corrects Cohen's d for small-sample bias—essential for meta-analysis and studies with n < 50. Glass's delta uses only the control group SD, appropriate when treatment affects variability. All three interpret similarly: 0.2 = small, 0.5 = medium, 0.8 = large. Use Hedges' g by default for publication; use Glass's Δ when groups have unequal variances due to treatment effects.