Hedges' g vs Cohen's d: When, Why, and How to Choose (With Formulas)

Direct Summary (TL;DR)

Hedges' g is simply Cohen's d adjusted with a small-sample bias correction.

In small samples (N < 40 or n < 20 per group), Cohen's d systematically overestimates the true population effect size. Hedges' g applies a mathematical correction factor (J) that removes this upward bias:

• When N > 50: Cohen's d and Hedges' g are virtually identical (difference < 1%).
• When N < 40: Report Hedges' g to avoid inflating your effect size.
• In Meta-Analyses: Always use Hedges' g (it is the standard metric in R metafor, CMA, and Cochrane reviews).

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Why Is Cohen's d Biased in Small Samples?

The formula for Cohen's d divides the observed mean difference by the sample pooled standard deviation:

d = (M₁ − M₂) / SD_pooled

Because sample standard deviations (s) are calculated from limited data points, they tend to slightly underestimate the true population standard deviation (σ) when sample sizes are small. Dividing the mean difference by an underestimated denominator produces an inflated effect size estimate.

This mathematical phenomenon is identical to why we use n − 1 instead of n when computing sample variance (s²): unadjusted estimators carry systematic bias.

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The Mathematical Correction Factor (J)

Larry Hedges (1981) derived the exact bias-correction factor. The exact multiplier is based on gamma functions, but Hedges provided a standard, highly accurate approximation used across statistical software:

J = 1 − 3 / (4(n₁ + n₂) − 9) = 1 − 3 / (4 · df − 1)

g = d × J

Where:

• d = Uncorrected Cohen's d
• n₁, n₂ = Sample sizes of Group 1 and Group 2
• df = Degrees of freedom (n₁ + n₂ − 2)
• g = Bias-corrected Hedges' g

Because J is always less than 1.0, Hedges' g will always be slightly more conservative (smaller in magnitude) than Cohen's d.

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Impact of Sample Size on Bias: Comparison Table

The table below demonstrates how the correction factor J impacts an observed Cohen's d = 0.80 across sample sizes:

Key Takeaway from the Data

At n = 5 per group, Cohen's d inflates the true effect by more than 11%, potentially causing an author to classify a medium effect as "large". At n = 50 per group, the inflation drops to under 1%.

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Decision Framework: When to Report Which

                            [ Your Study ]
                                  │
                   Is total N < 40 OR meta-analysis?
                                 / \
                               /     \
                             YES      NO
                             /         \
                 [ Report Hedges' g ]  [ Report Cohen's d or Both ]

1. Always Report Hedges' g When:

• Total sample size N < 40 (e.g., pilot studies, specialized clinical populations, neuroimaging cohorts).
• You are writing or contributing data to a systematic review or meta-analysis.
• Target journal reviewers emphasize statistical conservatism and replication accuracy.

2. Cohen's d Is Acceptable When:

• Total sample size N > 50 per group, where numerical divergence between d and g is negligible.
• Following specific subfield historical conventions (though reporting both is modern best practice).

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APA 7th Edition Reporting Style

Modern APA reporting recommends presenting exact values for both statistics when sample sizes are modest:

Example Writeup

An independent-samples t-test indicated that the intervention group (n = 18, M = 42.10, SD = 6.40) demonstrated significantly higher resilience scores than the control group (n = 17, M = 36.80, SD = 6.90), t(33) = 2.35, p = .025, Cohen's d = 0.80, Hedges' g = 0.78, 95% CI [0.09, 1.47].

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Summary Comparison Matrix

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Key Takeaways

1. Hedges' g and Cohen's d share the same numerator and pooled denominator; Hedges' g simply multiplies by correction factor J.
2. For small studies (N < 40), reporting Cohen's d alone exaggerates effect magnitudes.
3. Both metrics use the identical Cohen (1988) heuristic tiers (0.20, 0.50, 0.80).
4. When writing empirical papers with small samples, report both d and g to maintain full methodological transparency.