About

Reporting statistical significance without effect size is incomplete. A p-value tells you whether an effect exists. It does not tell you how large the effect is. Cohen's d quantifies the standardized mean difference between two groups, expressed in pooled standard deviation units. Misreporting or omitting effect size inflates the risk of publishing trivially small effects as meaningful findings. The APA Publication Manual (7th ed.) mandates effect size reporting for all primary outcomes. This calculator computes Cohen's d, Hedges' g (bias-corrected for small samples where n < 20), and Glass's Δ (when group variances are unequal). It derives the 95% confidence interval using the large-sample variance approximation and reports Common Language Effect Size (CLES). Note: all formulas assume independent samples and approximately normal distributions. For heavily skewed data or unequal variances exceeding a 2:1 ratio, Glass's Δ is preferred over pooled d.

Formulas

The primary effect size is Cohen's d, computed as the difference in group means divided by the pooled standard deviation:

d = M₁ − M₂S_pooled

where the pooled standard deviation is:

S_pooled = √(n₁ − 1) ⋅ s₁² + (n₂ − 1) ⋅ s₂²n₁ + n₂ − 2

Hedges' g applies a correction factor J for small-sample bias:

g = d ⋅ J , J = 1 − 34 ⋅ df − 1

Glass's Δ uses only the control group's standard deviation:

Δ = M₁ − M₂s₂

The 95% confidence interval uses the variance approximation:

V_d = n₁ + n₂n₁ ⋅ n₂ + d²2 ⋅ (n₁ + n₂)

Common Language Effect Size (CLES) converts d to a probability via the normal CDF:

CLES = Φ(d√2)

where M₁, M₂ = group means; s₁, s₂ = group standard deviations; n₁, n₂ = sample sizes; df = n₁ + n₂ − 2; Φ = standard normal CDF.

Reference Data

Effect Size Metric	Cohen's Convention	Interpretation	Percentile Standing	% Non-Overlap (U₃)	CLES
Very Small	d = 0.01	Negligible difference	50.4%	0.8%	50.7%
Small	d = 0.20	Barely noticeable	57.9%	14.7%	55.6%
Small - Medium	d = 0.35	Detectable with care	63.7%	24.2%	59.7%
Medium	d = 0.50	Visible to careful observer	69.1%	33.0%	63.8%
Medium - Large	d = 0.65	Noticeable difference	74.2%	40.5%	67.7%
Large	d = 0.80	Obvious difference	78.8%	47.4%	71.4%
Very Large	d = 1.00	Substantial gap	84.1%	55.4%	76.0%
Very Large	d = 1.20	Very large gap	88.5%	62.2%	80.2%
Huge	d = 1.50	Dramatic separation	93.3%	70.7%	85.6%
Huge	d = 2.00	Near-total separation	97.7%	81.1%	92.1%
Percentile standing: the percentile rank of the average Group 1 member in Group 2's distribution. U₃: percentage of Group 1 above the Group 2 median. CLES: probability a randomly chosen Group 1 member scores higher.

Frequently Asked Questions

Use Hedges' g when either group has n < 20. Cohen's d has a positive bias in small samples - it systematically overestimates the population effect size. The Hedges correction factor J = 1 − 3 ÷ (4df − 1) shrinks d by approximately 0.75% at df = 100 but by 7.5% at df = 10. For meta-analyses, Hedges' g is the standard metric.

CLES is the probability that a randomly selected individual from Group 1 will score higher than a randomly selected individual from Group 2. For example, a Cohen's d of 0.80 yields a CLES of approximately 71.4%, meaning if you pick one person from each group at random, the Group 1 member will outscore the Group 2 member 71 times out of 100. It assumes both distributions are normal with equal variances.

When the ratio s₁ ÷ s₂ exceeds 2.0, the pooled SD becomes a poor estimate of spread because it weights the larger-variance group disproportionately. In this case, Glass's Δ is preferred because it uses only the control group's SD (s₂) as the standardizer. This calculator flags the variance ratio and recommends Glass's Δ when the ratio is extreme.

Yes. The sign of d depends on which group mean is subtracted from which. A negative d means Group 2's mean is higher than Group 1's. The sign is arbitrary and depends on labeling. The magnitude (absolute value) determines effect size strength. When reporting, always specify which direction favors which group.

The confidence interval width depends on both n and d. For a 95% CI of width ±0.20 around a medium effect (d = 0.50), you need approximately 100 participants per group. With n = 30 per group, the CI width is roughly ±0.38, making it difficult to distinguish small from medium effects. Plan sample sizes with the desired CI precision in mind.

It handles both. The pooled standard deviation formula weights each group's variance by its degrees of freedom (n − 1), so unequal group sizes are properly accommodated. However, severely unbalanced designs (e.g., n₁ = 10, n₂ = 200) produce wider confidence intervals and less stable estimates than balanced designs with the same total N.