Effect Size Calculator: Cohen’s d and Hedges’ g

Calculate effect size from group means and standard deviations, t-statistics, or correlation coefficients. See the two distributions side by side, with overlap shaded.

Group 1 Mean

Group 2 Mean

Identify Group 2 as control (compute Glass’s Δ)

Cohen’s d —

Hedges’ g —

Pearson r —

η² (Eta-squared) —

95% CI for d —

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Fig. 1 — Two normal distributions with effect size — standard deviations apart. Shaded area shows overlap (common-language effect size: — probability one group exceeds the other).

How This Was Calculated

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§1 The Formulas for Each Variant

Effect size quantifies how large the practical difference is, separate from whether the difference is statistically significant. There are many flavours; these are the most common in psychology and education.

Cohen’s d (independent samples)

d = (M₁ − M₂) / s_pooled
where s_pooled = √[(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2)]

d: Cohen's effect size (standardized mean difference)
M₁, M₂: Group means
s₁, s₂: Sample standard deviations
n₁, n₂: Sample sizes
s_pooled: Pooled standard deviation (weighted by degrees of freedom)

Hedges’ g (small-sample unbiased correction)

g = d × J
where J = 1 − [3 / (4(n₁+n₂) − 9)]

g: Hedges' unbiased effect size
J: Small-sample correction factor (standard approximation to the exact gamma-function form). When n is large, J approaches 1 and g ≈ d.

Glass’s Δ (control group SD only)

Δ = (M_treatment − M_control) / s_control

Δ: Glass's delta. Use this when you want to standardize by the control group’s variability only, especially if the control and treatment groups have different variances.

One-sample or paired: d = (M − μ₀) / s

M: Sample mean
μ₀: Comparison value (hypothesized population mean)
s: Sample SD

From t-statistic

Independent samples: d = t × √(1/n₁ + 1/n₂)

Paired or one-sample: d = t / √n

Conversions among effect sizes

When you have correlation r and want d (assuming equal group sizes):

d = 2r / √(1 − r²)

When you have d and want r:

r = d / √(d² + 4)

From η² (eta-squared) to d:

d = 2√[η² / (1 − η²)]

From t-statistic and df to η²:

η² = t² / (t² + df)

Confidence interval for d (large-sample approximation)

The standard error of d is:

SE = √{[(n₁+n₂) / (n₁n₂)] + [d² / (2(n₁+n₂))]}

Then the 95% CI is d ± 1.96 × SE. Note: this is a large-sample approximation, not exact. It may be unreliable for very small samples.

Common-language effect size (overlap probability)

The probability that a randomly chosen member of Group 1 scores higher than one from Group 2:

Φ(d/√2)

Φ: Standard normal cumulative distribution function

This translates the abstract d into an intuitive statement: "there’s a 72% chance someone from Group 1 will have a higher score than someone from Group 2."

§2 Which One Do I Report: d, g, or η²?

In most contexts, report Hedges' g. It is the unbiased estimator and accounts for small-sample bias. If your sample is large (both n > 50), g and d are nearly identical.

Cohen's d is simple and widely recognized. It is slightly biased when samples are small, but the bias is usually negligible.
Glass's Δ (control group SD) is used when treatment and control groups have very different variances or when you specifically want to standardize by the control condition.
η² (eta-squared) is the proportion of variance in the outcome explained by group membership. It is especially common in ANOVA contexts. It is not a mean-difference effect size.
Pearson r is used for continuous two-variable associations and is widely understood in many fields.

§3 Worked Example

A researcher compares reading comprehension in two groups: Group 1 (M = 22.4, SD = 4.1, n = 48) and Group 2 (M = 20.1, SD = 3.8, n = 52).

Step 1: Pooled SD
s_pooled = √{[(48−1) × 4.1² + (52−1) × 3.8²] / (48+52−2)} = √{[767.47 + 754.68] / 98} = √15.53 = 3.94

Step 2: Cohen's d
d = (22.4 − 20.1) / 3.94 = 2.3 / 3.94 = 0.584

Step 3: Hedges' g
J = 1 − 3 / [4(48+52) − 9] = 1 − 3 / 391 = 0.9923
g = 0.584 × 0.9923 = 0.580

Interpretation: This is a small-to-medium effect. On the common-language scale, there is a 70% probability that a randomly chosen member of Group 1 will score higher than one from Group 2.

§4 Conventional Benchmarks and Important Caveats

Jacob Cohen proposed rough guidelines in the 1980s:

Magnitude	Cohen's d
Small	0.2
Medium	0.5
Large	0.8

Critical caveat

These are conventions, not universal thresholds. Cohen himself described them as arbitrary rules of thumb for use when no better basis is available. In some fields, a d of 0.2 is substantial; in others, 0.8 is trivial. Always ground effect size interpretation in your research domain, the cost of the intervention, and prior literature—never treat these numbers as fixed rules.

§5 Why a Significant p-Value with a Tiny Effect Size Usually Means a Large Sample

A small p-value tells you that an observed difference is unlikely under the null hypothesis. But likelihood and practical importance are different things. With a very large sample, you can detect a tiny, meaningless difference with high confidence. Example: if you survey 10,000 people and find that Group A's average height is 170.1 cm and Group B's is 170.0 cm, the p-value might be 0.02, but d ≈ 0.08—essentially no real-world difference. The small p-value reflects n, not importance.

Always report both. The p-value answers "Is this effect real?" The effect size answers "How big is it?" Together they tell the full story.

§6 How to Report Effect Size in APA Style

In your results section, include the effect size alongside the test statistic and p-value:

t(98) = 2.31, p = .021, g = 0.58, 95% CI [0.21, 0.95]

Or in a table of effect sizes for meta-analysis. Prefer Hedges' g over Cohen's d for publication.

§7 FAQ

What’s the difference between Cohen’s d and Hedges’ g?

Hedges' g applies a small-sample correction factor to d. When sample sizes are large, they are nearly identical. When samples are small, g is the preferred, unbiased estimate. In meta-analyses combining many studies, use g.

Can I compute effect size from a p-value alone?

No. A p-value is a function of both effect size and sample size; reversing it requires knowing n. If you know the p-value and n, you can recover the test statistic (t, z, F, etc.), and from that compute d.

Is effect size the same as correlation?

No, but they are related. Pearson r measures the linear association between two continuous variables. Cohen's d measures the standardized mean difference between groups. They can be converted to each other, but they answer different questions. Use r for "how tightly are two variables linked?" and d for "how much do the groups differ?"

What if one group has zero variance?

If a group’s SD is 0 (all members have identical scores), the pooled SD is undefined and effect size cannot be computed. This usually signals a data-entry error or a group with no variability. Check your data.

Can effect size be negative?

Yes. d is simply (Group 1 mean − Group 2 mean) / pooled SD, so the sign depends on which group you list first. A negative d means Group 2 has the higher mean. The magnitude is what matters; d = −0.5 is the same strength as d = 0.5, just in opposite directions.

Why does the confidence interval for d sometimes include zero?

When the 95% CI spans zero, the data are consistent with a null effect (or an effect in either direction). Even if the point estimate is d = 0.4, a wide CI signals sampling uncertainty. Larger samples yield narrower CIs and more precision.

Sources

Wikipedia: Effect size — definition, families (Cohen's d, Hedges' g, Glass's Δ), and interpretation caveats.
Statology: Cohen's d formula and interpretation — worked examples and benchmark guidance.
American Psychological Association — official APA style guidelines for reporting effect sizes in research.
NIH / PubMed Central: "Understanding the effect size" — comprehensive overview of effect sizes in medical research.
Khan Academy: Statistics and Probability — introductory lessons on statistical effect and inference.