Effect Size Calculator (Cohen's d, Hedges' g, r, η²)

Choose the input that matches what you have:

• **Means and SDs**: enter the mean, standard deviation, and sample size for each of two independent groups. This is the most accurate option.
• **t statistic**: enter the independent-samples t value and the two group sizes. Use this when a paper reports t but not SDs, for example when you are extracting data for a meta-analysis.

The calculator returns:

1. **Cohen's d**, the standardized mean difference using the pooled SD. 2. **Hedges' g**, d with a small-sample bias correction. 3. **r** (point-biserial correlation) and **η²** (eta squared), converted from the same result. 4. A verbal label based on Cohen's conventional benchmarks, plus an APA 7 sentence.

The sign of d follows the order of entry (Group 1 minus Group 2). Keep the order consistent with how you describe the comparison.

A p-value tells you whether an observed difference is larger than you would expect by chance; an effect size tells you **how large** the difference is. With a big enough sample, a trivial difference can be statistically significant, and with a small sample an important difference can fail to reach significance. That is why the APA Publication Manual (7th edition) asks authors to report effect sizes alongside significance tests.

Standardized effect sizes put results on a common scale:

• **Cohen's d** expresses a mean difference in standard-deviation units. d = 0.5 means the group means are half a standard deviation apart.
• **Hedges' g** is d multiplied by a correction factor. d slightly overestimates the population effect in small samples; g removes most of that bias. Meta-analyses usually use g.
• **r** expresses the same group difference as a correlation between group membership and the outcome.
• **η²** is the proportion of total variance in the outcome explained by group membership. For two groups it equals r².

Because they are unit-free, these indices can be compared across studies that used different scales, and they are the inputs needed for power analysis and meta-analysis.

From means, SDs, and sample sizes:

• Pooled SD: `sp = sqrt( ((n1 − 1)·s1² + (n2 − 1)·s2²) / (n1 + n2 − 2) )`
• Cohen's d: `d = (M1 − M2) / sp`
• Hedges' correction (common approximation): `J = 1 − 3 / (4·(n1 + n2) − 9)`
• Hedges' g: `g = J · d`

From an independent-samples t:

• `d = t · sqrt(1/n1 + 1/n2)`
• `r = sqrt( t² / (t² + df) )`, with `df = n1 + n2 − 2`
• `η² = t² / (t² + df)`

From d to r (when only d and the group sizes are known): `r = d / sqrt(d² + (n1 + n2)² / (n1·n2))`, which is close to `d / sqrt(d² + 4)` when the groups are equal in size.

Approximate standard error of d, for a 95% CI: `SE = sqrt( (n1 + n2)/(n1·n2) + d² / (2·(n1 + n2)) )`, and `CI = d ± 1.96·SE`.

A study compares exam scores after two teaching methods.

• Method A: M = 78.4, SD = 10.2, n = 30
• Method B: M = 72.1, SD = 11.5, n = 32

Step by step:

• `sp = sqrt((29·10.2² + 31·11.5²) / 60) = 10.89`
• `d = (78.4 − 72.1) / 10.89 = 0.58`
• `J = 1 − 3 / (4·62 − 9) = 0.987`, so `g = 0.987 · 0.58 = 0.57`
• The matching Student's t test gives t(60) = 2.28, p = .026. Entering that t with n1 = 30 and n2 = 32 returns the same d = 0.58.
• `r = sqrt(2.28² / (2.28² + 60)) = .28`, and `η² = .08`
• Approximate 95% CI for d: 0.07 to 1.09

All values were checked in Python with SciPy (`scipy.stats.ttest_ind_from_stats`) and NumPy.

Cohen (1988) proposed rough benchmarks for when nothing better is available:

| Index | Small | Medium | Large | |---|---|---|---| | d or g | 0.2 | 0.5 | 0.8 | | r | .10 | .30 | .50 | | η² | .01 | .06 | .14 |

By these labels, the example (g = 0.57) is a medium effect. Treat the labels with caution:

• Cohen described them as conventions for when field-specific information is missing, not as universal standards.
• What counts as meaningful depends on the outcome, the cost of the intervention, and what other studies in your area find. A small effect on mortality can matter a great deal; a large effect on a lab task may not.
• The CI matters as much as the point estimate. In the example the interval runs from almost zero to large, so the true effect is uncertain.

Report descriptive statistics, the test, and the effect size with its CI when possible:

> Students taught with Method A scored higher (M = 78.4, SD = 10.2) than students taught with Method B (M = 72.1, SD = 11.5), t(60) = 2.28, p = .026, d = 0.58, 95% CI [0.07, 1.09].

If you report g, say so and note that it includes the small-sample correction. In the final manuscript, italicize statistical symbols as APA style requires.

If you have the full dataset, Data2Paper's [Generate Paper](/generate-paper) tool can draft a complete paper from a CSV or Excel file, with the analysis, effect sizes, tables, and figures written up; you can preview the first half free before deciding to unlock it, and you should verify the numbers yourself.

• **Using the formula for independent groups with paired data.** Repeated-measures designs need a different standardizer (for example d based on the SD of difference scores, or d_av), and the choice changes the value. State which one you used.
• **Mixing up d and g** in meta-analyses or not saying which was reported.
• **Using η² from a multi-factor ANOVA as if it were partial η²**, or the other way round. They are different quantities once there is more than one factor.
• **Treating "small/medium/large" as verdicts** on practical importance.
• **Dropping the sign.** A negative d simply means Group 2 scored higher; keep the direction clear in the text.
• **Converting from t when SDs are available.** It works, but entering means and SDs avoids rounding error from a reported t.