5
15
0.333333
0.333333
0.329004
1
0
0
0
58
0.987013
5
15
0.333333
0.333333
0.329004
1
0
0
0
58
0.987013
The Effect Size Calculator (Cohen's d) computes the standardized difference between two group means, providing a measure of practical significance that complements p-values. While a p-value tells you whether an effect is statistically significant, Cohen's d tells you how large the effect is in standard deviation units. This distinction is crucial because with large enough samples, even trivially small effects can be statistically significant.
This calculator computes the pooled standard deviation, Cohen's d, and Hedges' g (a bias-corrected version for small samples). It also provides an interpretation based on Cohen's widely-used conventions. Effect sizes are essential for meta-analysis, power analysis, and making meaningful comparisons across studies that used different measurement scales.
Cohen's d is calculated as the difference between two group means divided by the pooled standard deviation:
$$d = \frac{M_1 - M_2}{S_p}$$
The pooled standard deviation weights each group's variance by its degrees of freedom:
$$S_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$
Where:
Cohen's d has a slight upward bias in small samples. Hedges' g corrects this using the approximation:
$$g = d \times \left(1 - \frac{3}{4(n_1 + n_2 - 2) - 1}\right)$$
This correction factor (sometimes called J) approaches 1 as sample size increases, so for large samples, d and g are virtually identical. The exact correction involves the gamma function, but this approximation is accurate to several decimal places for all practical sample sizes.
Cohen's d is measured in standard deviation units. A d of 0.5 means the two group means differ by half a standard deviation. The sign indicates direction: positive means Group 1 has the higher mean, negative means Group 2 does. The absolute value indicates magnitude regardless of direction.
Cohen (1988) proposed the following conventions for interpreting d, though he emphasized that context matters more than arbitrary thresholds:
| |d| Range | Interpretation | Overlap |
|---|---|---|
| < 0.20 | Negligible | ~92% overlap between groups |
| 0.20 - 0.49 | Small | ~85% overlap |
| 0.50 - 0.79 | Medium | ~67% overlap |
| 0.80 - 1.19 | Large | ~53% overlap |
| ≥ 1.20 | Very Large | < 45% overlap |
The "overlap" column shows what percentage of the two distributions share common area (Cohen's U₃ complement), providing an intuitive sense of how much the groups differ.
Inputs
Results
Treatment group (M₁ = 105, s₁ = 15, n₁ = 30) vs. placebo (M₂ = 100, s₂ = 15, n₂ = 30). Sp = √((29×225 + 29×225)/58) = 15.0. d = (105-100)/15 = 0.333, a small effect. Hedges' g = 0.333 × (1 - 3/231) = 0.329. The treatment raises scores by about a third of a standard deviation.
Inputs
Results
Intervention group (M₁ = 82, s₁ = 12, n₁ = 50) vs. control (M₂ = 68, s₂ = 14, n₂ = 45). Sp = √((49×144 + 44×196)/93) = √168.19 ≈ 12.97. d = (82-68)/12.97 ≈ 1.08, a large effect. The intervention improved scores by over one standard deviation.
Cohen's d and Hedges' g are both standardized mean difference effect sizes, but Hedges' g includes a bias correction for small samples. Cohen's d slightly overestimates the population effect size, especially when group sizes are small (n < 20). Hedges' correction factor (approximately 1 - 3/(4df - 1)) shrinks d slightly toward zero. For large samples, the difference is negligible. Hedges' g is preferred for meta-analysis because it provides unbiased estimates that can be properly weighted and combined across studies.
P-values only tell you whether an effect is statistically significant (unlikely under the null hypothesis), not whether it is practically meaningful. With a large enough sample, even a tiny, meaningless difference can be statistically significant. Effect sizes quantify how large the difference is, independent of sample size. This information is essential for: judging practical importance, planning future studies (power analysis), comparing results across studies with different scales, and conducting meta-analyses.
A negative d simply means that Group 2 has a higher mean than Group 1. The sign depends on which group you label as Group 1. The magnitude (absolute value) of d is what matters for interpreting effect size. A d of -0.8 has the same practical importance as d = +0.8; only the direction differs. When reporting, you can either note the direction explicitly or arrange the groups so d is positive, as long as you clearly state which group was subtracted from which.
Cohen's d is inappropriate when: (1) group variances are very different (ratio > 2:1) -- consider Glass's delta, which uses only the control group's SD; (2) the outcome is binary -- use odds ratios or risk ratios instead; (3) the data are ordinal or non-normal -- consider rank-biserial correlation or Cliff's delta; (4) you're comparing correlated or paired observations -- use Cohen's d_z for paired designs; (5) you're measuring association rather than group differences -- use r, R², or η².
Several practical translations help: (1) Percentage of non-overlap: d = 0.5 means about 33% of the distributions don't overlap. (2) Probability of superiority: d = 0.5 means a randomly chosen person from Group 1 has about a 64% chance of scoring higher than a random person from Group 2. (3) Percentile shift: d = 0.5 means the average person in Group 1 exceeds about 69% of Group 2. (4) Number needed to treat (NNT): for clinical contexts, NNT ≈ 1/(Φ(d/√2) - 0.5)/2, where Φ is the normal CDF.
The pooled standard deviation combines the variability from both groups into a single estimate by weighting each group's variance by its degrees of freedom (n-1). It assumes homogeneity of variance -- that both populations have similar spread. This is analogous to the pooled variance used in the independent-samples t-test. If group variances are very unequal, the pooled SD may not be appropriate, and alternatives like Glass's Δ (using only the control group SD) or the unweighted average SD should be considered.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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