Effect Size Calculator: Cohen’s d & More

This effect size calculator computes Cohen’s d, Hedges’ g, and the correlation coefficient r from the means, standard deviations, and sample sizes of two groups. You enter the summary statistics for each group, and the calculator returns the standardized mean difference (effect size) along with its confidence interval, giving you a measure of the practical significance of your results.

Formula

Cohen’s d is the difference between two group means divided by the pooled standard deviation. The formula is:

d = (Mâ‚� – Mâ‚‚) / s_pooled

where:

• Mâ‚� = mean of group 1 (same unit as the data)
• Mâ‚‚ = mean of group 2 (same unit as the data)
• s_pooled = pooled standard deviation, calculated as √[((nâ‚�-1)SDâ‚�² + (nâ‚‚-1)SD₂²) / (nâ‚� + nâ‚‚ – 2)]
• nâ‚�, nâ‚‚ = sample sizes of group 1 and group 2
• SDâ‚�, SDâ‚‚ = standard deviations of group 1 and group 2

The pooled SD is a weighted average of the two group variances, giving more weight to the larger sample. This assumes the two groups have similar variances (homogeneity of variance).

Worked Example

Using the default values: Group 1 has mean 100, SD 15, n=30; Group 2 has mean 110, SD 15, n=30.

1. Compute the pooled variance: ((30-1)×15² + (30-1)×15²) / (30+30-2) = (29×225 + 29×225) / 58 = (6525 + 6525) / 58 = 13050 / 58 = 225.
2. Pooled SD = √225 = 15.
3. Cohen’s d = (100 – 110) / 15 = -10 / 15 = -0.667.
4. Hedges’ g = -0.667 × (1 – 3 / (4×60 – 9)) = -0.667 × (1 – 3/231) = -0.667 × 0.987 = -0.658.
5. r equivalent = -0.667 / √((-0.667)² + 4) = -0.667 / √(0.445 + 4) = -0.667 / √4.445 = -0.667 / 2.108 = -0.316.

The calculator shows Cohen’s d = -0.667, Hedges’ g = -0.658, and r = -0.316. The negative sign indicates Group 1 has the lower mean; the magnitude is what matters for interpretation.

When to Use Effect Size

Effect size quantifies the magnitude of a difference or relationship, independent of sample size. While p-values tell you if an effect exists, effect sizes tell you how large it is. Use it when:

• You want to report the practical significance of a study, not just statistical significance.
• You are conducting a meta-analysis and need a standardized metric to combine studies.
• You are planning a study and need to estimate sample size (using a target effect size).
• You are comparing results across studies that used different measurement scales.

Common benchmarks for Cohen’s d (Cohen, 1988): 0.2 = small, 0.5 = medium, 0.8 = large. These are rules of thumb, not absolute thresholds; context matters.

How to Choose the Right Effect Size Measure

Several effect size indices exist; the choice depends on your data and design.

• Cohen’s d – for comparing two independent groups with similar variances and normally distributed data. Most common in psychology and education.
• Hedges’ g – same as Cohen’s d but with a correction for small sample sizes (n < 20 per group). Use when sample sizes are small.
• Glass’s Δ – uses only the control group’s SD. Use when the treatment group’s variance is very different (e.g., in clinical trials).
• Pearson’s r – for correlation or regression; can be converted from d. Values: 0.1 = small, 0.3 = medium, 0.5 = large.
• Eta-squared (η²) – for ANOVA, proportion of variance explained.

If you have more than two groups, use η² or ω². If you have paired data (pre-post), use Cohen’s d_z (mean difference / SD of differences).

Common Mistakes and Misconceptions

A frequent error is using the wrong standard deviation. For Cohen’s d, you must use the pooled SD, not the SD of one group. Another mistake is ignoring the sign: d = -0.5 and d = 0.5 have the same magnitude, but the sign indicates direction. Also, effect size is not affected by sample size, but its precision (confidence interval) is – larger samples give narrower CIs.

Misconception: “An effect size of 0.8 is always large.� In some fields (e.g., medicine), 0.8 might be huge; in others (e.g., social psychology), it might be rare. Always interpret in context.

Rounding: keep at least two decimals in your calculations to avoid rounding errors in the final result.

Limitations and Assumptions

The formulas assume:

• Independent groups (not paired).
• Approximately normal distributions.
• Homogeneity of variance (for Cohen’s d). If variances are very different, use Welch’s correction or Glass’s Δ.

Effect size does not tell you if the result is statistically significant – that’s what p-values are for. A large effect size from a small sample may be non-significant; a tiny effect from a huge sample may be significant. Report both.

Frequently Asked Questions

What is a good effect size? There is no universal “goodâ€� value. Cohen’s benchmarks (0.2, 0.5, 0.8) are rough guides. In your field, compare with published studies. A “goodâ€� effect size is one that is meaningful for your research question, not just a number.

Can I use this calculator with unequal sample sizes? Yes. The pooled SD formula weights by sample size, so it works with unequal n. However, if the group sizes are very different and variances are unequal, consider a different effect size (e.g., Glass’s Δ).

How do I convert Cohen’s d to r? Use the formula r = d / √(d² + 4) for two equal-sized groups. For unequal groups, a more complex formula exists, but the simple one is a good approximation. The calculator provides this conversion.

What is the difference between Cohen’s d and Hedges’ g? Hedges’ g includes a correction factor (1 – 3/(4(nâ‚�+nâ‚‚)-9)) that reduces bias for small samples. For large samples, they are nearly identical. Use g when n < 20 per group.

Do I need to report effect size in my paper? Most journals now require it. It helps readers understand the magnitude of your findings and facilitates meta-analysis. Always report the effect size measure, its value, and a confidence interval.

Can effect size be negative? Yes. The sign indicates the direction of the difference (which group has the higher mean). When reporting, you can take the absolute value, but keep the sign if direction is important.

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in m1 | Mean of Group 1 | - | 100 | 0
in sd1 | SD of Group 1 | - | 15 | 0
in n1 | Sample Size Group 1 | - | 30 | 2
in m2 | Mean of Group 2 | - | 110 | 0
in sd2 | SD of Group 2 | - | 15 | 0
in n2 | Sample Size Group 2 | - | 30 | 2
out cohend | Cohen's d | - | (m1 - m2) / sqrt(((n1-1)*sd1^2 + (n2-1)*sd2^2) / (n1 + n2 - 2)) | 3
out hedgesg | Hedges' g (corrected) | - | cohend * (1 - 3 / (4 * (n1 + n2) - 9)) | 3
out r_equiv | Correlation r equivalent | - | cohend / sqrt(cohend^2 + 4) | 3