Effect Size Calculator
Calculate effect size instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
Effect Size Calculator
Calculator
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Formula: Cohen d = (M1 - M2) / SD_pooled; r = d / sqrt(d^2 + 4); eta^2 = SS_between / SS_total
Additional inputs: Total N, Min Dimension (k), SS Between, SS Total.
Worked example — Cohen d: 0.453 | Hedges g: 0.447 | CLES: 62.4% | Small-Medium Effect
Formula
Cohen d = (M1 - M2) / SD_pooled; r = d / sqrt(d^2 + 4); eta^2 = SS_between / SS_total
Cohen d divides the mean difference by the pooled standard deviation. Correlation r can be derived from d. Eta-squared is the ratio of between-group variance to total variance. Each measures effect magnitude in a different context.
Worked Examples
Example 1: Treatment vs Control Group
Problem:A treatment group (n=30, mean=75, SD=10) is compared to a control group (n=30, mean=70, SD=12). Calculate Cohen d and interpret.
Solution:Mean difference = 75 - 70 = 5 Pooled SD = sqrt(((29 x 100) + (29 x 144)) / 58) = sqrt(7076/58) = sqrt(122.0) = 11.045 Cohen d = 5 / 11.045 = 0.4528 Hedges g = 0.4528 x (1 - 3/(4x58-1)) = 0.4528 x 0.987 = 0.4469 CLES = 62.4% Interpretation: Small to Medium effect
Result:Cohen d: 0.453 | Hedges g: 0.447 | CLES: 62.4% | Small-Medium Effect
Example 2: Correlation Effect Size
Problem:A study finds r = 0.45 between study hours and exam scores. Convert to Cohen d and interpret.
Solution:r = 0.45 r-squared = 0.2025 (20.25% variance explained) Cohen d = 2 x 0.45 / sqrt(1 - 0.2025) = 0.90 / 0.893 = 1.008 Interpretation: Large effect A correlation of 0.45 explains about 20% of the variance in exam scores.
Result:r = 0.45 | r-squared: 20.25% | Equivalent d: 1.008 | Large Effect
Frequently Asked Questions
What is effect size and why is it important in statistics?
Effect size is a quantitative measure of the magnitude of a phenomenon or the strength of the relationship between variables, independent of sample size. While p-values tell you whether an effect exists (statistical significance), effect size tells you how large that effect is (practical significance). This distinction is crucial because with a large enough sample, even trivially small differences can be statistically significant. For example, a study with 10000 participants might find a statistically significant difference in test scores of 0.5 points on a 100-point scale, which while real is practically meaningless. Effect sizes allow researchers to compare results across studies, conduct meta-analyses, and determine whether findings have real-world importance beyond mere statistical detection.
What is Cohen d and how do I interpret it?
Cohen d is the most commonly used effect size measure for comparing two group means. It is calculated by dividing the difference between group means by the pooled standard deviation. Jacob Cohen proposed benchmark interpretations: d equals 0.2 is a small effect, meaning the two group distributions overlap about 85 percent. A d of 0.5 is a medium effect with about 67 percent overlap. A d of 0.8 is a large effect with about 53 percent overlap. Values above 1.0 indicate very large effects. For context, the effect of psychotherapy versus no treatment is typically d equals 0.8. The gender difference in height is approximately d equals 1.7. These benchmarks are guidelines, not rules. In some fields like pharmacology, a small effect size may still be clinically important if the condition is severe.
What is the difference between Cohen d and Hedges g?
Both Cohen d and Hedges g measure the standardized mean difference between two groups, but Hedges g includes a correction factor for small sample bias. Cohen d slightly overestimates the population effect size when sample sizes are small, typically below 20 per group. Hedges g multiplies Cohen d by a correction factor of approximately 1 minus 3 divided by 4 times the total degrees of freedom minus 1. For large samples of 30 or more per group, the difference between d and g is negligible, usually less than one percent. However, for small samples, the correction can be meaningful. Most meta-analyses use Hedges g because it provides an unbiased estimate of the population effect size. In practice, you should report Hedges g when either group has fewer than 20 participants.
How do I convert between different effect size measures?
Effect size measures can be converted between different types using established formulas. Cohen d converts to correlation r using the formula r equals d divided by the square root of d squared plus 4. Correlation r converts to d using d equals 2r divided by the square root of 1 minus r squared. Eta squared converts to Cohen f using f equals the square root of eta squared divided by 1 minus eta squared. R squared equals r times r, so a correlation of 0.3 means 9 percent of variance is explained. These conversions allow comparison across studies using different designs. For example, Cohen d of 0.5 corresponds to r of 0.243 and r squared of 5.9 percent. Understanding these relationships helps researchers interpret findings from different methodological approaches within the same framework.
What is the Common Language Effect Size and how should I use it?
The Common Language Effect Size, also known as the probability of superiority, translates Cohen d into a probability that is much easier for non-statisticians to understand. It represents the probability that a randomly chosen individual from the higher-scoring group will score higher than a randomly chosen individual from the lower-scoring group. For example, a Cohen d of 0.8 corresponds to a CLES of approximately 71 percent, meaning there is a 71 percent chance that a random person from the treatment group outperforms a random person from the control group. A CLES of 50 percent indicates no effect because it equals chance. This measure is particularly valuable when communicating results to clinicians, policymakers, or the general public who may not understand standardized mean differences but intuitively grasp probability statements.
How do I determine the right sample size for a study?
Sample size depends on the desired confidence level (typically 95%), margin of error (e.g., plus or minus 3%), and expected variability. The formula is n = (Z^2 * p * (1-p)) / E^2, where Z is the z-score for your confidence level, p is the expected proportion, and E is the margin of error. Larger populations need surprisingly similar sample sizes.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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