Effect Size Calculator
Calculate effect sizes including Cohen d, Hedges g, correlation r, Cramer V, and eta-squared. Interpret practical significance with benchmarks.
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 is the pooled standard deviation in Cohen d actually computed?
The pooled standard deviation is the square root of the weighted average of the two group variances, weighting each by its degrees of freedom: SD_pooled = sqrt([(n1-1)s1^2 + (n2-1)s2^2] / (n1 + n2 - 2)). With n1 = n2 = 30, s1 = 10 and s2 = 12, that is sqrt((29 x 100 + 29 x 144) / 58) = sqrt(7076 / 58) = sqrt(122.0) = 11.045. Pooling assumes roughly equal population variances; when they differ markedly, Glass delta, which divides by the control group SD alone, is the safer denominator.
What is eta-squared, and how does partial eta-squared differ from it?
Eta-squared is the between-groups sum of squares divided by the total sum of squares, so it reports the share of total variance in the outcome attributable to a factor. Partial eta-squared instead divides by the sum of the effect and error sums of squares, excluding variance explained by other factors in the model. In a multi-factor ANOVA partial values are therefore larger, sometimes much larger, and they do not sum to 1 across effects. Always state which version you report, because the two are frequently confused in write-ups.
Why does the interpretation of Cramer V depend on the size of the contingency table?
Cramer V is computed as sqrt(chi-square / (n x (k - 1))) where k is the smaller of the row and column counts, which is what rescales it onto a 0 to 1 range regardless of table dimensions. Because the denominator grows with k, the same absolute departure from independence yields a smaller V in a large table than in a 2x2. Cohen's benchmarks for V are consequently stated per degrees of freedom, and a V of 0.2 in a 5x5 table represents a stronger association than the same value in a 2x2.
Why should an effect size be reported with a confidence interval?
A point estimate such as d = 0.45 hides how precisely it was measured. Small samples produce wide intervals that may span from negligible to large, so a study reporting d = 0.45 with an interval of roughly -0.06 to 0.96 offers far weaker evidence than the same d from a large trial with a narrow interval. Reporting the interval also makes clear whether the estimate is compatible with zero, which is the same information a significance test gives, plus the range of magnitudes still plausible.
Which effect size measure should I choose for my study design?
Match the measure to the analysis. Two independent group means call for Cohen d, or Hedges g when either group is small. A linear relationship between two continuous variables calls for r and r-squared. An ANOVA with three or more groups calls for eta-squared or Cohen f. Two categorical variables in a contingency table call for Cramer V. Reporting a measure that does not match the design, such as d for a chi-square result, makes the number impossible to pool in a later meta-analysis.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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