Effect Size Calculator
Calculate effect size instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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.
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.
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.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Does Effect Size Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.