Effect Size Calculator
Calculate effect size instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Calculator
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Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Treatment vs Control Group
Example 2: Correlation Effect Size
Background & Theory
The Effect Size Calculator applies the following established principles and formulas. Statistics and probability provide the mathematical framework for drawing conclusions from data under uncertainty. The measures of central tendency describe where data cluster. The mean is the arithmetic average, computed as the sum of all values divided by the count. The median is the middle value of an ordered dataset, robust to extreme outliers. The mode is the most frequent value. Spread is quantified by variance, the average squared deviation from the mean, and by its square root, the standard deviation. For a sample, variance uses n minus one in the denominator to correct for bias in estimation. The normal distribution, defined by its mean and standard deviation, is the cornerstone of parametric statistics. Its bell-shaped probability density follows the formula f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2). The empirical rule states that approximately 68 percent of observations fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. A z-score standardizes a data point by subtracting the mean and dividing by the standard deviation, expressing how many standard deviations an observation lies from the mean. In hypothesis testing, the p-value is the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. Confidence intervals express the range within which the true population parameter falls with a specified probability, typically 95 percent. Correlation measures linear association between two variables, with Pearson's r ranging from negative one to positive one. Correlation does not imply causation. Linear regression fits a line of the form y = a + bx to minimize the sum of squared residuals. Bayes' theorem relates conditional probabilities: P(A|B) = P(B|A) * P(A) / P(B), allowing prior beliefs to be updated on new evidence. The law of large numbers guarantees that the sample mean converges to the population mean as sample size grows. The central limit theorem states that the distribution of sample means approaches normality regardless of the population distribution, provided the sample size is sufficiently large, typically 30 or more.
History
The history behind the Effect Size Calculator traces back through the following developments. The mathematical study of probability emerged in the 17th century from correspondence between Blaise Pascal and Pierre de Fermat in 1654. Their exchange, prompted by a gambling problem posed by the Chevalier de Mere, established the foundations of probability theory by calculating expected outcomes through systematic enumeration of cases. Jacob Bernoulli formalized the law of large numbers in his posthumously published Ars Conjectandi of 1713, proving rigorously that empirical frequencies converge to theoretical probabilities with increasing observations. His work laid the groundwork for inferential statistics by connecting mathematical probability to observed data. Carl Friedrich Gauss developed the method of least squares around 1795 while adjusting astronomical observations, and he recognized the bell-shaped error distribution that now bears his name. Pierre-Simon Laplace independently worked on the normal distribution and proved an early version of the central limit theorem around 1810, demonstrating why errors in measurement tend toward normality. The late 19th century saw statistics emerge as a distinct scientific discipline. Francis Galton introduced regression and correlation in the 1880s while studying heredity. Karl Pearson formalized these concepts, developed the chi-squared test, and founded the journal Biometrika in 1901, establishing statistics as a rigorous academic field. Ronald Fisher transformed statistical practice in the early 20th century. His 1925 book Statistical Methods for Research Workers introduced significance testing, analysis of variance, and the concept of the p-value as a decision threshold, establishing the framework still used in scientific research. Fisher and Jerzy Neyman engaged in a prolonged methodological dispute over the interpretation of hypothesis tests. The Bayesian approach, rooted in the 18th century work of Thomas Bayes and Laplace, was largely eclipsed by frequentist methods through much of the 20th century but experienced a revival after World War II and accelerated with computational advances. The late 20th and early 21st centuries brought statistics into every domain through big data, machine learning, and the routine availability of software capable of processing millions of observations.
Frequently Asked Questions
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.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy