# Effect Size Calculator

Group 1:
M1
SD1
Group 2:
M2
SD2
Cohen's d  =  4.2422
Effect Size (r)  =  0.9045
CALCULATE

Effect Size calculator calculates the value of effect size correlation and the Cohen's-D using the means and standard deviations of two groups. This calculator finds the effect size using Cohen's d formula. There are two types of methods to perform the error size calculation. The first method uses the input values of means and standard deviations and second one uses the inputs of t-value and degrees of freedom value to estimate the effect of size. It is necessary to follow the next steps:

1. Enter means and standard deviations of two groups in the boxes. These values must be real numbers;
2. Press the "CALCULATE" button to make the computation;
3. Effect size calculator will find the Cohen's-D and the effect size for two groups.
Input : Four real numbers;
Output: Two real numbers

Cohen's d & Effect Size Formula:
Cohen's d Formula: Cohen's-d & is determined by the formula $$d=\frac{\bar X-\bar Y}{s}$$ where $\bar X$, $\bar Y$ are sample means and $s=\sqrt{\frac{s_X^2+s_Y^2}{2}}$ is the pooled standard deviation.

Effect Size Formula: The effect size is determined by the formula $$r=\frac{d}{\sqrt{d^2+4}}$$ where $d$ is a Cohen's d.

## What is Effect Size?

An effect size is a measure of the magnitude of a phenomenon. For instance, the correlation between two variables, the regression coefficient in a regression, the mean difference, are examples of effect size. The effect size is independent of sample sizes. In other words, it measures a treatment effect and it is independent of the sizes of samples. In meta analysis, in addition to the statistical hypothesis, we consider also effect sizes. The standard error of the effect size is of great importance. The effect size can be measured in the following ways:

• as the standardized difference between two means;
• as the correlation between the independent variable and the dependent variable.
One of the simplest methods for measuring effect size is Cohen's-D. Cohen (1988) used a mean difference in terms of the standard deviation of either group to measure the effect size. In practice, the pooled standard deviation is commonly used (Rosnow, R. L., Rosenthal, R. (1996), Computing contrasts, effect sizes, and counternulls on other people's published data: General procedures for research consumers, Psychological Methods, 1(4), 331-340). The pooled standard deviation is the root mean square of the two standard deviations.
Cohen's D is an effect size indicates the standardized difference between two means. It can be used, for example, to accompany the reporting of t-test and ANOVA results.
Cohen's D is the ratio of the difference between the means and the pooled standard deviation: $$d=\frac{\bar X-\bar Y}{s}$$ where $s=\sqrt{\frac{s_X^2+s_Y^2}{2}}$. Cohen's d better works for sample sizes largen than $50$. If the sample sizes smaler than $50$, it tends to over-inflate results.
The effect size can be found from Cohen's D in the following way $$r=\frac{d}{\sqrt{d^2+4}}$$ The effect size for t-Test indicates whether or not the difference between two means is statistically significant. The effect size and Cohen's D from t-Test and df values are determined by the formulas: $$r^2 = {\frac{t^2}{t^2+df}},\quad d=\frac{2t}{\sqrt{df}}$$ where $t$ is the t-Test value and $df$ is the degrees of freedom.
The effect size calculator work with steps shows the complete step-by-step calculation for finding the effect size and Cohen's-D for two groups with the means of $\bar X=12$ and $\bar Y=8$ and standard deviations of $s_X=0.3468$ and $s_Y=1.2876$ using the effect size and Cohen's d formulas. For any other values of means and standard deviations, just supply four real numbers and click on the "CALCULATE" button. The grade school students may use this normal distribution calculator to generate the work, verify the results derived by hand or do their homework problems efficiently.

### Practice Problems for Effect Size

The effect size is also used for ANOVA (analysis of variance) and it is determined by measure variability that is accounted for by the treatment effect. The effect size is applicable in social science and in medical sciences (where size of treatment effect is important). The values of effect sizes from different experiments can be compared and combined to give a broader picture. It can interpret results of studies with large samples and show statistical significance for small differences.

Practice Problem 1:
In a school, teaches evaluate students knowledge in mathematics. Some of the obtained results were:

 mean standard deviation Trigonometry 2.56 1.09 Statistics 4.56 2.54 Probability 3.56 1.98
Twenty students participated in the study. Find Cohen's-D for the comparison of trigonometry to statistics and for trigonometry to probability.

Practice Problem 2:
A sample of $18$ persons are selected from a population with mean of $89$. After treatment, the sample mean is $85$ with standard deviation of $9$. Find the Cohen's d and the effective size for test with $\alpha=0.05$.

The effect size calculator, formula, work with steps and practice problems would be very useful for grade school students (K-12 education) to understand the concept of the effect size and Cohen's-D. This concept can be of benefit in statistics and probability, especially in meta analysis.