Effect Size Calculator: Cohen's d and r

What is Effect Size?

Effect size is a statistical concept that measures the strength of the relationship between two variables in a statistical population, or a sample-based estimate of that quantity. It quantifies the magnitude of a phenomenon, such as the difference between two groups or the correlation between variables.

Formulas and Their Meanings

1. Cohen's d: \[d = \frac{M_1 - M_2}{s_p}\] Where \(M_1\) and \(M_2\) are the means of the two groups, and \(s_p\) is the pooled standard deviation.

2. Pooled Standard Deviation: \[s_p = \sqrt{\frac{s_1^2 + s_2^2}{2}}\] Where \(s_1\) and \(s_2\) are the standard deviations of the two groups.

3. Effect Size r: \[r = \frac{d}{\sqrt{d^2 + 4}}\] Where \(d\) is Cohen's d.

4. Cohen's d from t-value: \[d = t \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\] Where \(t\) is the t-value and \(n_1\) and \(n_2\) are the sample sizes (often equal, simplifying to \(2/n\)).

5. Effect Size r from t-value: \[r = \sqrt{\frac{t^2}{t^2 + df}}\] Where \(t\) is the t-value and \(df\) are the degrees of freedom.

Calculation Steps

1. If using means and standard deviations:
   • Calculate the pooled standard deviation.
   • Calculate Cohen's d using the means and pooled standard deviation.
   • Calculate the effect size r using Cohen's d.
2. If using t-value and degrees of freedom:
   • Calculate Cohen's d using the t-value and degrees of freedom.
   • Calculate the effect size r using the t-value and degrees of freedom.

Example Calculation

Let's calculate the effect size for two groups with the following data:

• Group 1: Mean (M1) = 75, Standard Deviation (SD1) = 10
• Group 2: Mean (M2) = 70, Standard Deviation (SD2) = 8

1. Calculate pooled standard deviation: \[s_p = \sqrt{\frac{10^2 + 8^2}{2}} = \sqrt{\frac{100 + 64}{2}} = \sqrt{82} \approx 9.06\]
2. Calculate Cohen's d: \[d = \frac{75 - 70}{9.06} \approx 0.55\]
3. Calculate Effect Size r: \[r = \frac{0.55}{\sqrt{0.55^2 + 4}} \approx 0.27\]

Visual Representation

This visualization represents the effect size between two groups. The overlap between the circles indicates the magnitude of the effect. Less overlap suggests a larger effect size.