In statistics, we often use p-values to determine if there is a statistically significant difference between the mean of two groups.

However, while a p-value can tell us whether or not there is a statistically significant difference between two groups, an effect size can tell us how large this difference actually is.

One of the most common measurements of effect size is **Cohen’s d**, which is calculated as:

Cohen’s d = (x_{1} – x_{2}) / √(s_{1}^{2 }+ s_{2}^{2}) / 2

where:

- x
_{1}, x_{2}: mean of sample 1 and sample 2, respectively - s
_{1}^{2}, s_{2}^{2}: variance of sample 1 and sample 2, respectively

Using this formula, here is how we interpret Cohen’s d:

- A
*d*of**0.5**indicates that the two group means differ by 0.5 standard deviations. - A
*d*of**1**indicates that the group means differ by 1 standard deviation. - A
*d*of**2**indicates that the group means differ by 2 standard deviations.

And so on.

Here’s another way to interpret cohen’s d: An effect size of 0.5 means the value of the average person in group 1 is 0.5 standard deviations above the average person in group 2.

We often use the following rule of thumb when interpreting Cohen’s d:

- A value of
**0.2**represents a small effect size. - A value of
**0.5**represents a medium effect size. - A value of
**0.8**represents a large effect size.

The following example shows how to calculate Cohen’s d in R.

**Example: How to Calculate Cohen’s d in R**

Suppose a botanist applies two different fertilizers to plants to determine if there is a significant difference in average plant growth (in inches) after one month.

There are two methods we can use to quickly calculate Cohen’s d in R:

**Method 1: Use lsr Package**

library(lsr) #define plant growth values for each group group1 <- c(8, 9, 11, 11, 12, 14, 15, 16, 16, 18, 20, 21) group2 <- c(7, 9, 10, 10, 11, 11, 12, 14, 14, 16, 20, 23) #calculate Cohen's d cohensD(group1, group2) [1] 0.2635333

**Method 2: Use effsize Package**

library(effsize) #define plant growth values for each group group1 <- c(8, 9, 11, 11, 12, 14, 15, 16, 16, 18, 20, 21) group2 <- c(7, 9, 10, 10, 11, 11, 12, 14, 14, 16, 20, 23) #calculate Cohen's d cohen.d(group1, group2) Cohen's d d estimate: 0.2635333 (small) 95 percent confidence interval: lower upper -0.5867889 1.1138555

Notice that both methods produce the same result: Cohen’s d is **0.2635**.

We interpret this to mean that the average height of plants that received fertilizer #1 is **0.2635** standard deviations greater than the average height of plants that received fertilizer #2.

Using the rule of thumb mentioned earlier, we would interpret this to be a small effect size.

In other words, whether or not there is a statistically significant difference in the mean plant growth between the two fertilizers, the actual difference between the group means is trivial.

**Additional Resources**

The following tutorials offer additional information on effect size and Cohen’s d:

Effect Size: What It Is and Why It Matters

How to Calculate Cohen’s d in Excel