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.

The following table shows the percentage of individuals in group 2 that would be below the average score of a person in group 1, based on cohen’s d.

Cohen’s d |
Percentage of Group 2 who would be below average person in Group 1 |
---|---|

0.0 | 50% |

0.2 | 58% |

0.4 | 66% |

0.6 | 73% |

0.8 | 79% |

1.0 | 84% |

1.2 | 88% |

1.4 | 92% |

1.6 | 95% |

1.8 | 96% |

2.0 | 98% |

2.5 | 99% |

3.0 | 99.9% |

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 interpret Cohen’s d in practice.

**Example: Interpreting Cohen’s d**

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.

Here is a summary of the plant growth for each group:

**Fertilizer #1:**

- x
_{1}: 15.2 - s
_{1}: 4.4

**Fertilizer #2:**

- x
_{2}: 14 - s
_{2}: 3.6

Here is how we would calculate Cohen’s d to quantify the difference between the two group means:

- Cohen’s d = (x
_{1}– x_{2}) / √(s_{1}^{2 }+ s_{2}^{2}) / 2 - Cohen’s d = (15.2 – 14) / √(4.4
^{2 }+ 3.6^{2}) / 2 - Cohen’s d = 0.2985

Cohen’s d is **0.2985**.

Here’s how to interpret this value for Cohen’s d: The average height of plants that received fertilizer #1 is **0.2985** 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