SESSION:10 Hypothesis Testing with Two Samples10.2 Cohen’s Standards for Small, Medium, and Large Effect Sizes

Session:10 Hypothesis Testing with Two Samples

10.2 Cohen’s Standards for Small, Medium, and Large Effect Sizes

Introductory Business Statistics | Leadership Development – Micro-Learning Session

Rice University 2020 | Michael Laverty, Colorado State University Global Chris Littel, North Carolina State University| https://openstax.org/details/books/introductory-business-statistics

Cohen’s d is a measure of “effect size” based on the differences between two means. Cohen’s d, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.

Size of effect	d
Small	0.2
Medium	0.5
Large	0.8

Table 10.2 Cohen’s Standard Effect Sizes

Cohen’s d is the measure of the difference between two means divided by the pooled standard deviation: $d = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{s_{p o o l e d}}$

$� = \frac{{\bar{�}}_{1} - {\bar{�}}_{2}}{�_{� � � � � �}}$ where $s_{p o o l e d} = \sqrt{\frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2}}$

$�_{� � � � � �} = \sqrt{\frac{(�_{1} - 1) �_{1}^{2} + (�_{2} - 1) �_{2}^{2}}{�_{1} + �_{2} - 2}}$

It is important to note that Cohen’s d does not provide a level of confidence as to the magnitude of the size of the effect comparable to the other tests of hypothesis we have studied. The sizes of the effects are simply indicative.

EXAMPLE 10.4

Problem

Calculate Cohen’s d for example 10.2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.

Solution

x̅₁ = 4 s₁ = 1.5 n₁ = 11

x̅₂ = 3.5 s₂ = 1 n₂ = 9

d = 0.384

The effect is small because 0.384 is between Cohen’s value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences of the means for the two companies is small indicating that there is not a significant difference between them.