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 |
Cohen's \(d\) is the measure of the difference between two means divided by the pooled standard deviation:
\[d=\dfrac{\bar{x}_1-\bar{x}_2}{s_{\text {pooled }}} \text { where } s_{\text {pooled }}=\sqrt{\dfrac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}}\]
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.
A study is done to determine if Hospital A retains its nurses longer than Hospital B. It is believed that Hospital A has a higher retention than Hospital B. The study finds that in a sample of 11 workers at Hospital A their average time with the company is four years with a standard deviation of 1.5 years. A sample of 9 workers at Hospital B finds that the average time with the company was 3.5 years with a standard deviation of 1 year. Test this proposition at the 1% level of significance.
Calculate Cohen’s d. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.
- Answer
-
\[\bar{x}_1=4 s_1=1.5 n_1=11\]
\[\bar{x}_2=3.5 s_2=1 n_2=9\]
\[d=0.384\]
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 hospitals is small indicating that there is not a significant difference between them.