Learning Outcomes
- Calculate and interpret the measure of effect size, Cohen’s d
Cohen’s Standards for Small, Medium, and Large Effect Sizes
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.
Cohen’s Standard 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:
[latex]\displaystyle{d}=\dfrac{{\overline{{x}}_{{1}}-\overline{{x}}_{{2}}}}{{{s}_{{\text{pooled}}}}} \text{ where } {s}_{{\text{pooled}}}=\sqrt{{\dfrac{{{({n}_{{1}}-{1})}{{s}_{{1}}^{{2}}}+{({n}_{{2}}-{1})}{{s}_{{2}}^{{2}}}}}{{{n}_{{1}}+{n}_{{2}}-{2}}}}} [/latex]
Example 4
Calculate Cohen’s d for Example 2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.
Example 5
Calculate Cohen’s d for Example 3. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.
Try It 3
Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013, are listed in the two tables below.
Northeast
| 94.2 | 75.2 | 69.6 | 52.0 | 48.0 | 41.9 | 36.4 | 33.4 | 31.5 | 27.6 |
| 77.3 | 71.9 | 67.5 | 50.6 | 46.2 | 38.4 | 35.2 | 33.0 | 28.7 | 26.5 |
| 76.3 | 71.7 | 56.3 | 48.7 | 43.2 | 37.6 | 33.7 | 31.8 | 28.5 | 26.0 |
West
| 126.0 | 70.6 | 65.2 | 51.4 | 45.5 | 37.0 | 33.0 | 29.6 | 23.7 | 22.6 |
| 116.1 | 70.6 | 58.2 | 51.2 | 43.2 | 36.0 | 31.4 | 28.7 | 23.5 | 21.6 |
| 78.2 | 68.2 | 55.6 | 50.3 | 39.0 | 34.1 | 31.0 | 25.3 | 23.4 | 21.5 |
Is there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:
- Is this a test of two means or two proportions?
- Are the population standard deviations known or unknown?
- Which distribution do you use to perform the test?
- What is the random variable?
- What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
- Is this test right-, left-, or two-tailed?
- What is the p-value?
- Do you reject or not reject the null hypothesis?
- At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.
- Calculate Cohen’s d and interpret it.
