At a recent scotch-tasting party, 20 faculty members and their (opposite sex) spouses were guzzling it back pretty well. One person quite correctly noticed that the women seemed to be drinking more than the men. If the mean number of drinks had by the men was 5, and that for the women was 7.2, and the standard deviation for the whole group was 2 drinks, what was the correlation between gender and the number of drinks consumed?
Cohen's d is the difference between two means divided by a standard deviation.
Calculate Cohen's d (d) and the effect-size correlation (r) using the following formulas:
d = (M1 - M2) / s
r = d / √(d^2 + 4)
With your data:
d = (7.2 - 5) / 2 = 1.1
r = 1.1 / √(1.1^2 + 4) = 0.48
Check these formulas and calculations.
To calculate the correlation between gender and the number of drinks consumed, we can use the Pearson correlation coefficient. This coefficient measures the strength and direction of the linear relationship between two variables.
In this case, the two variables are gender and number of drinks consumed. We can assign numerical values to the genders, such as 0 for men and 1 for women. Then, we can calculate the correlation coefficient using the following formula:
r = Cov(X, Y) / (σX * σY)
where:
- r is the correlation coefficient
- Cov(X, Y) is the covariance between the two variables
- σX is the standard deviation of variable X
- σY is the standard deviation of variable Y
First, let's calculate the covariance between gender and the number of drinks consumed:
Cov(X, Y) = E[(X - μX)(Y - μY)]
Here, X represents the gender (0 for men, 1 for women) and Y represents the number of drinks consumed. μX and μY represent the means of X and Y, respectively.
μX = (0 * 5 + 1 * 7.2) / (20 + 20) = 0.36
μY = (5 + 7.2) / 2 = 6.1
Cov(X, Y) = [(0 - 0.36)(5 - 6.1) + (1 - 0.36)(7.2 - 6.1)] / 40
= (-0.36 * -1.1 + 0.64 * 1.1) / 40
= (-0.396 + 0.704) / 40
= 0.308 / 40
= 0.0077
Next, let's calculate the standard deviations of X and Y:
σX = √[((0 - 0.36)^2 + (1 - 0.36)^2) / 40]
= √[(0.1296 + 0.1296) / 40]
= √(0.2592 / 40)
= √0.00648
= 0.0804
σY = √[((5 - 6.1)^2 + (7.2 - 6.1)^2) / 40]
= √[(1.21 + 1.344) / 40]
= √(2.554 / 40)
= √0.06385
= 0.2527
Finally, let's substitute these values into the correlation coefficient formula:
r = 0.0077 / (0.0804 * 0.2527)
= 0.0077 / 0.02036
= 0.3786
The correlation between gender and the number of drinks consumed is approximately 0.3786.