In an extensive study involving thousands of British children, Arden and Plomin (2006) found
significantly higher variance in the intelligence scores for males than for females. Following are
hypothetical data, similar to the results obtained in the study. Note that the scores are not regular IQ
scores but have been standardized so that the entire sample has a mean of M = 10 and a standard
deviation of s = 2.
Female Male
9 8
11 10
10 11
13 12
8 6
9 10
11 14
9 9
a. Calculate the mean and standard deviation for the sample of n = 8 females and for the sample of
n = 8 males.
b. Based on the means and the standard deviation, describe the differences in intelligence scores for
males and females.
To calculate the mean and standard deviation for the sample of females, we need to add up all the scores and divide by the total number of scores. Here are the steps to calculate the mean for the females:
1. Add up all the female scores: 9 + 11 + 10 + 13 + 8 + 9 + 11 + 9 = 80.
2. Divide the sum by the total number of female scores: 80 / 8 = 10.
So, the mean for the sample of females is 10.
To calculate the standard deviation for the sample of females, we need to calculate the variance first. Here are the steps to calculate the variance for the females:
1. Calculate the squared difference between each female score and the mean: (9 - 10)^2, (11 - 10)^2, (10 - 10)^2, (13 - 10)^2, (8 - 10)^2, (9 - 10)^2, (11 - 10)^2, (9 - 10)^2.
2. Add up all the squared differences: (1)^2 + (1)^2 + (0)^2 + (3)^2 + (2)^2 + (1)^2 + (1)^2 + (1)^2 = 16.
3. Divide the sum by the total number of female scores minus one: 16 / (8 - 1) = 16 / 7 = 2.29 (rounded to two decimal places).
To calculate the standard deviation, we take the square root of the variance. So, the standard deviation for the sample of females is √2.29 ≈ 1.51 (rounded to two decimal places).
Similarly, let's calculate the mean and standard deviation for the sample of males.
1. Add up all the male scores: 8 + 10 + 11 + 12 + 6 + 10 + 14 + 9 = 80.
2. Divide the sum by the total number of male scores: 80 / 8 = 10.
So, the mean for the sample of males is also 10.
Calculating the variance for the males:
1. Calculate the squared difference between each male score and the mean: (8 - 10)^2, (10 - 10)^2, (11 - 10)^2, (12 - 10)^2, (6 - 10)^2, (10 - 10)^2, (14 - 10)^2, (9 - 10)^2.
2. Add up all the squared differences: (2)^2 + (0)^2 + (1)^2 + (2)^2 + (4)^2 + (0)^2 + (4)^2 + (1)^2 = 32.
3. Divide the sum by the total number of male scores minus one: 32 / (8 - 1) = 32 / 7 = 4.57 (rounded to two decimal places).
The standard deviation for the sample of males is √4.57 ≈ 2.14 (rounded to two decimal places).
Now, let's compare the means and standard deviations for males and females:
- Mean: The mean for both males and females is 10, indicating that on average, males and females in this sample have the same intelligence scores.
- Standard deviation: The standard deviation for males (2.14) is larger than that for females (1.51), indicating that there is higher variance in intelligence scores for males than females. This means that the scores for males are more spread out or diverse compared to the scores for females.