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.

a. Calculate the mean and the 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 deviations,
describe the differences in intelligence scores for
males and females.
Female Male
9 8
11 10
10 11
13 12
8 6
9 10
11 14
9 9

Mean 10 but standard deviation female id 1.5 and male is 2.29

a. To calculate the mean and standard deviation for the sample of females, use the following steps:

Step 1: Add up all the scores for females
9 + 11 + 10 + 13 + 8 + 9 + 11 + 9 = 80

Step 2: Divide the sum by the number of females (n = 8)
80 / 8 = 10

The mean for the sample of females is 10.

Step 3: Calculate the sum of the squared differences from the mean for each female score
(9 - 10)^2 + (11 - 10)^2 + (10 - 10)^2 + (13 - 10)^2 + (8 - 10)^2 + (9 - 10)^2 + (11 - 10)^2 + (9 - 10)^2 = 18

Step 4: Divide the sum by (n - 1) to get the variance
18 / (8 - 1) = 2.57

Step 5: Take the square root of the variance to get the standard deviation
√2.57 ≈ 1.60

The standard deviation for the sample of females is approximately 1.60.

Repeat the same steps for the sample of males:

Step 1: Add up all the scores for males
8 + 10 + 11 + 12 + 6 + 10 + 14 + 9 = 80

Step 2: Divide the sum by the number of males (n = 8)
80 / 8 = 10

The mean for the sample of males is 10.

Step 3: Calculate the sum of the squared differences from the mean for each male score
(8 - 10)^2 + (10 - 10)^2 + (11 - 10)^2 + (12 - 10)^2 + (6 - 10)^2 + (10 - 10)^2 + (14 - 10)^2 + (9 - 10)^2 = 40

Step 4: Divide the sum by (n - 1) to get the variance
40 / (8 - 1) = 5.71

Step 5: Take the square root of the variance to get the standard deviation
√5.71 ≈ 2.39

The standard deviation for the sample of males is approximately 2.39.

b. Comparing the means and standard deviations, we can see that the mean intelligence score for both males and females is the same (10). However, the standard deviation for males (approximately 2.39) is larger than the standard deviation for females (approximately 1.60). This indicates that there is higher variability in intelligence scores for males compared to females. In other words, there is greater diversity of intelligence scores among males than among females in the sample.

To calculate the mean and standard deviation, you can follow these steps:

Step 1: List all the intelligence scores for females and males separately:
Female: 9, 11, 10, 13, 8, 9, 11, 9
Male: 8, 10, 11, 12, 6, 10, 14, 9

Step 2: Calculate the sum of intelligence scores for each group:
Sum of female scores = 9 + 11 + 10 + 13 + 8 + 9 + 11 + 9 = 80
Sum of male scores = 8 + 10 + 11 + 12 + 6 + 10 + 14 + 9 = 80

Step 3: Calculate the mean (average) for each group:
Mean of female scores = Sum of female scores / Number of females = 80 / 8 = 10
Mean of male scores = Sum of male scores / Number of males = 80 / 8 = 10

Step 4: Calculate the squared difference from the mean for each intelligence score:
For females:
(9 - 10)^2 = 1
(11 - 10)^2 = 1
(10 - 10)^2 = 0
(13 - 10)^2 = 9
(8 - 10)^2 = 4
(9 - 10)^2 = 1
(11 - 10)^2 = 1
(9 - 10)^2 = 1

For males:
(8 - 10)^2 = 4
(10 - 10)^2 = 0
(11 - 10)^2 = 1
(12 - 10)^2 = 4
(6 - 10)^2 = 16
(10 - 10)^2 = 0
(14 - 10)^2 = 16
(9 - 10)^2 = 1

Step 5: Calculate the sum of squared differences for each group:
Sum of squared differences for females = 1 + 1 + 0 + 9 + 4 + 1 + 1 + 1 = 18
Sum of squared differences for males = 4 + 0 + 1 + 4 + 16 + 0 + 16 + 1 = 42

Step 6: Calculate the variance for each group:
Variance for females = Sum of squared differences / (Number of females - 1) = 18 / (8 - 1) = 18 / 7 ≈ 2.57
Variance for males = Sum of squared differences / (Number of males - 1) = 42 / (8 - 1) = 42 / 7 = 6

Step 7: Calculate the standard deviation (square root of variance) for each group:
Standard deviation for females = √2.57 ≈ 1.60
Standard deviation for males = √6 ≈ 2.45

b. Based on the means and standard deviations for the scores, we can make the following observations:

1. The mean intelligence score for both males and females is 10, which indicates that both genders have similar average intelligence scores.
2. The standard deviation for males (2.45) is higher than the standard deviation for females (1.60), suggesting greater variability in intelligence scores among males compared to females.
3. The higher standard deviation for males implies that there is a wider range of intelligence scores among males, with some males scoring significantly higher or lower than the mean, compared to females who have a narrower range of scores.

In summary, while the mean intelligence scores are similar for both genders, the greater variance among males' scores suggests more variability in intelligence levels for males compared to females.