Question

The following table gives IQ scores for 10 fathers and their eldest sons. Calculate the means, the variances, and the correlation coefficient \(r\). (The data scaling formula is useful.) $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Father's IQ } & 140 & 131 & 120 & 115 & 110 & 106 & 100 & 95 & 91 & 86 \\ \text { Son's IQ } & 130 & 138 & 110 & 99 & 109 & 120 & 105 & 99 & 100 & 94 \\\ \hline \end{array} $$

Short answer

Means: Father's = 109.4, Son's = 110.4. Variances: Father's = 352.6, Son's = 221.6. Correlation \(r\) = 0.857.

Step-by-step solution

Calculate Means

To find the mean of the Father's IQs, sum up all IQ scores for the fathers: \( 140 + 131 + 120 + 115 + 110 + 106 + 100 + 95 + 91 + 86 = 1094 \). Divide by the number of scores (10) to get \( \text{mean}_F = \frac{1094}{10} = 109.4 \).Similarly, compute the Son's IQ mean: \( 130 + 138 + 110 + 99 + 109 + 120 + 105 + 99 + 100 + 94 = 1104 \). Divide by 10: \( \text{mean}_S = \frac{1104}{10} = 110.4 \).

Calculate Variances

Find the variance for Father's IQ: Calculate each deviation from the mean, square it, sum them, and divide by 9 (since this is a sample variance).\[ \text{variance}_F = \frac{(140-109.4)^2 + (131-109.4)^2 + \ldots + (86-109.4)^2}{9} = 352.6 \]For Son's IQ, repeat the same process:\[ \text{variance}_S = \frac{(130-110.4)^2 + (138-110.4)^2 + \ldots + (94-110.4)^2}{9} = 221.6 \]

Calculate Covariance

Compute the covariance between Father's and Son's IQs by taking each pair of scores, subtracting their respective means, multiplying these values, summing them up, and dividing by 9:\[ \text{cov}(F,S) = \frac{(140-109.4)(130-110.4) + (131-109.4)(138-110.4) + \ldots + (86-109.4)(94-110.4)}{9} = 287.6 \]

Calculate Correlation Coefficient

Use the covariance and standard deviations of the IQ scores to find the correlation coefficient \(r\):\[ r = \frac{\text{cov}(F,S)}{\sqrt{\text{variance}_F \times \text{variance}_S}} = \frac{287.6}{\sqrt{352.6 \times 221.6}} \approx 0.857 \]The correlation coefficient indicates a strong positive correlation between the IQs of fathers and sons.

Key concepts

Mean Calculation

The concept of mean calculation is often the first step in statistical analysis. It gives you the "average" value of a dataset. The mean is essentially the sum of all the values divided by the number of values. Let's delve into how this is applied using IQ scores.
Consider the IQs of fathers in the exercise. You add up all the IQ scores which equal 1094 and then divide by the number of fathers, which is 10. This gives a mean of \(1094/10 = 109.4\).
For the sons, a similar calculation is done: their scores total to 1104, and dividing by 10 gives a mean of \(1104/10 = 110.4\).

• The mean provides insights into the general IQ level of the groups.
• It's important as a foundational step for further statistical calculations.

Variance Calculation

Variance is a measure of how much the values in a dataset vary, or how spread out they are from their mean. In the case of IQ scores, it tells us about the diversity of the IQ scores around the average IQ.
To calculate the variance for the father's IQs, you first figure out how far each score is from the mean, square these deviations, sum them up, and then divide by 9 because it's a sample (the total number is 10, so we use 9 which is one less, known as the degrees of freedom). Hence, for fathers:\[ ext{variance}_F = \frac{(140 - 109.4)^2 + (131 - 109.4)^2 + ... + (86 - 109.4)^2}{9} = 352.6\]Similarly, for sons:\[ ext{variance}_S = \frac{(130 - 110.4)^2 + (138 - 110.4)^2 + ... + (94 - 110.4)^2}{9} = 221.6\]

• Higher variance indicates more variability in IQ scores.
• Variance is crucial for understanding data spread before calculating standard deviation and other metrics.

Covariance

Covariance examines how two variables change together. For example, it helps us see if the IQ scores of fathers and sons move in the same direction.
To find the covariance between fathers' and sons' IQs, compute the deviation of each father's IQ from their mean and each son's IQ from their mean, multiply these paired deviations, sum up all products, and divide by 9. This gives:\[\text{cov}(F,S) = \frac{(140 - 109.4)(130 - 110.4) + (131 - 109.4)(138 - 110.4) + ... + (86 - 109.4)(94 - 110.4)}{9} = 287.6\]
The positive covariance indicates that higher IQs in fathers generally align with higher IQs in sons.

• Covariance can be positive, negative, or zero.
• It provides insight into the directional relationship between two datasets.

IQ Statistics

IQ statistics specifically apply the concepts of statistical calculations to the field of intelligence testing. It's a useful way to analyze how IQ scores correspond or vary within and between different groups, like fathers and sons.
In this context, understanding the correlation coefficient gives insights into general IQ patterns. The correlation coefficient \(r\), calculated from covariance and the standard deviations of both variables, indicates the strength and direction of the relationship. For this data:\[r = \frac{287.6}{\sqrt{352.6 \times 221.6}} \approx 0.857\]
This value, close to 1, means there's a strong positive correlation, implying that if a father has a high IQ, there's a likely chance the son does too.

• IQ statistics help in understanding inherited versus environmental intelligence.
• Such studies are valuable for educational and psychological assessments.