"In studies of monozygotic (identical) twins, the correlation between intelligence (IQ) scores is 0.85.


c. What fraction of the variation in one twin’s IQ can be accounted for by the other twin’s IQ?"

Verified step by step guidance

1. Understand that the correlation coefficient, denoted as \(r\), measures the strength and direction of a linear relationship between two variables. Here, \(r = 0.85\) represents the correlation between the IQ scores of identical twins.
2. Recall that the fraction of variation in one variable explained by the other variable is given by the coefficient of determination, which is \(r^2\).
3. Calculate the coefficient of determination by squaring the correlation coefficient: \(r^2 = (0.85)^2\).
4. Interpret \(r^2\) as the proportion (or fraction) of the variance in one twin's IQ that can be explained by the IQ of the other twin.
5. Express the result as a percentage or fraction to clearly communicate how much of the variation in one twin's IQ is accounted for by the other twin's IQ.

Key Concepts

Correlation Coefficient

The correlation coefficient measures the strength and direction of a linear relationship between two variables, ranging from -1 to 1. A value of 0.85 indicates a strong positive relationship between the IQ scores of identical twins, meaning as one twin’s IQ increases, the other’s tends to increase as well.

Coefficient of Determination (R-squared)

The coefficient of determination is the square of the correlation coefficient and represents the proportion of variance in one variable explained by the other. In this case, squaring 0.85 gives the fraction of variation in one twin’s IQ accounted for by the other twin’s IQ.

Variance and Variation in Statistics

Variance measures how much data points differ from the mean, representing the spread or variability in a dataset. Understanding variance is essential to interpret how much of the total variation in IQ scores can be explained by the relationship between twins’ IQs.