Which statistic describes the average cross-product deviation between two variables?

Which statistic describes the average cross-product deviation between two variables?

• A. Covariance
• B. Correlation coefficient
• C. Regression coefficient
• D. Standard deviation

Answer: (A)

Covariance describes the average cross-product deviation between two variables. It captures how X and Y move together: when deviations from their means have the same sign, the product is positive and the covariance is positive; when they move in opposite directions, the product is negative and the covariance is negative. The average of these cross-products across observations is what covariance measures. For a sample, Cov(X,Y) = Σ (Xi − X̄)(Yi − Ȳ) / (n − 1). The value is in units of X times Y and depends on the scales of the variables. The correlation coefficient is the standardized form of covariance, r = Cov(X,Y)/(sX sY), which scales Cov to a dimensionless range from −1 to 1. The standard deviation describes dispersion of a single variable, not the joint variability of two variables, and the regression coefficient reflects the slope of the line predicting Y from X (it uses covariance in its calculation but is not itself the average cross-product deviation).

Covariance describes the average cross-product deviation between two variables. It captures how X and Y move together: when deviations from their means have the same sign, the product is positive and the covariance is positive; when they move in opposite directions, the product is negative and the covariance is negative. The average of these cross-products across observations is what covariance measures. For a sample, Cov(X,Y) = Σ (Xi − X̄)(Yi − Ȳ) / (n − 1). The value is in units of X times Y and depends on the scales of the variables. The correlation coefficient is the standardized form of covariance, r = Cov(X,Y)/(sX sY), which scales Cov to a dimensionless range from −1 to 1. The standard deviation describes dispersion of a single variable, not the joint variability of two variables, and the regression coefficient reflects the slope of the line predicting Y from X (it uses covariance in its calculation but is not itself the average cross-product deviation).