Which statement correctly describes the relationship between covariance and correlation?

Which statement correctly describes the relationship between covariance and correlation?

• A. Correlation measures how two variables vary together; covariance standardizes this to a -1 to 1 scale.
• B. Covariance is always non-negative.
• C. Covariance measures how two variables vary together; correlation standardizes this measure to a -1 to 1 scale and indicates strength and direction.
• D. Correlation is simply covariance divided by the sum of the variances.

Answer: (C)

Understanding the relationship between covariance and correlation helps you see how two variables relate on both direction and strength. Covariance tells you whether the variables tend to move together (positive covariance) or in opposite directions (negative covariance) and by how much, but its value depends on the units and scales of the variables, so it isn’t easy to compare across different data sets. Correlation takes that covariance and standardizes it by dividing by the product of the variables’ standard deviations. This makes the result a dimensionless number between -1 and 1, where the sign shows direction (positive means they move together, negative means they move in opposite directions) and the magnitude shows strength (values near -1 or 1 indicate a strong linear relationship, near 0 a weak one). That is exactly what the statement conveys: covariance measures how two variables vary together; correlation standardizes this measure to a -1 to 1 scale and indicates strength and direction. The other ideas aren’t correct: covariance can be negative, not always non-negative; correlation is not covariance divided by the sum of variances, but by the product of standard deviations; and standardizing covariance is what yields the -1 to 1 range, not any other divisor.

Understanding the relationship between covariance and correlation helps you see how two variables relate on both direction and strength. Covariance tells you whether the variables tend to move together (positive covariance) or in opposite directions (negative covariance) and by how much, but its value depends on the units and scales of the variables, so it isn’t easy to compare across different data sets.

Correlation takes that covariance and standardizes it by dividing by the product of the variables’ standard deviations. This makes the result a dimensionless number between -1 and 1, where the sign shows direction (positive means they move together, negative means they move in opposite directions) and the magnitude shows strength (values near -1 or 1 indicate a strong linear relationship, near 0 a weak one).

That is exactly what the statement conveys: covariance measures how two variables vary together; correlation standardizes this measure to a -1 to 1 scale and indicates strength and direction.

The other ideas aren’t correct: covariance can be negative, not always non-negative; correlation is not covariance divided by the sum of variances, but by the product of standard deviations; and standardizing covariance is what yields the -1 to 1 range, not any other divisor.