What are the typical ranges of correlation values?

What are the typical ranges of correlation values?

• A. -1 to 1
• B. 0 to 1
• C. -2 to 2
• D. 1 to 2

Answer: (A)

Correlation values are always between -1 and 1. The coefficient measures both the strength and direction of a linear relationship between two variables, and this bound arises from the Cauchy–Schwarz inequality: the absolute covariance cannot exceed the product of the standard deviations, so after standardizing, |r| ≤ 1. A value of 1 indicates a perfect positive linear relationship, meaning one variable increases in fixed proportion with the other. A value of -1 indicates a perfect negative linear relationship, where one variable increases as the other decreases in fixed proportion. A value near 0 means little or no linear relationship. Values outside -1 to 1 aren’t possible, and ranges like 0 to 1 would ignore negative relationships, while -2 to 2 would imply impossible magnitudes.

Correlation values are always between -1 and 1. The coefficient measures both the strength and direction of a linear relationship between two variables, and this bound arises from the Cauchy–Schwarz inequality: the absolute covariance cannot exceed the product of the standard deviations, so after standardizing, |r| ≤ 1. A value of 1 indicates a perfect positive linear relationship, meaning one variable increases in fixed proportion with the other. A value of -1 indicates a perfect negative linear relationship, where one variable increases as the other decreases in fixed proportion. A value near 0 means little or no linear relationship. Values outside -1 to 1 aren’t possible, and ranges like 0 to 1 would ignore negative relationships, while -2 to 2 would imply impossible magnitudes.