What is the valid range of the Pearson correlation coefficient?

• A. 0 to 1
• B. -∞ to ∞
• C. -1 to 1
• D. 0 to 100

Answer: (C)

The main concept here is how the Pearson correlation coefficient behaves as a measure of linear association. It is defined as r = cov(X,Y) / (σ_X σ_Y), and because of the Cauchy–Schwarz inequality, the absolute value of the covariance cannot exceed the product of the standard deviations: |cov(X,Y)| ≤ σ_X σ_Y. This makes the ratio r always lie between -1 and 1, inclusive. An r of 1 means a perfect positive linear relationship, -1 means a perfect negative linear relationship, and 0 means no linear relationship. The other ranges listed aren’t possible: 0 to 1 would miss negative correlations, -∞ to ∞ would be unbounded, and 0 to 100 isn’t within the valid [-1, 1] interval.