Which of the following are properties of the linear correlation coefficient $r$?

• A. It always satisfies $-1\le r\le 1$.
• B. It is unaffected by outliers in the data.
• C. It measures the strength of a nonlinear relationship.
• D. It can only take positive values.

Answer: (A)

Verified step by step guidance

1. Recall the definition of the linear correlation coefficient \(r\), which measures the strength and direction of a linear relationship between two variables.
2. Understand the range of \(r\): by definition, the correlation coefficient always satisfies the inequality \(-1 \leq r \leq 1\). This means \(r\) can be negative, zero, or positive, but never less than -1 or greater than 1.
3. Consider the effect of outliers: \(r\) is sensitive to outliers because extreme values can disproportionately affect the calculation of covariance and standard deviations used in \(r\).
4. Recognize what \(r\) measures: it quantifies the strength and direction of a linear relationship, not nonlinear relationships. Nonlinear relationships may have a low or zero \(r\) even if there is a strong nonlinear association.
5. Note the sign of \(r\): since \(r\) can be negative or positive, it is incorrect to say it can only take positive values.