Which of the following is a characteristic of the correlation coefficient?

• A. It always takes values between $-1$ and $1$ inclusive
• B. It is affected by changing the units of measurement of the variables
• C. It can only be positive
• D. It measures the strength of a nonlinear relationship

Answer: (A)

Verified step by step guidance

1. Recall the definition of the correlation coefficient, often denoted as \(r\), which measures the strength and direction of a linear relationship between two variables.
2. Understand that the correlation coefficient \(r\) is calculated using the formula: \[r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\] where \(X_i\) and \(Y_i\) are data points, and \(\bar{X}\) and \(\bar{Y}\) are their respective means.
3. Recognize that the value of \(r\) is always between \(-1\) and \$1\( inclusive, where \)r = 1\( indicates a perfect positive linear relationship, \)r = -1\( indicates a perfect negative linear relationship, and \)r = 0$ indicates no linear relationship.
4. Note that the correlation coefficient is not affected by changing the units of measurement of the variables because it is a standardized measure (it is unitless).
5. Understand that \(r\) measures the strength of a linear relationship, not a nonlinear one, and it can be both positive or negative, so it is not restricted to only positive values.