In the context of a scatterplot and correlation, what does it mean when the slope of a line (or the slope between two points) is undefined?

• A. There is no linear relationship, so the correlation is $r=0$.
• B. The line is vertical, so the change in $x$ is $0$ and the slope $\frac{\mathrm{rise}}{\mathrm{run}}$ would require dividing by $0$.
• C. The line is horizontal, so the change in $y$ is $0$ and the slope is $0$.
• D. The points show a perfect positive linear relationship, so the correlation is $r=1$.

Answer: (B)

Verified step by step guidance

1. Recall that the slope of a line between two points is calculated as the ratio of the change in y-values (rise) to the change in x-values (run), expressed as \(\text{slope} = \frac{\Delta y}{\Delta x}\).
2. Understand that an undefined slope occurs when the denominator in this ratio, \(\Delta x\), is zero, meaning there is no horizontal change between the two points.
3. Recognize that when \(\Delta x = 0\), the line connecting the points is vertical, because the x-values are the same but the y-values differ.
4. Note that a vertical line does not represent a function in terms of y as a function of x, and the slope cannot be assigned a numerical value since division by zero is undefined.
5. Conclude that an undefined slope indicates a vertical line, which means the correlation coefficient \(r\) is not defined in the usual sense for linear relationships, and it does not imply zero correlation.