Question

calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. then calculate it again, letting row 2 represent the x - values and row 1 the y - values. what effect does switching the variables have on r?
row 1: 19 20 37 46 60 63 73
row 2: 132 172 152 169 113 109 131
calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values.
r = - 0.582 (round to three decimal places as needed.)
calculate the correlation coefficient r, letting row 2 represent the x - values and row 1 the y - values.
r = (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for correlation coefficient

The formula for the correlation coefficient $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. But we can also use statistical software or a calculator with a correlation - function. When we switch the roles of $x$ and $y$, the value of the correlation coefficient remains the same. This is because the correlation coefficient measures the strength and direction of the linear relationship between two variables, and it is symmetric with respect to the two variables.
Let $x_1 = [19,20,37,46,60,63,73]$, $y_1=[132,172,152,169,113,109,131]$.
Let $x_2 = [132,172,152,169,113,109,131]$, $y_2=[19,20,37,46,60,63,73]$.

Step2: Use a calculator or software

Using a scientific calculator with a correlation - coefficient function or software like Excel (CORREL function) or Python (numpy.corrcoef), when we calculate the correlation coefficient for the first case with $x = x_1$ and $y = y_1$ and then for the second case with $x = x_2$ and $y = y_2$, we will get the same result.
Since when $x$ is from Row 1 and $y$ is from Row 2, $r=- 0.582$, when we switch them, $r=-0.582$.

Answer:

$-0.582$