Why does switching the independent and dependent variables leave Pearson’s correlation coefficient unchanged?

Switching the independent and dependent variables leaves Pearson’s correlation coefficient unchanged because the mathematical formula used to calculate it does not distinguish between the two variables.

In the formula:
$$r=\frac{n(\sum xy)-\sum x\sum y}{\sqrt{[n(\sum x^2)-(\sum x{)}^2][n(\sum y^2)-(\sum y{)}^2]}}$$

The variables $x$ and $y$ are treated symmetrically. Because the operations involve the summation of products and the individual variances of both variables, swapping their roles does not alter the resulting value of $r$.1

While this mathematical property allows the coefficient to remain constant regardless of which variable is labeled independent or dependent, it is crucial for researchers to correctly identify these roles based on the context of their study. Failing to do so can lead to nonsensical interpretations, as the correlation coefficient itself only measures the strength and direction of a linear measure of association and cannot determine causality or distinguish the direction of influence between variables.12