如果有两个变量$X$和$Y$，并且它们的尺度都是-1到1的，那么$X$和$Y$的皮尔逊相关系数就是$X$和$Y$做一元线性回归的回归系数（解释），从这点我们可以看明显的看出，皮尔逊相关系数就是描述的线性关系。

另外一个反例是，如果$Y=X^2$，并且$X$的范围是-1到1，尽管它们是有关联的，但是它们的皮尔逊系数为0，因为它们并没有任何线性关系。

"If a relationship between two variables is not linear, the rate of increase or decrease can change as one variable changes, causing a "curved pattern" in the data. This curved trend might be better modeled by a nonlinear function, such as a quadratic or cubic function, or be transformed to make it linear. Plot 4 shows a strong relationship between two variables. However, because the relationship is not linear, the Pearson correlation coefficient is only +0.244."

source: https://support.minitab.com/en-us/minitab/18/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/linear-nonlinear-and-monotonic-relationships/