Introduction

Beyond just analyzing the association of two variables by looking at a scatter plot, it’s important to be able to numerically analyze bivariate quantitative data. This topic will go over correlation and the correlation coefficient, a way to numerically determine the linear association between two variables.

Correlation Coefficient

Correlation is a description of the linear association between two quantitative variables. The correlation coefficient, or $r$, is the way we actually quantify how correlated the two variables are. The formula for $r$ is $r=\frac{1}{n-1}\sum \left(\frac{x_i-\bar{x}}{s_x}\right)\left(\frac{u_i-\bar{u}}{s_u}\right)$, but you will not need to calculate this value on the AP exam, and instead just understand what it means. $r$ is a unitless number that ranges from -1 to 1. The closer that $|r|$ is to $1$, the stronger the correlation. The closer $|r|$ is to $0$, the weaker the correlation. A positive $r$ value means a positive association and a negative $r$ value means a negative association. The sign of $r$ will always coincide with the direction of the association.

An important thing to keep in mind when looking at $r$, is that while a high $r$ value does imply a strong linear association, it does not imply that a linear form is the best fit for the data set. 

This scatterplot, for example, has a very high correlation, around $0.95$, but is clearly curved, and so the association would be non-linear (in this case, quadratic).

As always, it is also very important to remember that correlation does not equal causation. Just because a scatterplot has a very high correlation does not mean that one variable causes the other, or even that the variables are related at all!

Practice