Calculate and interpret Pearson's correlation coefficient r and the coefficient of determination r squared, and state their limitations.

What this dot point is asking

You must compute $r$ (almost always with technology), describe what its sign and size mean, find and interpret $r^2$, and know when neither number should be trusted.

Reading Pearson's r

The correlation coefficient $r$ satisfies $-1 \le r \le 1$.

• The sign matches the direction: positive $r$ for a positive association, negative for a negative one.
• The size (distance from zero) matches the strength: values near $\pm 1$ are strong, near $0$ are weak.

In practice you read $r$ from your calculator's regression output after entering the two lists; you are rarely asked to compute it by hand.

The coefficient of determination

The coefficient of determination is $r^2$. It is the fraction (or percentage) of the variation in the response variable that is explained by the linear relationship with the explanatory variable.

Because it is squared, $r^2$ is always between $0$ and $1$ and loses the sign. You quote the direction from $r$ and the explained proportion from $r^2$.

Limitations of both measures

Both $r$ and $r^2$ describe only linear association and only over the range of the data.

• A curved relationship can give a small $r$ even though the variables are strongly related; the relationship is just not linear.
• An outlier can pull $r$ towards or away from zero, so always check the scatterplot.
• Neither number proves causation. A strong $r$ between two variables can be produced by a lurking third variable.