A residual plot tells you whether a line is the right shape. The next question is how much the line helps. The answer is $r^2$, the square of the correlation, which has a clear meaning of its own.
Key idea
$$r^2 = 1 - \dfrac{\text{SSE}}{\text{SST}},$$ where $\text{SST} = \sum (y - \bar{y})^2$ is the variation in y with no model and $\text{SSE} = \sum (y - \hat{y})^2$ is the variation left after the line. $r^2$ is the fraction of the variation in y explained by the linear relationship with x.
Why it works
Without x, the best single guess for every y is $\bar{y}$, and its total squared error is SST. With the line, the squared error drops to SSE. The fraction removed, $(\text{SST} - \text{SSE})/\text{SST} = 1 - \text{SSE}/\text{SST}$, is how much knowing x helped. For the least-squares line this fraction is exactly the square of $r$ (a fact that follows from the z-score derivation of the line).
Worked example: From r to r²
The correlation between hours of sunlight and plant growth is $r = 0.8$. Interpret $r^2$.
- $r^2 = 0.8^2 = 0.64$.
- About 64% of the variation in plant growth is explained by the linear relationship with hours of sunlight.
- The other 36% comes from other things (water, soil, genetics) and chance.
Answer: $r^2 = 0.64$: 64% of the variation in growth is explained.
Worked example: From sums of squares
For a data set, $\text{SST} = 200$ and $\text{SSE} = 50$. The slope is negative. Find $r^2$ and $r$.
- $r^2 = 1 - \dfrac{50}{200} = 0.75$.
- $r = \pm\sqrt{0.75} \approx \pm 0.866$.
- r has the same sign as the slope: $r \approx -0.866$.
Answer: $r^2 = 0.75$, $r \approx -0.866$.
Common mistake
Reading $r^2 = 0.64$ as "64% of the points are on the line" or "predictions are right 64% of the time." $r^2$ is about variation in y, not about individual points. Check your sentence: it should say "of the variation in [y] is explained by the linear relationship with [x]."
Squaring hides the sign: $r = 0.8$ and $r = -0.8$ both give $r^2 = 0.64$. To recover r from $r^2$, take the square root and use the sign of the slope.