A made-up science club measures how far a toy car rolls down ramps of six different heights…

Step 1. The line. $b = \dfrac{87.5}{17.5} = 5$ m per unit, and $a = 12.17 - 5 \times 3.5 = -5.333$ m. So $\hat{y} = -5.333 + 5x$.

Step 2. r. $S_{yy} = 474.83$, so $r = \dfrac{87.5}{\sqrt{17.5 \times 474.83}} = 0.9599$.

Step 3. Residuals. They are $3.333$, $-0.667$, $-2.667$, $-2.667$, $-0.667$ and $3.333$ m.

What it means. The residuals are positive at both ends and negative in the middle: a U. The distances bend upwards, and a straight line is the wrong model, though $r$ is high. In fact these distances follow $y = x^2 - 2x + 4$ exactly.