R² and the sums of squares

Splitting the variation in y

Before the line, the only summary of credit score was its mean, 687.5, and the total variation around it:

$$SST=\sum (y-\bar{y}{)}^2=\sum y^2-n{\bar{y}}^2=S{S}_{yy}$$

After the line, each $y$ is split into the part the line predicts and the part it misses. The variation splits the same way:

$$SST=SSR+SSE,SSR=\sum (\hat{y}-\bar{y}{)}^2,SSE=\sum (y-\hat{y}{)}^2=\sum e^2$$

$SSE$ is the quantity least squares minimised; $SSR$ is what the line accounts for. The coefficient of determination is the explained share:

$$R^2=\frac{SSR}{SST}=1-\frac{SSE}{SST}$$

and for a straight line with one predictor, $R^2=r^2$ exactly.

Worked example — Northfield's decomposition

$SST=3,801,150-8\times 687.5^2=3,801,150-3,781,250=19,900$.

$SSE$ needs the eight residuals. With $\hat{y}=544.18+2.0659x$: −6.5, 1.7, −11.6, 9.8, 11.2, 2.3, 4.6, −11.4 (they sum to zero, as least-squares residuals must). Squared and summed: $SSE=557.8$.

$SSR=19,900-557.8=19,342.2$.

$$R^2=\frac{19,342.2}{19,900}=0.972\text{and}{r}^2=0.98589^2=0.972$$

The sentence

> About 97.2 % of the variation in credit score among these eight clients is explained by the linear relationship with income.

Every word matters: variation in the response, explained by the linear relationship with the explanatory variable. Not "97.2 % of the points are on the line" (none are), not "income explains 97.2 % of credit score" (meaningless), not "the line is 97.2 % accurate".

What R² is not

It has no units and no sign — it cannot tell you the direction, which is why $r$ is still reported. It is not a test: a high $R^2$ from eight points can arise by chance, and whether the slope is distinguishable from zero is Module 13's question, answered with the standard error and a $t$. And it does not certify the form: a curved relationship can carry a respectable $R^2$ from a straight line while the residuals show a clear bend.

A second run, from SST and SSE alone

A question may give only the two sums. If $SST=6,000$ and $SSE=450$, then $R^2=1-450/6,000=0.925$: 92.5 % of the variation in the response is explained by the line, and $r=\pm \sqrt{0.925}=\pm 0.962$, with the sign taken from the slope.