Residual sum of squares in simple linear regression
= Residual sum of squares in simple linear regression
For a nonconstant predictor, put
$$
S_{xx}=\sum_i(x_i-\bar x)^2,
\quad
S_{xy}=\sum_i(x_i-\bar x)(y_i-\bar y),
\quad
S_{yy}=\sum_i(y_i-\bar y)^2.
$$
The minimized residual sum of squares is
$$
\operatorname{RSS}=S_{yy}-\frac{S_{xy}^2}{S_{xx}}.
$$
Each centered sum can be recovered in constant time from the five raw sums $\sum x_i$, $\sum y_i$, $\sum x_i^2$, $\sum x_iy_i$, and $\sum y_i^2$.