### R SQUARED: SST, SSE AND SSR:

From these Wikipedia definitions:

\begin{align} \text{SST}_{\text{otal}} &= \color{red}{\text{SSE}_{\text{xplained}}}+\color{blue}{\text{SSR}_{\text{esidual}}}\\ \end{align}

or, equivalently,

\begin{align} \sum(y_i-\bar y)^2 &=\color{red}{\sum(\hat y_i-\bar y)^2}+\color{blue}{\sum(y_i-\hat y_i)^2} \end{align}

and

$$\large \text{R}^2 = 1 - \frac{\text{SSR}_{\text{esidual}}}{\text{SST}_{\text{otal}}}$$

So if the model explained all the variation, $$\text{SSR}_{\text{esidual}}=\sum(y_i-\hat y_i)^2=0$$, and $$\bf R^2=1.$$

From Wikipedia:

Suppose $$r = 0.7$$ then $$R^2 = 0.49$$ and it implies that $$49\%$$ of the variability between the two variables have been accounted for and the remaining $$51\%$$ of the variability is still unaccounted for.

Proof by example:

fit = lm(mpg ~ wt, mtcars)
summary(fit)$r.square [1] 0.7528328 > sse = sum((fitted(fit) - mean(mtcars$mpg))^2)
> ssr = sum((fitted(fit) - mtcars\$mpg)^2)
> 1 - (ssr/(sse + ssr))
[1] 0.7528328