What is the most efficient way to calculate $R^2$? Hello I am working on a question from an old exam paper and wondered what is the best way to tackle parts ii and iii. Given the data it is easy to find $\hat{\beta_0}=-1.071$ and $\hat{\beta_1}=2.741$.
Now for part ii) I have the formula $R^2=1-SSE/SST$ where $SST=\sum(y_i-\bar{y})^2$ (easy to work out) and $SSE=\sum e_i^2=\sum (y_i-\hat{y_i})$.
Likewise I have for part iii) An unbiased estimate of $\sigma^2$ is $\sum e_i^2/(n-2)$.

Question: I wondered if there is a nice and more efficient way to work
  out $\sum e_i^2$ or do I have to calculate each predicted value based
  on the model take it away from the actual value square that value and
  then sum all the values up?



There are many perspectives on 'efficiency' of computation for regression.
Often a data file and software are available. Then computer output will
give all or most of what you need, depending on the output format of the
software. For programmers of software, 'efficiency' includes speed of
computation and protection against underflow and overflow for data values
that are very small or very large (respectively).
I think your definition of 'efficiency' is ease of computation during an exam.
Even then, exactly the best path depends on what kind of calculator or computer you
are allowed to use. If the data are summarized as in the table you give,
then the Answer by @V.V. (+1) seems on the right track.
I assume you already know how to use the 'Total' row to find the sample means $\bar X$ and $\bar Y$
and the the sample variances $S_X^2$ and $S_Y^2$. Then all you need is
the sample covariance 
$$S_{XY} = \frac{1}{n-1}\left(\sum X_iY_i - n\bar X \bar Y\right)
= \frac{1003 - (55)(151)/10}{9},$$
in order to get $r = \frac{S_{XY}}{S_XS_Y}.$
One often says that $r^2$ (or $R^2$or R-SQ, possibly legacy notation from days when
computer terminals did not use lower-case letters) is the proportion of the
total variability of Y explained by regression on X. This is due to the equation
$$S_{Y|x}^2 = \frac{n-1}{n-2}S_Y^2(1 - r^2),$$
which is equivalent to the equation $r^2 = 1 - SSE/SST$ in your question. Here
$S_Y^2$ is interpreted as the 'total variability of Y',  $S_{Y|x}^2$ is interpreted as
the 'variability of the residuals about the regression line' or 'the variability of Y unexplained
by regression', and the factor $\frac{n-1}{n-2} \approx 1$ is ignored. (If $r^2 = 1,$
then the data fit the regression line perfectly, and there is no unexplained
variability in the Y's; if $r^2 \approx 0,$ then there is essentially no
linear component of association between X and Y and regression is not useful
for using x's to predict Y's.)
You are correct that $\hat{\sigma^2} = S_{Y|x}^2 = \frac{1}{n-2}\sum_i e_i^2$
is an unbiased estimator of $\sigma^2.$
