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A made-up gardener keeps only the totals of eight pairs of values: the age of a hedge in years (x)…

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Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A made-up gardener keeps only the totals of eight pairs of values: the age of a hedge in years ( $x$ ) and the number of new shoots counted on it in spring ( $y$ ). She has $n = 8$ , $\sum x = 76$ , $\sum y = 322$ , $\sum x^2 = 820$ , $\sum y^2 = 13\,320$ and $\sum xy = 2873$ . Find $S_{xx}$ , $S_{yy}$ , $S_{xy}$ and $r$ , to four decimal places. Describe the pattern in words.

Need a reminder? The key ideaShow

$S_{xy} = \sum (x - \bar{x})(y - \bar{y})$ , $S_{xx} = \sum (x - \bar{x})^2$ , $S_{yy} = \sum (y - \bar{y})^2$ and

\displaystyle r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}.

With sample standard deviations, $r$ is also $\dfrac{1}{n - 1}\sum z_x z_y$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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