Question

For $10$ pairs of observation on two variables $x$ and $y$, the following data are available.
$\sum (x-2)=30,\sum (y-5)=40,\sum (x-2{)}^2=900$.
$\sum (y-5{)}^2=800,\sum (x-2\left)\right(y-5)=480$.
Find the correlation coefficient between $x$ and $y$

Answer 1

Let $x=x_i$ and $y=y_i$
Let $u_i=x_i-2$ and $v_i=y_i-5$
Then it is given that,
$n=10,\sum u_i=30,\sum v_i=40,\sum u_i^2=900,\sum v_i^2=800,\sum u_i{v}_i=480$
$\therefore \overline{u}=\frac{\sum u_i}{n}=\frac{30}{10}=3$
$\overline{v}=\frac{\sum vi}{n}=\frac{40}{10}=4$
Since the correlation coefficient is independent of change of origin and scale, correlation coefficient between $x$ and $y$ is
$r(x,y)=r(u,v)$
$=\frac{\frac{1}{n}\sum u_i{v}_i-\overline{u}\overline{v}}{\sqrt{\frac{1}{n}\sum u_i^2-(\overline{u}{)}^2}\cdot \sqrt{\frac{1}{n}\sum v_i^2-\left(\overline{v^2}\right)}}$
$=\frac{\frac{480}{10}-3\times 4}{\sqrt{\frac{900}{10}-(3{)}^2}\cdot \sqrt{\frac{800}{10}-(4{)}^2}}$
$=\frac{48-12}{\sqrt{81}\cdot \sqrt{64}}=\frac{36}{9\times 8}=\frac{4}{8}=0.5$