Question

Calculate the correlation coefficient between $x$ and $y$ for the following data:

$x$	$5$	$9$	$13$	$17$	$21$
$y$	$12$	$20$	$25$	$33$	$35$

Answer 1

$x_i$	$y_i$	$u_i$	$v_i$	$u_i^2$	$v_i^2$	$u_i{v}_i$
5	12	-8	-13	64	169	104
9	20	-4	-5	16	25	20
13	25	0	0	00	00	00
17	33	4	8	16	64	32
21	35	8	10	64	100	80
$\sum x_i=65$	$\sum y_i=125$	$\sum u_i=0$	$\sum v_i=0$	$\sum u_i^2=160$	$\sum v_i^2=358$	$\sum u_i{v}_i=236$

$\overline{x}=\frac{1}{n}\sum x_i=\frac{1}{5}\times 65=13$
$\overline{y}=\frac{1}{n}\sum y_i=\frac{1}{5}\times 125=25$
$\rho (x,y)=\rho (u,v)=\frac{n\sum u_i{v}_i-\sum u_i\sum v_i}{\sqrt{n\sum u_i^2-{(\sum u_i^2)}^2}\sqrt{n\sum v_i^2-{(\sum v_i^2)}^2}}$
$=\frac{5\times 236-0\times 0}{\sqrt{5\times 160-0^2}\sqrt{5\times 358-0^2}}=\frac{236}{239.33}$
$=0.98$