Question

The following results were obtained with respect to two variable x and y:

$\sum x=30,\sum y=42,\sum xy=199,\sum x^2=184,\sum y^2=318,n=6.$ Find the correlation coefficient between x and y.

Answer 1

$b_{yx}=\frac{\sum xy-\frac{1}{n}\sum x.\sum y}{\sum x^2-\frac{1}{n}(\sum x{)}^2}$
$=\frac{199-\frac{1}{6}\left(30\right)\left(42\right)}{184-\frac{1}{6}(30{)}^2}$
$=\frac{199-210}{184-150}=\frac{-11}{34}=-0.323$
$b_{xy}=\frac{\sum xy-\frac{1}{n}\sum x\sum y}{\sum y^2-\frac{1}{n}(\sum y{)}^2}$
$=\frac{199-\frac{1}{6}\left(30\right)\left(42\right)}{318-\frac{1}{6}(42{)}^2}$
$=\frac{199-210}{318-294}=\frac{-11}{24}=-0.458$

Correlation coefficient between x and y.
$r^2=b_{yx}.b_{xy}$
$=(-0.323)\times (-0.458)$
$=0.1479$
$\therefore r=\sqrt{0.1479}=-0.384$
Since $b_{xy}$ and $b_{yx}$ have the same sign, that is why we have used negative sign with r.