Question

Calculate the correlation coefficient between $X$ and $Y$ and comment on their relationship.

$\begin{array}{ccccccc}X& 1& 3& 4& 5& 7& 8\\ Y& 2& 6& 8& 10& 14& 16\end{array}$

Answer 1

$\begin{array}{ccccc}X& Y& XY& X^2& Y^2\\ 1& 2& 2& 1& 4\\ 3& 6& 18& 9& 36\\ 4& 8& 32& 16& 64\\ 5& 10& 50& 25& 100\\ 7& 14& 98& 49& 196\\ 8& 16& 128& 64& 256\\ \sum X=28& \sum Y=56& \sum XY=328& \sum X^2=164& \sum Y^2=656\end{array}$

Now, Correlation coefficient between $X$ and $Y$ is

$\left(r\right)=\frac{\sum XY-\frac{(\sum X)(\sum Y)}{N}}{\sqrt{\sum X^2-\frac{(\sum X{)}^2}{N}}\sqrt{\sum Y^2-\frac{(\sum Y{)}^2}{N}}}$

$=\frac{328-\frac{28\times 56}{6}}{\sqrt{164-\frac{(28{)}^2}{6}}\sqrt{656-\frac{(56{)}^2}{6}}}$

$=\frac{328-\frac{1568}{6}}{\sqrt{164-\frac{784}{6}}\sqrt{656-\frac{3136}{6}}}$

$=\frac{328-261.33}{\sqrt{164-130.67}\sqrt{655-522.67}}$

$=\frac{66.67}{5.77\times 11.55}$

$=\frac{66.6}{66.6}$

$=1$

As the correlation coefficient between the two variables is $+1$, so the two variables are perfectly positive correlated.