Question

Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y).

$\begin{array}{ccccccccc}X& 65& 66& 57& 67& 68& 69& 70& 72\\ Y& 67& 56& 65& 68& 72& 72& 69& 71\end{array}$

Answer 1

$\begin{array}{ccccc}X& Y& XY& X^2& Y^2\\ 65& 67& 4355& 4255& 4489\\ 66& 56& 3696& 4356& 3136\\ 57& 65& 3705& 3249& 4255\\ 67& 68& 4556& 4489& 4624\\ 68& 72& 4896& 4624& 5184\\ 69& 72& 4968& 4761& 5184\\ 70& 69& 4830& 4900& 4761\\ 72& 71& 5112& 5184& 5041\\ \sum X=534& \sum Y=540& \sum XY=36118& \sum X^2=35788& \sum Y^2=36644\end{array}$

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

$=\frac{36118-\frac{534\times 540}{8}}{\sqrt{35788-\frac{(534{)}^2}{8}}\sqrt{36644-\frac{(540{)}^2}{8}}}$

$=\frac{36118-\frac{288360}{8}}{\sqrt{35788-\frac{285156}{8}}\sqrt{36644-\frac{291600}{8}}}$

$=\frac{36118-36045}{\sqrt{35788-356445}\sqrt{36644-36450}}$

$=\frac{73}{\sqrt{143.5}\sqrt{194}}=\frac{73}{11.98\times 13.93}=\frac{73}{166.88}=0.44$