Question

Calculate coefficient of correlation between age group and rate of mortality from the following data.

$\begin{array}{cccccc}AgeGroup& 0-20& 20-40& 40-60& 60-80& 80-100\\ RateofMortality& 350& 280& 540& 760& 900\end{array}$

Answer 1

Since the class interval is given for age, mid value should be used for the calculation of r.

Calculate of Coefficient of Correlation
$\begin{array}{cccccccccc}AgeGroup& MidValue\left(X\right)& dx=(X-A)& d{x}^{\prime }=\frac{dx}{C_1}& d{x}^{\prime 2}& RateofMortalityy\left(Y\right)& dy=(Y-A)& C_2=10,& d{y}^{\prime 2}& d{x}^{\prime }d{y}^{\prime }\\ 0-20& 10& -40& -2& 4& 350& -190& -19& 361& 38\\ 20-40& 30& -20& -1& 1& 280& -260& -26& 676& 26\\ 40-60& 50& 0& 0& 0& 540& 0& 0& 0& 0\\ 60-80& 70& 20& 1& 1& 760& 220& +22& 484& 2\\ 80-100& 90& 40& 2& 4& 900& 360& +36& 1296& 72\\ & & & \sum d{x}^{\prime }=0& \sum d{x}^{\prime }=0& & & \sum d{y}^{\prime }=13& \sum d{y}^{\prime 2}=2817& \sum d{x}^{\prime }dy=158\end{array}$
Here, $N=5,\sum d{x}^{\prime }=0,\sum d{x}_i^2=10,\sum d{y}^{\prime }=13,\sum d{y}^{\prime 2}=2817$ and $\sum d{x}^{\prime }d{y}^{\prime }=158$
Now, $r=\frac{\sum d{x}^{\prime }d{y}^{\prime }-\frac{\sum d{x}^{\prime }\times \sum d{y}^{\prime }}{N}}{\sqrt{\sum d{x}^{\prime 2}-\frac{(\sum d{x}^{\prime }{)}^2}{N}}\times \sqrt{\sum d{y}^{\prime 2}-\frac{(\sum d{y}^{\prime }{)}^2}{N}}}$
$=\frac{158-\frac{0\times 13}{5}}{\sqrt{10-\frac{(0{)}^2}{5}}\times \sqrt{2817-\frac{(13{)}^2}{5}}}$
$=\frac{158}{\sqrt{10-0}\times \sqrt{2817-\frac{169}{5}}}=\frac{158}{\sqrt{10}\times \sqrt{2817-33.8}}$
$=\frac{158}{\sqrt{10}\times \sqrt{2783.2}}=\frac{158}{3.16\times 52.8}=\frac{158}{166.8}=+0.95$
There is high degree of positive correlation between age group and rate of mortality.