Question
From the following data, calculate coefficient of correlation between age and playing habits.
Age Group20 -3030 - 4040 - 5050 - 6060 - 70
Number of students2560402020
Number of Regular Players10301221

Answer

 First, we are required to calculate the percentage of regular players.
Calculation of Percentage of Regular Players:
Number of StudentsNumber of Regular PlayersPercentage of Regular Players
2510$\frac{10}{25} \times 100=40$
6030$\frac{30}{60} \times 100=50$
4012$\frac{12}{40} \times 100=30$
202$\frac{2}{20} \times 100=10$
201$\frac{1}{20} \times 100=5$
Denoting mid value of age as X and percentage of regular players as Y. 
Age GroupXdx(X - A), A = 45$\begin{array}{c}d x^{\prime}\left(\frac{d z}{a}\right),  c_1=10\end{array}$$dx ^{\prime 2}$Ydy(Y - A), A = 30$\begin{array}{c} dy ^{ \prime }\left(\frac{d y}{c_2}\right),  c _2=5\end{array}$$dy ^{\prime 2}$dx'dy'
20 -3025-20-24401024-4
30-4035-10-115020416-4
40-5045000300000
50-6055101110-20-416-4
60 -706520245-25-525-10
   $\Sigma d x^{\prime}=0$\begin{array}{c}\Sigma d x^2=10\end{array}  $\Sigma d y^{\prime}=-3$\begin{array}{c}\Sigma \Sigma d^{\prime 2}=61\end{array}\begin{array}{c}\Sigma dx^{\prime} dy y^{\prime}=-22\end{array}
Here, $n =5, \Sigma dx ^{\prime}=0, \Sigma dx ^{\prime 2}=10, \Sigma dy ^{\prime}=-3, \Sigma dx ^{\prime} dy ^{\prime}=-22 \Sigma dy ^{\prime 2}=61$
Now,Putting the values in the given formula:
$\begin{array}{l}r=\frac{\Sigma d x^{\prime} d y^{\prime}-\frac{\Sigma d x^{\prime} \times \Sigma d y^{\prime}}{n}}{\sqrt{\Sigma d x^{\prime 2}-\frac{\left(\Sigma d x^{\prime}\right)^2}{n}} \times \sqrt{\Sigma d d y^2-\frac{\left(\Sigma d y^{\prime}\right)^2}{n}}}= \\ =\frac{-22-\frac{0 \times-3}{5}}{\sqrt{10-\frac{(0)^2}{5}} \times \sqrt{61-\frac{(-3)^2}{5}}} \\ =\frac{-22}{\sqrt{10} \times \sqrt{61-1.8}}=\frac{-22}{\sqrt{10} \times \sqrt{59.2}} \\ =\frac{-22}{3.16 \times 7.69}=\frac{-22}{24.3}=-0.90\end{array}$
- Therefore, Karl Pearson's coefficient of correlation between age and playing habits is $=0.90$.
- Interpretation of $r$
1. It indicates that there is a high degree of negative correlation between age and playing habits.
2. It indicates that as age increases, the tendency to play decreases.

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