Question

From the following data, relating to playing habits in various age group of 900 students. Calculate coefficient of correlation between age group and playing habits.

Age Group	15−16	16−17	17−18	18−19	19−20	20−21
Number of Students	250	200	150	120	100	80
Regular Players	200	150	90	48	30	12

Answer 1

Age Group	Number of People	Number of Players	Percentage of Players (%)
15-16	250	200	$\frac{200}{250}\times 100=80\%$
16-17	200	150	$\frac{150}{200}\times 100=75\%$
17-18	150	90	$\frac{90}{150}\times 100=60\%$
18-19	120	48	$\frac{48}{120}\times 100=40\%$
19-20	100	30	$\frac{30}{100}\times 100=30\%$
20-21	80	12	$\frac{12}{80}\times 100=15\%$

Age Group	Mid Value (X)	Percentage of Players (%) (Y)	${\mathit{d}}_{\mathit{X}}\mathbf{=}\mathit{X}\mathit{-}\mathit{A}\mathbf{=}\mathit{X}\mathbf{-}\mathbf{17}\mathbf{.}\mathbf{5}$	${\mathit{d}}_{\mathit{Y}}\mathbf{=}\mathit{Y}\mathit{-}\mathit{B}\mathbf{=}\mathit{Y}\mathbf{-}\mathbf{40}$	dXdY	dX2	dY2
15-16 16-17 17-18 18-19 19-20 20-21	15.5 16.5 $\boxed{A=17.5}$ 18.5 19.5 20.5	80 75 60 $\boxed{B=40}$ 30 15	–2 –1 0 1 2 3	40 35 20 0 –10 –25	–80 –35 0 0 –20 –75	4 1 0 1 4 9	1600 1225 400 0 100 625
	N = 6	N = 6	∑dX= 3	∑dY = 60	∑dXdY== –210	∑dX2 = 19	∑dY2=∑3950

$$\begin{gathered} r=\frac{{\sum }_{}{d}_X{d}_Y-{\displaystyle \frac{\left({\sum }_{}^{}{d}_X\right)\times \left({\sum }_{}^{}{d}_Y\right)}{N}}}{\sqrt{{\sum }_{}^{}{d}_X^{\mathit{2}}-{\displaystyle \frac{{\left({\sum }_{}^{}{d}_X\right)}^2}{N}}}\times \sqrt{{\sum }_{}^{}{d}_Y^{\mathit{2}}-{\displaystyle \frac{{\left({\sum }_{}^{}{d}_Y\right)}^2}{N}}}} \\ \mathrm{or},r=\frac{-210-{\displaystyle \frac{3\times \left(60\right)}{6}}}{\sqrt{19-{\displaystyle \frac{(3{)}^2}{6}}}\times \sqrt{3950-{\displaystyle \frac{(60{)}^2}{6}}}}=\frac{-240}{\sqrt{17.5}\times \sqrt{3350}} \\ \mathrm{or},r=\frac{-240}{4.18\times 57.88}=\frac{-240}{241.94}=-0.992 \\ \therefore r=-0.992 \end{gathered}$$
Hence, the coefficient of correlation between age group and playing habits is – 0.992