Question

Following data relates to age group and percentage of regular players. Calculate Karl Pearson's coefficient of correlation.

Age Group	20−25	25−30	30−35	35−40	40−45	45−50
% of Regular Players	40	35	28	20	15	5

Answer 1

Age Group	Mid Value (X)	% of Regular Players (Y)	${{\mathit{d}}^{\mathbf{\text{'}}}}_{\mathbf{X}}\mathbf{=}\frac{X\mathit{-}A}{h}\mathbf{=}\frac{\mathit{X}\mathbf{-}\mathbf{37}\mathbf{.}\mathbf{5}}{\mathbf{5}}$	${\mathit{d}}_{\mathit{Y}}^{\mathbf{\text{'}}}\mathbf{=}\frac{\mathit{Y}\mathit{-}\mathit{B}}{\mathit{i}}\mathbf{=}\frac{\mathit{Y}\mathbf{-}\mathbf{28}}{\mathbf{5}}$	(dX') (dY')	(dX')2	(dY')2
20-25 25-30 30-35 35-40 40-45 45-50	22.5 27.5 32.5 37.5 42.5 47.5	40 35 28 20 15 5	–3 –2 –1 0 1 2	2.4 1.4 0 –1.6 –2.6 –4.6	–7.2 –2.8 0 0 –2.6 –9.2	9 4 1 0 1 4	5.76 1.96 0 2.56 6.76 21.16
			∑dX' = –3	∑dY' = –5	∑(dX') (dY') = – 21.8	∑(dX')2 = 19	∑(dX')2 = 38.2

$$\begin{gathered} r=\frac{\sum d_X^{\text{'}}{d}_Y^{\text{'}}-{\displaystyle \frac{\left(\sum d_X^{\text{'}}\right)\times \left(\sum d_Y^{\text{'}}\right)}{N}}}{\sqrt{\left[\sum {d_X^{\text{'}}}^2-{\displaystyle \frac{{\left(\sum d_X^{\text{'}}\right)}^2}{N}}\right]\times \left[\sum {d_Y^{\text{'}}}^2-{\displaystyle \frac{{\left(\sum d_Y^{\text{'}}\right)}^2}{N}}\right]}} \\ \mathrm{or},r=\frac{\left(-21.8\right)-{\displaystyle \frac{\left(-3\right)\times \left(-5\right)}{6}}}{\sqrt{\left[19-{\displaystyle \frac{{\left(-3\right)}^2}{6}}\right]\times \left[38.2-{\displaystyle \frac{{\left(-5\right)}^2}{6}}\right]}}=\frac{-24.3}{\sqrt{\left[19-{\displaystyle \frac{9}{6}}\right]\times \left[38.2-{\displaystyle \frac{25}{6}}\right]}} \\ \mathrm{or},r=-\frac{-24.3}{\sqrt{17.5\times 34.03}}=-\frac{24.3}{\sqrt{595.525}}=-\frac{24.3}{24.403} \\ \mathrm{Hence},r=-0.996 \end{gathered}$$