Question

A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are:
$$\begin{array}{cccccc}X& 25& 85& 40& 80& 120\\ Y& 50& 70& 65& 45& 80\end{array}$$
(a) Calculate coefficient of standard deviation, variance and coefficient of variation.
(b) Which batsman should be selected if we want:
(i) A high run scorer,
(ii) A more reliable batsman in the team.

Answer 1

$$\begin{array}{cccccc}\text{Batsman}& X-\overline{X}& (x-\overline{X}{)}^2& \text{Batsman}& X-\overline{Y}& (X-\overline{Y}{)}^2\\ \text{Score X}& x& x^2& Y& y& y^2\\ 25& -45& 2025& 50& -12& 144\\ 85& 15& 225& 70& 8& 64\\ 40& -30& 900& 65& 3& 9\\ 80& 10& 100& 45& -17& 289\\ 120& 50& 2500& 80& 18& 324\\ \sum X=350& & \sum x^2=5750& \sum Y=310& & \sum y^2=830\end{array}$$
$$\begin{array}{cc}\text{Batsman X}& \text{Batsman Y}\\ \text{Arithmetic mean}& \\ \overline{X}=\frac{\sum X}{N}& \overline{Y}=\frac{\sum Y}{N}\\ \sum X=350,N=5& \sum Y=310N=5\\ \overline{X}=\frac{350}{5}=70& \overline{Y}=\frac{310}{5}=62\\ \text{Average score = 70 Runs}& \text{Average Score = 62 Runs}\\ \text{Standard Deviation}& \\ {\sigma }_x=\sqrt{\frac{\sum x^2}{N}}& {\sigma }_y=\sqrt{\frac{\sum y^2}{N}}\\ \sum x^2=5750andN=5& \sum y^2=830andN=5\\ {\sigma }_X=\sqrt{\frac{5750}{5}}=\sqrt{1150}& {\sigma }_Y=\sqrt{\frac{830}{5}}=\sqrt{166}\\ {\sigma }_X=33.91& {\sigma }_Y=12.88\\ \text{Coefficient of S.D}& \\ \text{Coefficient of}{\sigma }_X=\frac{\text{Coefficient of x}}{\overline{X}}& \text{Coefficient of}{\sigma }_Y=\frac{\text{Coefficient of y}}{\overline{Y}}\\ \text{=}\frac{33.91}{70}=0.484& =\frac{12.88}{62}=0.207\\ \text{Variance}& \\ {\sigma }_X^2=\frac{\sum x^2}{N}=\frac{5750}{5}=1150& {\sigma }_Y^2=\frac{\sum y^2}{N}=\frac{830}{5}=166\\ \text{Coefficient of variation}& \\ C.V.=\frac{{\sigma }_X}{\overline{X}}\times 100=\frac{33.91}{70}\times 100& C.V.=\frac{{\sigma }_Y}{\overline{Y}}\times 100=\frac{12.88}{62}\times 100\\ =48.44\%& =20.77\%\end{array}$$
(b) (i) Batsman X should be selected as a higher run scorer as his average score (70 runs) is greater than the average score of Y (62 runs).
(ii) Batsman Y is a more reliable batsman in the team because his coefficient of variation (C.V = 20.77%) is less than that of batsman X (C.V = 48.44%).