Question

From the prices of shares $X\text{and}Y$ below, find out which is more stable in value:

$X$	$35$	$54$	$52$	$53$	$56$	$58$	$52$	$50$	$51$	$49$
$Y$	$108$	$107$	$105$	$105$	$106$	$107$	$104$	$103$	$104$	$101$

Answer 1

Number of observations $N=10$

$\left(x_i\right)$	$x_i^2$
$35$	$1225$
$54$	$2916$
$52$	$2704$
$53$	$2809$
$56$	$3136$
$58$	$3364$
$52$	$2704$
$50$	$2500$
$51$	$2601$
$49$	$2401$
$\sum x_i=510$	$\sum x_i^2=26360$

Mean $\left(\overline{x}\right)=\frac{\sum x_i}{N}=\frac{510}{10}=51$
Finding variance of share prices of $X$
Variance ${\sigma }_x^2=\frac{1}{N^2}[N\sum x_i^2-{(\sum x_i)}^2]$
$\Rightarrow {\sigma }_x^2=\frac{1}{(10{)}^2}[10\times 26360-(510{)}^2]$
$\Rightarrow {\sigma }_x^2=\frac{1}{100}[263600-260100]$
$\Rightarrow {\sigma }_x^2=\frac{1}{100}\times 3500=35$
Finding standard deviation of share prices of $X$
Standard deviation ${\sigma }_x=\sqrt{\text{Variance}}$
$\Rightarrow {\sigma }_x=\sqrt{35}=5.91$

Finding Coefficient of Variation of share prices of $X$
Coefficient of Variation
$(C.V{.}_x)=\frac{{\sigma }_x}{\overline{x}}\times 100$
$\Rightarrow C.V{.}_x=\frac{5.91}{51}\times 100=11.58$
$(C.V{.}_x=11.58)$

Finding mean of share prices of $Y$

$\left(y_i\right)$	$y_i^2$
$108$	$11664$
$107$	$11449$
$105$	$11025$
$105$	$11025$
$106$	$11236$
$107$	$11449$
$104$	$10816$
$103$	$10609$
$104$	$10816$
$101$	$10201$
$\sum y_i=1050$	$\sum y_i^2=110290$

Mean $\left(\overline{y}\right)=\frac{\sum y_i}{N}=\frac{1050}{10}=105$
Finding variance of share prices of $Y$
Variance ${\sigma }_y^2=\frac{1}{N^2}[N\sum y_i^2-{(\sum y_i)}^2]$
$\Rightarrow {\sigma }_y^2=\frac{1}{(10{)}^2}[10\times 110290-(1050{)}^2]$
$\Rightarrow {\sigma }_y^2=\frac{1}{100}[1102900-1102500]$
$\Rightarrow {\sigma }_y^2=\frac{400}{100}=4$

Finding standard deviation of share prices of $Y$
Standard deviation ${\sigma }_y=\sqrt{\text{Variance}}$
$\Rightarrow {\sigma }_y=\sqrt{4}=2$

Finding Coefficient of Variation of share prices of $Y$
Coefficient of Variation
$(C.V{.}_y)=\frac{{\sigma }_y}{\overline{y}}\times 100$
$\Rightarrow C.V{.}_y=\frac{2}{105}\times 100=1.904$
$\therefore C.V{.}_x>C.V{.}_y$
Group $Y$ is more stable that $X$.