Question

Find the variance and standard deviation of the following scores: $68,72,80,84,92,100$

Answer 1

Score $x$	Deviation $d=\frac{x-84}{4}$	$d^2$
$68$ $72$ $80$ $84$ $92$ $100$	$\begin{array}{c}-4\\ -3\\ -1\end{array}\}-8$ $0$ $\begin{array}{c}+2\\ +4\end{array}\}+6$	$16$ $9$ $1$ $0$ $4$ $16$
$\sum x=496$	$\sum d=-2$	$\sum d^2=46$

$Mean=\frac{\sum x}{n}=\frac{496}{6}=82.6$

We observe that all the scores have $4$ as common factor.
Let us take $84$ as assumed mean i.e., $A=84$
Since, the data has a common factor and the mean value $\left(82.6\right)$ is not an integer, let us use step deviation method to find ${\sigma }^2$ and $\sigma$.
$\therefore d=\frac{x-A}{C}=\frac{x-84}{4}$
i.e., $\frac{68-84}{4}=\frac{-16}{4}=-4$ and so on.
(i) $Variance=\frac{\sum d^2}{n}-{\left(\frac{\sum d}{n}\right)}^2=\frac{46}{6}-{\left(\frac{-2}{6}\right)}^2=7.7-0.1$
${\sigma }^2=7.6\times 4^2=122.88$
(ii) Standard deviation $=\sigma =\sqrt{Variance}=\sqrt{7.6}=2.76\times 4=11.04$