Question

The number of books bought at a book fair by $200$ students from a school are given in the following table.

x	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$
f	$4$	$4$	$5$	$15$	$8$	$5$	$4$	$5$

Calculate the standard deviation.

Answer 1

$x$	$f$	$xf$	$(x-\overline{x})$	$(x-\overline{x}{)}^2$	$(x-\overline{x}{)}^{2}f$
$2$	$4$	$8$	$-7$	$49$	$196$
$4$	$4$	$16$	$-5$	$25$	$100$
$6$	$5$	$30$	$-3$	$9$	$45$
$8$	$15$	$120$	$-1$	$1$	$15$
$10$	$8$	$80$	$1$	$1$	$8$
$12$	$5$	$60$	$3$	$9$	$45$
$14$	$4$	$56$	$5$	$25$	$100$
$16$	$5$	$80$	$7$	$49$	$245$
	$\sum f=50$	$\sum xf=450$			$\sum (x-\overline{x}{)}^{2}f=754$

$\Rightarrow$ $\overline{x}=\frac{\sum xf}{\sum f}=\frac{450}{50}=9$

Now, calculate the standard deviation,

$S=\sqrt{\frac{(x-\overline{x}{)}^2}{\sum f}}=\sqrt{\frac{754}{50}}=\sqrt{15.08}=3.88$