Question

The following table gives the number of goals scored by $71$ leading players in International Football matches. Find the standard deviation of the data.

Class Interval	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$
Frequency	$8$	$12$	$17$	$14$	$9$	$7$	$4$

Answer 1

Let A$=35$. In the $4$th column, the common factor of all items, $c=10$.

Class interval	x mid value	f	$x-A$	$d=\frac{x-A}{c}$	fd	$f{d}^2$
$0-10$	$5$	$8$	$-30$	$-3$	$-24$	$72$
$10-20$	$15$	$12$	$-20$	$-2$	$-24$	$48$
$20-30$	$25$	$17$	$-10$	$-1$	$-17$	$17$
$30-40$	$35$	$14$	$0$	$0$	$0$	$0$
$40-50$	$45$	$9$	$10$	$1$	$9$	$9$
$50-60$	$55$	$7$	$20$	$2$	$14$	$28$
$60-70$	$65$	$4$	$30$	$3$	$12$	$36$
		$\sum f=71$			$\sum fd=-30$	$\sum f{d}^2=210$

Stardard deviation, $\sigma =\sqrt{\frac{\sum f{d}^2}{\sum f}-{\left(\frac{\sum fd}{\sum f}\right)}^2}\times c$
$=\sqrt{\frac{210}{71}-{\left(\frac{-30}{71}\right)}^2}\times 10$
$=\sqrt{\frac{210}{71}-\frac{900}{5041}}\times 10$
$=\sqrt{\frac{14910-900}{5041}}\times 10$
$=\sqrt{\frac{14010}{5041}}\times 10=\sqrt{2.7791}\times 10$
Standard deviation, $\sigma \simeq 16.67$.