Question

The following table shows the marks obtained by $48$ students in a Quiz competition in Mathematics. Calculate the standard deviation.

Data x	$6$	$7$	$8$	$9$	$10$	$11$	$12$
Frequency f	$5$	$6$	$9$	$13$	$8$	$5$	$4$

Answer 1

Let us form the following table using the given data.

x	f	fx	$d=x-\overline{x}=x-9$	fd	$f{d}^2$
$6$	$3$	$18$	$-3$	$-9$	$27$
$7$	$6$	$42$	$-2$	$-12$	$24$
$8$	$9$	$72$	$-1$	$-9$	$9$
$9$	$13$	$117$	$0$	$0$	$0$
$10$	$8$	$80$	$1$	$8$	$8$
$11$	$5$	$55$	$2$	$10$	$20$
$12$	$4$	$48$	$3$	$12$	$36$
	$\sum f=48$	$\sum fx=432$	$\sum d=0$	$\sum fd=0$	$\sum f{d}^2=124$

Arithmetic mean, $\overline{x}=\frac{\sum fx}{\sum f}=\frac{432}{48}=9$
Standard deviation, $\sigma =\sqrt{\frac{\sum f{d}^2}{\sum f}}$
$=\sqrt{\frac{124}{48}}$
$\sigma =\sqrt{2.58}\simeq 1.61$