Question

Find the standard deviation of the following distribution.

x	$70$	$74$	$78$	$82$	$86$	$90$
f	$1$	$3$	$5$	$7$	$8$	$12$

Answer 1

Let us take the assumed mean A$=82$.

x	f	$d=x-82$	fd	$f{d}^2$
$70$	$1$	$-12$	$-12$	$144$
$74$	$3$	$-8$	$-24$	$192$
$78$	$5$	$-4$	$-20$	$80$
$82$	$7$	$0$	$0$	$0$
$86$	$8$	$4$	$32$	$128$
$90$	$12$	$8$	$96$	$768$
	$\sum f=36$		$\sum fd=72$	$\sum f{d}^2=1312$

Standard deviation $\sigma =\sqrt{\frac{\sum f{d}^2}{\sum f}-{\left(\frac{\sum fd}{\sum f}\right)}^2}$
$=\sqrt{\frac{1312}{36}-{\left(\frac{72}{36}\right)}^2}$
$=\sqrt{\frac{328}{9}-2^2}$
$=\sqrt{\frac{328-36}{9}}$
$=\sqrt{\frac{292}{9}}=\sqrt{32.44}$
$\therefore \sigma \simeq 5.7$.