Question

Find the mean, variance and standard deviation for the following frequency distribution.

Classes	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	15	16	6

Answer 1

Classes	Class marks $\left(x_i\right)$	$f_i$	$x_i{f}_i$
0-10	5	5	25
10-20	15	8	120
20-30	25	15	305
30-40	35	6	560
40-50	45	6	250

$\begin{array}{c}\overline{x}=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{1350}{50}=27\\ Now,\\ {\sigma }^2=\frac{\sum {\left(x-\overline{x}\right)}^2}{N}=\frac{36}{5}=7.2\end{array}$

$\begin{array}{c}\mathrm{var}iance={\sigma }^2=\frac{\sum f_i{\left(x-\overline{x}\right)}^2}{N}\\ u=\sum {\left(x-\overline{x}\right)}^2\end{array}$

$x_i$ $f_i$ $u={\left(x-\overline{x}\right)}^2$ $f_i\left(u\right)$

5 5 ${22}^2=484$ 2420

15 8 ${12}^2=144$ 1152

25 15 $2^2=4$ 60

35 16 ${12}^2=144$ 2304

45 6 ${18}^2=324$ 1344

${\sigma }^2=\frac{\sum f_{i}u}{N}=\frac{7880}{50}=157.6$