Question

Find the arithmetic mean of each of the following frequency distributions using step-deviation method:

Marks	0-10	10-20	20-30	30-40	40-50	50-60
Number of students	12	18	27	20	17	6

Answer 1

Class	Frequency$\left(f_i\right)$	Mid Values$\left(x_i\right)$	$u_i=\frac{(x_i-A)}{h}$ $\text{=}\frac{(x_i-25)}{10}$	$(f_i\times u_i)$
0-10	12	5	−2	−24
10-20	18	15	−1	−18
20-30	27	25=A	0	0
30-40	20	35	1	20
40-50	17	45	2	34
50-60	6	55	3	18
	$\sum f_i=100$			$\sum (f_i\times u_i)=30$

$$\begin{gathered} \text{Now,}A=25,h=10,\sum f_i=100\mathrm{and}\sum (f_{{}_i}\times u_i)=30 \\ \therefore \text{Mean,}\overline{x}=A+\left(h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\right) \\ \text{=25+}\left(10\times \frac{30}{100}\right) \\ \text{=25+3} \\ \text{=28} \\ \therefore \overline{x}=28 \end{gathered}$$