Question

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

Class	Number of students
4-8	2
8-12	12
12-16	15
16-20	25
20-24	18
24-28	12
28-32	13
32-36	3

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$u_i=\frac{(x_i-A)}{h}$ $\text{=}\frac{(x_i-18)}{4}$	$(f_i\times u_i)$
4-8	2	6	-3	-6
8-12	12	10	-2	-24
12-16	15	14	-1	-15
16-20	25	18=A	0	0
20-24	18	22	1	18
24-28	12	26	2	24
28-32	13	30	3	39
32-36	3	34	4	12
	$\sum f_i=100$			$\sum (f_i\times u_i)=48$

$$\begin{gathered} \text{Now,}A=18,h=4,\sum f_i=100\mathrm{and}\sum (f_{{}_i}\times u_i)=48 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=18+}\{4\times \frac{48}{100}\} \\ \text{=18+1}\text{.92} \\ \text{=19}\text{.92} \\ \therefore \overline{x}=19.92 \end{gathered}$$