Question

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

Class	10-15	15-20	20-25	25-30	30-35	35-40
Frequency	5	6	8	12	6	3

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-22.5)}{5} \end{gathered}$$	$(f_i\times u_i)$
10-15	5	12.5	−2	−10
15-20	6	17.5	−1	−6
20-25	8	22.5 = A	0	0
25-30	12	27.5	1	12
30-35	6	32.5	2	12
35-40	3	37.5	3	9
	$\sum f_i=40$			$\sum (f_i\times u_i)=17$

$$\begin{gathered} \text{Now,}A=22.5,h=5,\sum f_i=40\mathrm{and}\sum (f_{{}_i}\times u_i)=17 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=22}\text{.5+}\{5\times \frac{17}{40}\} \\ \text{=22}\text{.5+2}\text{.125} \\ \text{=24}\text{.625} \\ \therefore \overline{x}=24.625 \end{gathered}$$