Question

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

Class	0-20	20-40	40-60	60-80	80-100	100-120	120-140
Frequency	12	18	15	25	26	15	9

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-70)}{20} \end{gathered}$$	$(f_i\times u_i)$
0-20	12	10	−3	−36
20-40	18	30	−2	−36
40-60	15	50	−1	−15
60-80	25	70 = A	0	0
80-100	26	90	1	26
100-120	15	110	2	30
120-140	9	130	3	27
	$\sum f_i=120$			$\sum (f_i\times u_i)=-4$

$$\begin{gathered} \text{Now,}A=70,h=20,\sum f_i=120\mathrm{and}\sum (f_{{}_i}\times u_i)=-4 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=70+}\{20\times \frac{(-4)}{120}\} \\ \text{=70-0}\text{.67} \\ \text{=69}\text{.33} \\ \text{∴}\overline{x}=69.33 \end{gathered}$$