Question

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

Age (in years)	18-24	24-30	30-36	36-42	42-48	48-54
Number of workers	6	8	12	8	4	2

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-33)}{6} \end{gathered}$$	$(f_i\times u_i)$
18-24	6	21	−2	−12
24-30	8	27	−1	−8
30-36	12	33 = A	0	0
36-42	8	39	1	8
42-48	4	45	2	8
48-54	2	51	3	6
	$\sum f_i=40$			$\sum (f_i\times u_i)=2$

$$\begin{gathered} \text{Now,}A=33,h=6,\sum f_i=40\mathrm{and}\sum (f_{{}_i}\times u_i)=2 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=33+}\{6\times \frac{2}{40}\} \\ \text{=33+0}\text{.3} \\ \text{=33}\text{.3} \\ \therefore \overline{x}=33.3\text{years} \end{gathered}$$