Question

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

Class	500-520	520-540	540-560	560-580	580-600	600-620
Frequency	14	9	5	4	3	5

Answer 1

Class	Frequency$\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-550)}{20} \end{gathered}$$	$(f_i\times u_i)$
500-520	14	510	−2	−28
520-540	9	530	−1	−9
540-560	5	550 = A	0	0
560-580	4	570	1	4
580-600	3	590	2	6
600-620	5	610	3	15
	$\sum f_i=40$			$\sum (f_i\times u_i)=-12$

$$\begin{gathered} \text{Now,}A=550,h=20,\sum f_i=40\mathrm{and}\sum (f_{{}_i}\times u_i)=-12 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=550+}\{20\times \frac{(-12)}{40}\} \\ \text{=550-6} \\ \text{=544} \\ \therefore \overline{x}=544 \end{gathered}$$