Question

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

Marks	0-14	14-28	28-42	42-56	56-70
Number of students	7	21	35	11	16

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-35)}{14} \end{gathered}$$	$(f_i\times u_i)$
0-14	7	7	−2	−14
14-28	21	21	−1	−21
28-42	35	35 = A	0	0
42-56	11	49	1	11
56-70	16	63	2	32
	$\sum f_i=90$			$\sum (f_i\times u_i)=8$

$$\begin{gathered} \text{Now,}A=35,h=14,\sum f_i=90\mathrm{and}\sum (f_{{}_i}\times u_i)=8 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=35+}\{14\times \frac{8}{90}\} \\ \text{=35+1}\text{.24} \\ \text{=36}\text{.24} \\ \therefore \overline{x}=36.24\mathrm{cm} \end{gathered}$$