Question

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

Class	0-30	30-60	60-90	90-120	120-150	150-180
Frequency	12	21	34	52	20	11

Answer 1

Class	Frequency $\left(f_i\right)$	Mid values$\left(x_i\right)$	$u_i=\frac{(x_i-A)}{h}$ $\text{=}\frac{(x_i-75)}{30}$	$(f_i\times u_i)$
0-30	12	15	−2	−24
30-60	21	45	−1	−21
60-90	34	75 = A	0	0
90-120	52	105	1	52
120-150	20	135	2	40
150-180	11	165	3	33
	$\sum f_i=150$			$\sum (f_i\times u_i)=80$

$$\begin{gathered} \text{Now,}A=75,h=30,\sum f_i=150\mathrm{and}\sum (f_{{}_i}\times u_i)=80 \\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\} \\ \text{=75+}\{30\times \frac{80}{150}\} \\ \text{=75+16} \\ \text{=91} \\ \text{∴}\overline{x}=91 \end{gathered}$$