Question

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

Class	84-90	90-96	96-102	102-108	108-114	114-120
Frequency	15	22	20	18	20	25

Answer 1

Class	Frequency$\left(f_i\right)$	Mid values$\left(x_i\right)$	$$\begin{gathered} u_i=\frac{(x_i-A)}{h} \\ \text{=}\frac{(x_i-99)}{6} \end{gathered}$$	$(f_i\times u_i)$
84-90	15	87	−2	−30
90-96	22	93	−1	−22
96-102	20	99 = A	0	0
102-108	18	105	1	18
108-114	20	111	2	40
114-120	25	117	3	75
	$\sum f_i=120$			$\sum (f_i\times u_i)=81$

$\begin{array}{l}\text{Now,}A=99,h=6,\sum f_i=120\mathrm{and}\sum (f_{{}_i}\times u_i)=81\\ \therefore \text{Mean,}\overline{x}=A+\{h\times \frac{{\displaystyle \sum (f_{{}_i}\times u_i)}}{{\displaystyle \sum f_i}}\}\\ \text{=99+}\{6\times \frac{81}{120}\}\\ \text{=99+4}\text{.05}\\ \text{=103}\text{.05}\\ \text{∴}\overline{x}=103.05\end{array}$