Question

Find the mean of the following frequency distribution 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

Let us choose a = 105, h = 6, then d_i = x_i − 105 and u_i = $\frac{x_i-105}{6}$.

Using step-deviation method, the given data is shown as follows:

Class	Frequency (fi)	Class mark (xi)	di = xi − 105	ui = $\frac{{\mathbf{x}}_{\mathbf{i}}\mathbf{-}\mathbf{105}}{\mathbf{6}}$	fiui
84−90	15	87	−18	−3	−45
90−96	22	93	−12	−2	−44
96−102	20	99	−6	−1	−20
102−108	18	105	0	0	0
108−114	20	111	6	1	20
114−120	25	117	12	2	50
Total	∑ fi = 120				∑ fiui = −39

The mean of the given data is given by

$$\begin{gathered} \overline{x}=a+\left(\frac{{\displaystyle \sum _i}{f}_i{u}_i}{{\displaystyle \sum _i}{f}_i}\right)\times h \\ =105+\left(\frac{-39}{120}\right)\times 6 \\ =105-1.95 \\ =103.05 \end{gathered}$$

Thus, mean of the given data is 103.05.