Question

How do you find the arithmetic mean of grouped data $?$

Answer 1

Finding the arithmetic mean of grouped data:

There are three methods used to find the mean of grouped data. They are,
(a) Direct method
(b) Shortcut method
(c) Step deviation method

we can calculate the arithmetic mean by direct method formula as:

$\overline{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{}\frac{{\displaystyle \sum _{}^{}{x}_i{f}_i}}{{\displaystyle \sum _{}^{}{f}_i}}$

Where, $\overline{x}=$mean of group data

$x_i=$ Class mark

$f_i=$ frequency of class mark

Mean is calculated by shortcut method formula as:

$\overline{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\frac{\mathbf{1}}{\mathbf{N}}\mathbf{\sum }_{\mathbf{i}\mathbf{=}\mathbf{1}}^{\mathbf{n}}\mathbf{}{\mathit{f}}_{\mathbf{i}}{\mathit{d}}_{\mathbf{i}}$

where, $d_i=x_i-A$

$A=$ Assumed mean

$x_i=$ Class mark

$N=$ Total number of data

$f_i=$ frequency of class mark

$\overline{x}=$ mean of group data

To calculate the arithmetic mean using the step deviation method, we first find the deviations $d$ which are divisible by a number $h$. In such a case,

The arithmetic mean can be determined as:

$\overline{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{h}\left(\frac{1}{N}\sum _{i=1}^n{f}_i{u}_i\right)$

where, $u_i=\frac{x_i-A}{h}$

$x_i=$ Class mark

$N=$ Total number of class mark

$f_i=$ frequency of class mark

$\overline{x}=$ mean of group data

$h=$ Class size

Hence, we can find the arithmetic mean of group data by three different methods.