Question

Find the value of $f$ if the mean of the following distribution is $25.2$.

Class-interval	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequency	$8$	$12$	$f$	$11$	$9$

Answer 1

Step 1: Completing the table:

The value of $x_i$ can be found by taking the mid-value of the class intervals.

Class-interval	$x_i$	Frequency $f_i$	$f_i{x}_i$
$0-10$	$5$	$8$	$40$
$10-20$	$15$	$12$	$180$
$20-30$	$25$	$f$	$25f$
$30-40$	$35$	$11$	$385$
$40-50$	$45$	$9$	$405$
		$\sum _{}{f}_i=40+f$	$\sum _{}{f}_i{x}_i=1010+25f$

Step 2: Finding the value of $\mathit{f}$:

The formula of the mean is given by:

$\overline{x}=\left(\frac{\sum _{}{f}_i{x}_i}{\sum _{}{f}_i}\right)$

$f_i$ is the frequency of observations, and

$x_i$ is the mid-value of the class interval.

The mean of the given distribution is given as $25.2$.

Substituting the values we get,

$$\begin{gathered} 25.2=\left(\frac{1010+25f}{40+f}\right) \\ \Rightarrow 25.2\left(40+f\right)=1010+25f \\ \Rightarrow 1008+25.2f=1010+25f \\ \Rightarrow 0.2f=2 \\ \therefore f=10 \end{gathered}$$

Hence, the required value is $f=10$.