Question

Find the value of $f$ if the mean of the following distribution is $21$.

Class-interval	$0-8$	$8-16$	$16-24$	$24-32$	$32-40$
Frequency	$5$	$9$	$10$	$f+2$	$8$

Answer 1

Step 1: Finding the mean:

Completing the table, we get:

Class-interval	$x_i$	Frequency $f_i$	$f_i{x}_i$
$0-8$	$4$	$5$	$20$
$8-16$	$12$	$9$	$108$
$16-24$	$20$	$10$	$200$
$24-32$	$28$	$f+2$	$28f+56$
$32-40$	$36$	$8$	$288$
		$\sum _{}{f}_i=34+f$	$\sum _{}{f}_i{x}_i=672+28f$

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 $21$.

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

Substituting the values we get,

$$\begin{gathered} 21=\left(\frac{672+28f}{34+f}\right) \\ \Rightarrow 21\left(34+f\right)=672+28f \\ \Rightarrow 714+21f=672+28f \\ \Rightarrow 7f=42 \\ \therefore f=6 \end{gathered}$$

Hence, the required value is $f=6$.