Question

The arithmetic mean of the following frequency distribution is $50$. Determine the value of $p$.

Class-interval	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$
Frequency	$16$	$p$	$30$	$32$	$14$

Answer 1

Step 1: Finding mean:

Completing the table,

Class-interval	Mid-value $x_i$	Frequency $f_i$	$f_i{x}_i$
$0-20$	$10$	$16$	$160$
$20-40$	$30$	$p$	$30p$
$40-60$	$50$	$30$	$1500$
$60-80$	$70$	$32$	$2240$
$80-100$	$90$	$14$	$1260$
		$\Sigma f_i=92+p$	$\Sigma f_i{x}_i=5160+30p$

The formula for mean is $\overline{x}=\frac{\Sigma f_i{x}_i}{\Sigma f_i}$.

where $\Sigma f_i$ is the total frequency and $x_i$ is the mid-value of class intervals.

From the question, we have $\overline{x}=50$.

Step 2: Finding the value of $p$:

Substituting the values in the formula for mean,

$$\begin{gathered} \Rightarrow 50=\frac{5160+30p}{92+p} \\ \Rightarrow 4600+50p=5160+30p \\ \Rightarrow 20p=560 \\ \therefore p=28 \end{gathered}$$

Hence, the value of $p$ is obtained as $28$.