Question

Find the value of $p$ if the mean of the following distribution is $7.5$.

$x$	$3$	$5$	$7$	$9$	$11$	$13$
$f$	$6$	$8$	$15$	$p$	$8$	$4$

Answer 1

Step 1 Finding the mean:

The given table can be completed as

$x$	$f$	$fx$
$3$	$6$	$18$
$5$	$8$	$40$
$7$	$15$	$105$
$9$	$p$	$9p$
$11$	$8$	$88$
$13$	$4$	$52$
	$\sum _{}f=41+p$	$\sum _{}fx=303+9p$

The formula for mean is given by $\overline{x}=\frac{\sum _{}fx}{\sum _{}f}$.

where, $\sum _{}f$ is the total frequency and $x$ is the observation.

it is given in the question that $\overline{x}=7.5$.

Step 2 Finding the value of $p$:

Substituting the values in the formula,

$$\begin{gathered} \Rightarrow 7.5=\frac{303+9p}{41+p} \\ \Rightarrow 307.5+7.5p=303+9p \\ \Rightarrow 1.5p=4.5 \\ \therefore p=3 \end{gathered}$$

Hence, the value of $p$ is obtained as $3$.