Question

Consider the following data set:

$15$	$16$	$18$	$13$	$11$
$x$	$20$	$12$	$22$	$23$

If the median of the given data set is $16.5$, then $x$ = .

Answer 1

To find the median of the given data set, we first arrange the data points in ascending order, except the unknown $x$.

$11$	$12$	$13$	$15$	$\underset{–}{16}$
$\underset{–}{18}$	$20$	$22$	$23$	$x$

As it is an even-numbered data set, median is the mean of two central data points.

As per the current arrangement, the two middle-positioned data points are $16$ and $18$.

Mean of $16$ and $18$ is
$\frac{16+18}{2}=17$, which is not the median as per the question.

As per the question, the median is $16.5$, so we can place the unknown $x$ just after $16$, such that there is a possibility to get the mean of $16$ and $x$ as $16.5$.

Hence, the correct arrangement becomes

$11$	$12$	$13$	$15$	$\underset{–}{16}$
$\underset{–}{x}$	$18$	$20$	$22$	$23$

$\therefore \text{The median of the above data set}=\frac{16+x}{2}=16.5$
$\Rightarrow \frac{16+x}{2}=16.5$
$\Rightarrow \frac{16+x}{2}\times 2=16.5\times 2$
(multiplying $2$ both sides)
$\Rightarrow 16+x=33$
$\Rightarrow 16+x-16=33-16$
(substracting $16$ from both sides)
$\Rightarrow x=17$

Hence, the given data set becomes

$11$	$12$	$13$	$15$	$\underset{–}{16}$
$\underset{–}{17}$	$18$	$20$	$22$	$23$

And, the value of the unknown $x$ is $17$.