Question

The median of the following data set is $56.5$.

$91$	$49$	$83$	$49$	$76$
$9$	$76$	$x$	$23$	$63$

What is the value of $x$?

• A. $37$
• B. $56.5$
• C. $50$
  
• D. None of the above

Answer 1

The correct option is C $50$


Let's arrange the given data set in ascending order, except the unknown $x$.

$9$	$23$	$49$	$49$	$63$
$76$	$76$	$83$	$91$	$x$

As the data set has even number of elements, median is the mean of two central values.

As per the question, the median is $56.5$, which is less than $63$.

$\therefore$ $x$ must be placed just before $63$, such that the mean of $x$ and $63$ can be $56.5$.

The correct arrangement becomes

$9$	$23$	$49$	$49$	$\underset{–}{x}$
$\underset{–}{63}$	$76$	$76$	$83$	$91$

$\text{Median}=\text{Mean of}x\text{and}63=56.5$

$\Rightarrow \frac{x+63}{2}=56.5$

$\Rightarrow \frac{x+63}{2}\times 2=56.5\times 2$

(multiplying $2$ in both sides)

$\Rightarrow x+63=113$

$\Rightarrow x+63-63=113-63$

(substracting $63$ from both sides)

$\Rightarrow x=50$

Hence, the modified ordered data

$9$	$23$	$49$	$49$	$\underset{–}{50}$
$\underset{–}{63}$	$76$	$76$	$83$	$91$

And, the value of $x$ is $50$.