Question

The median of the following data is $---$.
$98,75,96,180,270,102,94,100,610,200,75,80$
A. $100$
B. $98$
C. $97$
D. $99$

Answer 1

Hint: For solving this kind of questions, we need to arrange the given elements in ascending or descending order. We know that Median is the Mid-term of the ascending or descending arrangement. After arranging the given elements in ascending or descending order we will remove one by one elements from the both sides i.e. element one element from right side of the arrangement and another element from the left side of arrangement simultaneously. For an even number of elements, we will get two terms in the mid position, hence we will calculate the mean of those two variables and declare it as the median of the given data. If only one element remains after eliminating one element from both sides simultaneously, then that variable is the median of the arrangement.

Complete step-by-step answer:
Given data,
$98,75,96,180,270,102,94,100,610,200,75,80$
Arranging the above data from smaller to larger elements. i.e. the ascending order of the above data is given by
\[\text{75,75,80,94,96,98,100,102,180,200,270,610}\]
The number of elements in the given data is $12$.
Eliminating the rightmost element$\left( 610 \right)$ and leftmost element$\left( 75 \right)$ from the arrangement, then we will get
\[\text{75,80,94,96,98,100,102,180,200,270}\]
Again, Eliminating the rightmost element$\left( 270 \right)$ and leftmost element$\left( 75 \right)$ from the arrangement, then we will get
\[\text{80,94,96,98,100,102,180,200}\]
Again, Eliminating the rightmost element$\left( 200 \right)$ and leftmost element$\left( 80 \right)$ from the arrangement, then we will get
\[\text{94,96,98,100,102,180}\]
Again, Eliminating the rightmost element$\left( 180 \right)$ and leftmost element$\left( 94 \right)$ from the arrangement, then we will get
\[\text{96,98,100,102}\]
Again, Eliminating the rightmost element$\left( 102 \right)$ and leftmost element$\left( 96 \right)$ from the arrangement, then we will get
\[\text{98,100}\]
We have a number of elements as $12$ which is an even number, so here the process of finding median is ended with $2$ number. To find the median of the arrangement we need to find the average of the obtained two numbers. Hence
$\begin{align}
  & \text{Median}=\dfrac{98+100}{2} \\
 & =\dfrac{198}{2} \\
 & =99 \\
\end{align}$
Hence the Median of the given data is $99$.

So, the correct answer is “Option d”.

Note: We can also get the same answer when we arranged the given data in descending order.
The descending order of the arrangement is
\[\text{610,270,200,180,102,100,98,96,94,80,75,75}\]
For the above arrangement the eliminated terms are
${{1}^{st}}$eliminated $\to 610,75$
${{2}^{nd}}$eliminated $\to 270,75$
${{3}^{rd}}$eliminated $\to 200,80$
${{4}^{th}}$eliminated $\to 180,94$
${{5}^{th}}$eliminated $\to 102,96$
The remaining elements are $100,98$, the average of these variables is $\dfrac{100+98}{2}=99$.
Hence from both the arrangements we got the same answer.