Question

Find the median of the following data
$31,38,27,28,36,25,35,40$

Answer 1

Hint: The median is the process of finding the mid-value in the given set of data. To calculate the median, we need to first arrange the values in order (increasing or decreasing) and then we have to count the number of values $\left( n \right)$present in the given data. Then we shall apply the given formula.

Formula to be used:
The formula to be used to find the median of the given set of values is as follows.
\[Median = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}observation\] , if $n$ is odd.
\[Median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}\], if $n$ is even.
Where, $n$ is the number of observations.

Complete step-by-step answer:
Let us consider the given set of values $\left\{ {31,38,27,28,36,25,35,40} \right\}$.
To calculate the median, we need to first arrange the values in order (increasing or decreasing) and then we have to count the number of values $\left( n \right)$ present in the given data.
First, we shall arrange the given data.
Thus, we have $25,27,28,31,35,36,38,40$
Here, the number of values $\left( n \right)$ =$8$, which is odd.
So, by the above-median formula, we need to apply \[Median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}\]
We shall substitute $n = 8$in the above formula.
Thus, \[Median = \dfrac{{{{\left( {\dfrac{8}{2}} \right)}^{th}}observation + {{\left( {\dfrac{8}{2} + 1} \right)}^{th}}observation}}{2}\]
\[ = \dfrac{{{{\left( 4 \right)}^{th}}observation + {{\left( {4 + 1} \right)}^{th}}observation}}{2}\]
\[ = \dfrac{{{4^{th}}observation + {5^{th}}observation}}{2}\]
\[ = \dfrac{{31 + 35}}{2}\] (Here the fourth observation is $31$ and the fifth observation is $35$)
$ = \dfrac{{66}}{2}$
$ = 33$
Hence, the median of the given set of values is $33$.

Note: The median can be found without using the formula. First, we shall arrange the given set of values either in ascending or descending order. Then we have to count the number of values $\left( n \right)$ present in the given data. If $\left( n \right)$ is odd, directly write the center value as the median. If $\left( n \right)$ is even, then find the average of two middle numbers to obtain the median.