Question

Find the median of: $17,23,36,12,18,23,40$ and 20.

Answer 1

According to the question given in the question we have to determine the median for the data $17,23,36,12,18,23,40$ and 20. So, first of all we have to arrange the given data into ascending order which means we have to arrange the given data from the minimum number to maximum number.
Now, we have to find the total number of observations with the help of the given series and then and if the obtained number is even so, we have to find the median for the given series which can be determined with the help of the formula to find the median which is as explained below:

Formula used: $\Rightarrow$Median$= {\dfrac{n}{2}^{th}}$and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$observation.
Where n are the even number of observations.
So, if the number of given observations are odd then we have to apply the formula (A) to determine the median. Then after determining the observation number we have to check for the median.

Complete step-by-step solution:
Step 1: First of all we have to rearrange all the given number in ascending order means we have to arrange all the numbers from minimum number to maximum number which is as below:
$\Rightarrow 12,17,18,20,23,23,36,40$
Step 2: Now, we have to count the total number of observations to determine that the given numbers are even in count or odd in count. Hence,
$\Rightarrow n = 8$ which is even.
Step 3: As from step 2 we have determined that the total number of observations are odd so, we have to use the formula (A) as mentioned in the solution hint.
$\Rightarrow$Median
$= {\dfrac{8}{2}^{th}}$
$= {4^{th}}$observation
And,
$= {\left( {\dfrac{8}{2} + 1} \right)^{th}} \\\ = {(4 + 1)^{th}}$
$=$ ${5^{th}}$observation
Step 4: Now, we have to determine the observation number from the obtained ascending number of series we have to check for the median. Hence,
$\Rightarrow$ ${4^{th}}$observation is 20 and,
$\Rightarrow$ ${5^{th}}$observation is 23.
Step 5: Now, we have to determine the median of the observations as obtained in the solution step 4 as mentioned in the solution hint. Hence,
$\Rightarrow \dfrac{{{4^{th}} + {5^{th}}}}{2}$observations
$= \dfrac{{20 + 23}}{2} \\\ = \dfrac{{43}}{2} \\\ = 21.5$

Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the median for the given data which is 21.5.

Note: To determine the median for the given data it is necessary that we have to find the total number of given observation and of the total number of given observation is odd then to find the median we have to use the formula ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$
If the total number of observations are even then there will be two median which can be obtain by finding the observations which are ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$observations.