Question

The median of the following data is:

$x:$	$10$	$20$	$30$	$40$	$50$
$f:$	$2$	$3$	$2$	$3$	$1$

A. $30$
B. $40$
C. $35$
D. $31$

Answer 1

Hint: To find the median of the given data, we will first arrange it in ascending or descending order. Then, find the cumulative frequency and finally find the median of the given data.
Cumulative frequency gives ranking after arranging in ascending or descending order. It is the sum of all the classes below it in distribution.

Complete step-by-step answer:
If the number of observations is odd
Then, median $ = {\left( {\dfrac{{\text{n}}}{{\text{2}}}} \right)^{{\text{th}}}}{\text{term}}$ .
If the number of observations is even
Then, median $ = {\left( {\dfrac{{{\text{n + 1}}}}{2}} \right)^{{\text{th}}}}{\text{ term}}$ .
Complete step by step solution:
Given data:

$x:$	$10$	$20$	$30$	$40$	$50$
$f:$	$2$	$3$	$2$	$3$	$1$

To find the median of the given data.
The first step is to arrange $x$ in ascending order or descending order. But in the given data it is already in ascending order.
Now, the total number of entities in the given data $ = 2 + 3 + 2 + 3 + 1 = 11$ i.e., odd.
Since in the given data frequency is given and it gives the number of times a number is repeating.
Therefore, we find cumulative frequency so that we can get the ranking of the terms after arranging them in ascending order.
So, cumulative frequency is given by:

$x$	$f$	$cf$
$10$	$2$	$2$
$20$	$3$	$2 + 3 = 5$
$30$	$2$	$5 + 2 = 7$
$40$	$3$	$7 + 3 = 10$
$50$	$1$	$10 + 1 = 11$

Now according to cumulative frequency $10$ is coming two times, then after 10 from third to the fifth position $20$ is coming, then after 20, from sixth to the seventh position $30$ is coming, then from eighth to the tenth position $40$ is coming and finally at eleventh position $50$ is coming.
So, the series becomes: $10,10,20,20,20,30,30,40,40,40,50$ .
Now, since a number of observations are odd.
So, median $ = {\left( {\dfrac{{{\text{n + 1}}}}{{\text{2}}}} \right)^{{\text{th}}}}{\text{term}}$
i.e., ${\text{ = }}{\left( {\dfrac{{{\text{11 + 1}}}}{{\text{2}}}} \right)^{{\text{th}}}}{\text{term = }}{{\text{6}}^{{\text{th}}}}{\text{ term}}$
And from the obtained series sixth term is $30$
Hence, the median of the given data is $30$ .
So, the correct answer is “Option A”.

Note: Here, a number of terms are a summation of frequency and not the sum of $x$ .Do remember frequency gives the number of times a term is repeating while cumulative frequency gives ranking. Arranging the given data in ascending or descending order is necessary.