Question

The following is a frequency table of the score obtained in a science quiz competition. Find the median score.

Score	$10$	$15$	$20$	$25$	$30$
Frequency	$2$	$3$	$5$	$6$	$4$

${\text{A) 22}}$
${\text{B) 22}}{\text{.5}}$
${\text{C) 20}}$
${\text{D) 25}}$

Answer 1

Hint: First we have to find the sum of scores \[\left( {\text{x}} \right)\] and the frequency \[\left( {\text{f}} \right)\]as this a discrete frequency distribution. Then by using the formula, whichever is around the answer i.e., round off is the required answer.

Formula used: When the number of observation is odd:
Median = ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term
When the number of observation is even:
First, find ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$ term
Then ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term
And find the average of two values i.e., average of ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$term and ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term

Complete step-by-step solution:
First we have to rearrange the given data as follows:

Score \[\left( {\text{x}} \right)\]	Frequency \[\left( {\text{f}} \right)\]
$10$	$2$
$15$	$3$
$20$	$5$
$25$	$6$
$30$	$4$
	${\text{N = 20}}$

Since the number of observation is even, we need to find average of ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$ term and ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term
On putting the value of N and we get,
${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}} = \dfrac{{20}}{2}$
Let us divide the term and we get
$ \Rightarrow {10^{th}}$ observation
Now we have to find:
${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}} = \dfrac{{20 + 1}}{2}$
On dividing the term and we get,
$ \Rightarrow 10.5$
Taking as round value and we get,
$ \Rightarrow {11^{th}}$ Observation
Now the changed data as formed as follows:

Score \[\left( {\text{x}} \right)\]	Frequency \[\left( {\text{f}} \right)\]	\[\left( {{\text{cf}}} \right)\]
$10$	$2$	$2$
$15$	$3$	$5$
$20$	$5$	$10$
$25$	$6$	$16$
$30$	$4$	$20$
	${\text{N = 20}}$

So here, ${10^{th}}$ term lies in $20$ and ${11^{th}}$ term lies in $25$
So we can write it as, by using the formula and find the median
Median = $\dfrac{{20 + 25}}{2}$
Let us add the numerator and we get,
Median = $\dfrac{{45}}{2}$
Let us divide the term and we get,
Median =$22.5$

Therefore the correct answer is ${\text{B) 22}}{\text{.5}}$.

Note: In this question we have an alternative method.
Alternative method:
We can also find median in a simple way.

Score \[\left( {\text{x}} \right)\]	Frequency \[\left( {\text{f}} \right)\]
$10$	$2$
$15$	$3$
$20$	$5$
$25$	$6$
$30$	$4$

Here we can also write this elaborately since it is discrete distribution,$10,10,15,15,15,20,20,20,20,20,25,25,25,25,25,25,30,30,30,30$
Median is the middle value of the given observation, so in this observation there are two numbers in middle (since the total numbers are even)
They are $20$ and $25$
Median = average of these two numbers
Median = $\dfrac{{20 + 25}}{2}$
Let us add the numerator and we get,
Median = $\dfrac{{45}}{2}$
Let us divide the term and we get,
Median =$22.5$
Therefore the correct answer is ${\text{B) 22}}{\text{.5}}$.