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# The following is a frequency table of the score obtained in a science quiz competition. Find the median score.Score $10$ $15$$20$$25$$30Frequency 2$$3$$5$$6$$4$${\text{A) 22}}$ ${\text{B) 22}}{\text{.5}}$${\text{C) 20}}$${\text{D) 25}}$

Last updated date: 06th Sep 2024
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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 ${\text{B) 22}}{\text{.5}}$.