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Finding Median for Grouped Data When Class Intervals Are Given

If the median...

Question

If the median of the following frequency distribution is 40 and total frequency is 60, then find the missing frequencies.
$\begin{matrix} Marks & 0 - 10 & 10 - 30 & 30 - 60 & 60 - 80 & 80 - 90 Number of students & 5 & f_{1} & 30 & f_{2} & 2 \end{matrix}$

A

f_{1} = 16, f_{2} = 9

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B

f_{1} = 15, f_{2} = 8

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C

f_{1} = 8, f_{2} = 15

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D

f_{1} = 18, f_{2} = 7

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Solution

The correct option is B $f_{1} = 15, f_{2} = 8$
$\begin{matrix} Marks & 0 - 10 & 10 - 30 & 30 - 60 & 60 - 80 & 80 - 90 Number of students & 5 & f_{1} & 30 & f_{2} & 2 \end{matrix}$

$∵ Total frequency = 60$ [given]
$∴ 5 + f_{1} + 30 + f_{2} + 2 = 60$
$\Rightarrow 37 + f_{1} + f_{2} = 60$
$\Rightarrow f_{1} + f_{2} = 23$ ...(i)
Now, N = 60
$∴ \frac{N}{2} = 30$
and median = 40
Since, it lies in the class interval 30-60
So, 30-60 is the median class.
Here, $l = 30, h = 30, f = 30, c f = 5 + f_{1}, n = 60$
$∴ Median = l + \frac{\frac{n}{2} - c f}{f} \times h$
$\Rightarrow 40 = 30 + \frac{30 - 5 - f_{1}}{30} \times 30$
$\Rightarrow 40 = 30 + 25 - f_{1}$
$\Rightarrow 40 = 55 - f_{1} \Rightarrow f_{1} = 15$
$\Rightarrow f_{2} = 23 - 15$
$a n d f_{2} = 23 - 15 [from Eq. (i)]$
$= 8$

$∴ f_{1} = 15 a n d f_{2} = 8.$

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