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Ungrouped Frequency Table

If the median...

Question

If the median of the following frequency distribution is $28.5$ . find the missing frequencies.
C.I. 0-10 10-20 20-30 30-40 40-50 50-60 Total
Freq. 5 $f_{1}$ 20 15 $f_{2}$ 5 60

A

f_{1}

= 7

,

f_{2}

= 8

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B

f_{1}

= 8

,

f_{2}

= 7

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C

f_{1}

= 6

,

f_{2}

= 9

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D

f_{1}

= 9

,

f_{2}

= 6

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Solution

The correct option is C $f_{1}$ $= 8$ , $f_{2}$ $= 7$
$x_{i}$ $f_{i}$ c.f.
0 - 10 5 5
10 - 20 $f_{1}$ 5 + $f_{1}$
20 - 30 20 25 + $f_{1}$
30 - 40 15 40 + $f_{1}$
40 - 50 $f_{2}$ 40 + $f_{1} + f_{2}$
50 - 60 5 45 + $f_{1} + f_{2}$
Given : Median = 28.5 So Median class = 20 - 30 and 45 + $f_{1} + f_{2} = 60$
$f_{1} + f_{2} = 15$ Here $l = 20, c = 5, f_{1}, f = 20, h = 10 x_{i}$
So Median $= l + ⎛ ⎝ \frac{\frac{N}{2} - c}{f} \times h ⎞ ⎠ \Rightarrow 8.5 = 20 + (\frac{30 - 5 - f_{1}}{20}) \times 10$
$8.5 = \frac{25 - f_{1}}{2} \Rightarrow 25 - f_{1} = 17 \Rightarrow f_{1} = 8$
when $f_{1} = 8.8 + f_{2} = 15 \Rightarrow f_{2} = 7$ So $f_{1} = 8$ and $f_{2} = 7$

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