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Find the missing frequencies and the median for the following distribution if the mean is 1.46 and the sum of all frequencies is 200.

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f_{1} = 38

f_{2} = 76

and median = 10

f_{1} = 16

f_{2} = 28

and median = 2

f_{1} = 76

f_{2} = 38

and median = 1

None of these

answer is A.

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Detailed Solution

Given that the mean is 1.46 and the sum of all frequencies is 200 of the given data,
Question Image

Then,

We have,

\underset{̲}{x} = 1.46,

\sum f_{i} = 200

Then,

\underset{̲}{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}}

\Rightarrow 1.46 = \frac{140 + f_{1} + 2 f_{2}}{200}

\Rightarrow 292 = 140 + f_{1} + 2 f_{2}

\Rightarrow 292 - 140 = f_{1} + 2 f_{2}

\Rightarrow f_{1} + 2 f_{2} = 152

……..(i)
Also,

\Rightarrow 86 + f_{1} + f_{2} = 200

\Rightarrow f_{1} + f_{2} = 200 - 86

\Rightarrow f_{1} + f_{2} = 114

…….(ii)
Subtracting equation (ii) from (i),

\Rightarrow f_{1} + 2 f_{2} - (f_{1} + f_{2}) = 152 - 114

\Rightarrow f_{2} = 38

Substituting

f_{2} = 38

in equation (ii),

{\Rightarrow f}_{1} + 38 = 114

{\Rightarrow f}_{1} = 114 - 38

{\Rightarrow f}_{1} = 76

So,

f_{1} = 76 and f_{2} = 38

.
For Median,
Question Image

N = 200

Median

= \frac{{(\frac{N}{2})}^{th} obs + {(\frac{N}{2} + 1)}^{th} obs}{2}

\Rightarrow Median = \frac{{(\frac{200}{2})}^{th} obs + {(\frac{200}{2} + 1)}^{th} obs}{2}

\Rightarrow Median = \frac{100^{th} obs + 101^{th} obs}{2}

\Rightarrow Median = \frac{1 + 1}{2}

\Rightarrow Median = 1

Thus,

f_{1} = 76

f_{2} = 38

and median = 1.
Hence, the correct option is 1.

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