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Direct Method of Finding Mean

If the mean o...

Question

If the mean of the following distribution be $50$ , then find the values of $f_{1}$ and $f_{2}$ .

Class 0-20 20-40 40-60 60-80 80-100 Total

Frequency 17 28 $f_{1}$ 24 $f_{2}$ 120

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Solution

For the given distribution firstly we have
$17 + 28 + f_{1} + 24 + f_{2} = 120$
or, $f_{1} + f_{2} = 51$ .........(1).
Now, middle value of the classes are $10, 30, 50, 70, 90$ respectively.
Now, mean $= \frac{17 \times 10 + 28 \times 30 + f_{1} \times 50 + 24 \times 70 + f_{2} \times 90}{120} = \frac{17 + 84 + 5 f_{1} + 168 + 9 f_{2}}{12}$ .
According to the problem we have
$\frac{17 + 84 + 5 f_{1} + 168 + 9 f_{2}}{12}$ $= 50$
or, $5 f_{1} + 9 f_{2} + 269 = 600$
or, $5 f_{1} + 9 f_{2} = 331$ .......(2).
Now, (2)-5x(1) gives
$4 f_{2} = 76$
or, $f_{2} = 19$ .
Using value of $f_{2}$ in (1) we get, $f_{1} = 32$ .

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