Find the missing frequencies in the following frequency distribution whose mean is $34$ .
$x$ $10$ $20$ $30$ $40$ $50$ $60$ $T o t a l$
$f$ $4$ $f_{2}$ $8$ $f_{1}$ $3$ $4$ $35$

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Solution

$x_{i}$ $f_{i}$ $x_{i} f_{i}$
$10$ $4$ $40$
$20$ $f_{2}$ $20 f_{2}$
$30$
$8$
$240$
$40$ $f_{1}$ $40 f_{1}$
$50$ $3$ $150$
$60$ $4$ $240$
$\sum f_{i} = 35$ $\sum f_{i} x_{i} = 670 + 20 f_{2} + 40 f_{1}$
$m e a n = \frac{\sum f_{i} x_{i}}{\sum f_{i}} \Rightarrow 34 = \frac{670 + 20 f_{2} + 40 f_{1}}{35}$

Or, $1190 = 670 + 20 f_{2} + 40 f_{1}$

Or, $40 f_{1} + 20 f_{2} = 520$

Or, $2 f_{1} + f_{2} = 26$ ....... $(1)$

Also, $Σ f_{i} = 19 + f_{1} + f_{2} = 35 \Rightarrow f_{1} + f_{2} = 16$ ........ $(2)$
Subtracting (2) from (1) we get,
$f_{1} = 10$
From eq.(2) we get,
$f_{2} = 16 - 10 = 6$

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