The mean of the following distribution is $31.4$ . Determine the missing frequency $x$ .
Class $0 - 10$ $10 - 20$ $20 - 30$ $30 - 40$ $40 - 50$ $50 - 60$
Frequency $5$ $x$ $10$ $12$ $7$ $8$

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Solution

$C l a s s$ $C l a s s - m a r k$ $F r e q u e n c y$
$f_{i}$ $f_{i} x_{i}$
$0 - 10$ $5$ $5$ $25$
$10 - 20$ $15$ $x$ $15 x$
$20 - 30$ $25$ $10$ $250$
$30 - 40$ $35$ $12$ $420$
$40 - 50$ $45$ $7$ $315$
$50 - 60$ $55$ $8$ $440$
$\sum f_{i} = 42 + x$ $\sum f_{i} x_{i} = 1450 + 15 x$
$M e a n = \frac{\sum f_{i} x_{i}}{\sum f_{i}}$

$\Rightarrow$ $31.4 = \frac{1450 + 15 x}{42 + x}$

$\Rightarrow$ $1318.8 + 31.4 x = 1450 + 15 x$
$\Rightarrow$ $16.4 x = 131.2$
$∴$ x=8$

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$C . I .$	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
frequency	$8$	$X$	$10$	$11$	$9$

Class Interval	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$
Frequency	$8$	$5$	$12$	$35$	$24$	$16$

Class	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80
frequency	5	8	7	12	28	20	10	10

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Class	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
Frequency	$5$	$25$	$?$	$18$	$7$

f_{1}

f_{2}

N = 100

32

Class	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$T o t a l$
Frequency	$10$	$f_{1}$	$25$	$30$	$f_{2}$	$10$	$100$