If the mean of the following frequency distribution is $8$ , then find the value of $^{'} k^{'}$ ?
$x$ $2$ $4$ $6$ $8$ $10$ $12$
$f r e q u e n c y = f$ $4$ $7$ $10$ $16$ $k$ $3$

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Solution

From the given frequency distribution table we can find,
$Σ f_{i} x_{i} = 8 + 28 + 60 + 128 + 10 k + 36 ∴ \sum f = 40 + k \sum f_{i} x_{i} = 260 + 10 k$
So, $∴ f = 40 + k$
and $f \cdot x = 260 + 10 k$
Also, mean = $8$
Using the formula,
$m e a n = \frac{\sum f_{i} x_{i}}{\sum f} \Rightarrow 8 = \frac{260 + 10 k}{40 + k} \Rightarrow 320 + 8 k = 260 + 10 k \Rightarrow 2 k = 60 \Rightarrow k = 30$

Hence, the value of $k$ is $30.$

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k

$x$	$2$	$4$	$6$	$8$	$10$
$f$	$5$	$3$	$7$	$6$	$k$

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Class	0-4	4-8	8-12	12-16	16-20	20-24	24-28	28-32	32-36
Frequency	6	8	17	23	16	15	-	4	3

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m

$X$	$12$	$4$	$6 + m$	$8$	$10$
$F$	$10$	$15$	$25$	$24$	$16$

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$x$	$2$	$4$	$6$	$8$	$10$	$12$
$f r e q u e n c y = f$	$4$	$7$	$10$	$16$	$k$	$3$