If the mean of the following frequency distribution is $15$ , find the value of $p$ .
Variable $(x_{i})$ $10$ $12$ $14$ $16$ $18$
Frequency $(f_{i})$ $13$ $p$ $15$ $32$ $28$

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Solution

$x_{i}$ $f_{i}$ $f_{i} x_{i}$
$10$ $13$ $130$
$12$ $p$ $12 p$
$14$ $15$ $210$
$16$ $32$ $512$
$18$ $28$ $504$
$\sum f_{i} = 88 + p$ $\sum f_{i} x_{i} = 1356 + 12 p$
$\Rightarrow$ $¯ ¯ ¯ x = \frac{\sum f_{i} x_{i}}{\sum f_{i}}$

$\Rightarrow$ $15 = \frac{1356 + 12 p}{88 + p}$

$\Rightarrow$ $1320 + 15 p = 1356 + 12 p$
$\Rightarrow$ $3 p = 36$
$∴$ $p = 12$

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16.6

$x_{i}$	$8$	$12$	$15$	$18$	$20$	$25$	$30$	Total
$f_{i}$	$12$	$16$	$p$	$24$	$16$	$q$	$4$	$100$

Variable (x_i)	10	15	20	25	30
Frequency (f_i)	6	8	p	10	6

16

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

Locate median of the following distribution :

$\begin{matrix} Variable (X) & 10 & 11 & 12 & 13 & 14 & 15 & 16 Frequency (f) & 8 & 15 & 25 & 20 & 12 & 10 & 5 \end{matrix}$

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Variable $(x_{i})$	$10$	$12$	$14$	$16$	$18$
Frequency $(f_{i})$	$13$	$p$	$15$	$32$	$28$