Find the missing frequency ( $p$ ) for the following distribution whose mean is $7.68$
$x$ : $3$ $5$ $7$ $9$ $11$ $13$
$f$ : $6$ $8$ $15$ $p$ $8$ $4$

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Solution

Mean = $\frac{\sum x_{i} f_{i}}{\sum f_{i}}$

$\Rightarrow$ mean = $\frac{(3 \times 6) + (5 \times 8) + (7 \times 15) + (9 \times p) + (11 \times 8) + (13 \times 4)}{8 + 6 + 15 + p + 8 + 4}$

$\Rightarrow$ mean = $\frac{18 + 40 + 105 + 9 p + 88 + 52}{8 + 6 + 15 + p + 8 + 4} = 7.68$

$\Rightarrow 314.88 + 7.68 p = 303 + 9 p$

$\Rightarrow 11.88 = 1.32 p$

$\Rightarrow p = 9$

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