The arithmetic mean of the following frequency distribution is $18.75$ . Determine the value of $p$ .
$x_{i}$ $10$ $15$ $p$ $25$ $30$
$f_{i}$ $5$ $10$ $7$ $8$ $2$

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Solution

Step 1 Finding mean:
Completing the table,
$x_{i}$ $f_{i}$ $f_{i} x_{i}$
$10$ $5$ $50$
$15$ $10$ $150$
$p$ $7$ $7 p$
$25$ $8$ $200$
$30$ $2$ $60$
$Σ f_{i} = 32$ $Σ f_{i} x_{i} = 460 + 7 p$
The formula for mean is $\bar{x} = \frac{Σ f_{i} x_{i}}{Σ f_{i}}$ .
where $Σ f_{i}$ is the total frequency and $x_{i}$ is the observations.
From the question, we have $\bar{x} = 18.75$ .
Step 2 Finding the value of $p$ :
Substituting the values in the formula for mean,
$\Rightarrow 18.75 = \frac{460 + 7 p}{32} \Rightarrow 600 = 460 + 7 p \Rightarrow 7 p = 140 ∴ p = 20$
Hence, the value of $p$ is obtained as $20$ .

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18.75

p

Class mark $(x_{i})$	$10$	$15$	$p$	$25$	$30$
Frequency $(f_{i})$	$5$	$10$	$7$	$8$	$2$

$x_{i}$	$10$	$12$	$15$	$25$
$f_{i}$	$2$	$3$	$7$	$8$

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