The arithmetic mean of the following frequency distribution is $25$ . Determine the value of $p$ .
Class-interval $0 - 10$ $10 - 20$ $20 - 30$ $30 - 40$ $40 - 50$
Frequency $5$ $18$ $15$ $p$ $6$

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Solution

Step 1: Finding mean:
Completing the table,
Class-interval Mid-value $x_{i}$ Frequency $f_{i}$ $f_{i} x_{i}$
$0 - 10$ $5$ $5$ $25$
$10 - 20$ $15$ $18$ $270$
$20 - 30$ $25$ $15$ $375$
$30 - 40$ $35$ $p$ $35 p$
$40 - 50$ $45$ $6$ $270$
$Σ f_{i} = 44 + p$ $Σ f_{i} x_{i} = 940 + 35 p$
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 mid-value of class intervals.
From the question, we have $\bar{x} = 25$ .
Step 2: Finding the value of $p$ :
Substituting the values in the formula for mean,
$\Rightarrow 25 = \frac{940 + 35 p}{44 + p} \Rightarrow 1100 + 25 p = 940 + 35 p \Rightarrow 10 p = 160 ∴ p = 16$
Hence, the value of $p$ is obtained as $16$ .

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