The Arithmetic Mean of the following frequency distribution is $47$ Determined the value of $p$
C.I. $0 - 20$ $20 - 40$ $40 - 60$ $60 - 80$ $80 - 100$
Frequency $8$ $15$ $20$ $p$ $3$

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Solution

Calculation of Arithmetic Mean
C.I. Frequency
$(f_{i})$
Class Mark
$(x_{i})$
$u_{i} = f_{i} x_{i}$
0 - 20 8 10 80
20 - 40 15 30 450
40 - 60 20 50 1000
60 - 80 p 70 70p
80 - 100 30 90 450
Total 48 + p 1980 + 70 p

$¯ x = \frac{\sum f_{i} x_{i}}{\sum f_{i}} = \frac{1980 + 70 p}{48 + p}$

$∴ \frac{1980 + 70 p}{48 + p} = 47$

$\Rightarrow 1980 + 70 p = 47 (48 + p)$
$\Rightarrow 1980 + 70 p = (47) (48) + 47 p$
$\Rightarrow 1980 + 70 p = 2256 + 47 p$
$\Rightarrow 70 p - 47 p = 2256 - 1980$
$\Rightarrow 23 p = 276$
$\Rightarrow p = \frac{276}{23} = 12$
Hence $p = 12$

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C.I.	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$
Frequency	$8$	$15$	$20$	$p$	$3$