The following distribution shows the daily pocket allowance of children of a locality.The mean pocket allowance is Rs 18.Find the missing frequency?

Question

Answer

Calculation of Missing Frequency in Pocket Allowance Distribution

Given Data:

Mean pocket allowance = Rs 18

Other frequencies and their corresponding pocket allowances are given in the table below

One of the frequencies is missing

Pocket Allowance	10	12	14	16	20	22
Frequency	5	10	15	20	?	5

Formula for Mean:

$$ Mean = \frac{\sum f_i imes x_i}{\sum f_i} $$

where,

$f_i$ = frequency of the i-th class

$x_i$ = midpoint of the i-th class

Calculation of Missing Frequency:

Let the missing frequency be 'm'.

Using the formula for mean, we can write:

$$ 18 = \frac{(5 imes 10) + (10 imes 12) + (15 imes 14) + (20 imes 16) + (m imes 20) + (5 imes 22)}{f} $$

where, 'f' is the total frequency (including the missing frequency)

Simplifying the above equation, we get:

$$ 18f = 1220 + 20m $$

Substituting the given values, we get:

$$ 18(f-1) = 1220 + 20 imes 22 $$

$$ 18f - 18 = 1660 $$

$$ 18f = 1678 $$

$$ f = 93.22 $$

Since 'f' represents the total frequency, it should be a whole number. Therefore, the missing frequency can be calculated as:

$$ m = \frac{18f - 1220}{20} $$

$$ m = \frac{(18 imes 93) - 1220}{20} $$

$$ m = 11.25 $$

Since the missing frequency must be a whole number, the closest whole number to 11.25 is 11.

Therefore, the missing frequency is 11.

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