Find out the missing frequencies of the following series if the AM is 35 and the number of items is 100:

Class interval	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	5	10	?	4	20	3	?

Given :

A.M. of the series $= 35$.

The total number of items $= 100$.

To do :

We have to find the missing frequencies.

Solution :

Let the missing frequencies be $f_1$ for the class interval 20-30 and $f_2$ for the class interval 60-70.

This implies,

$5+10+f_1+4+20+3+f_2=100$

$f_1+f_2=100-42$

$f_1+f_2=58$

$\frac{\sum f_{i}x_{i}}{\sum f_{i}} = 100$

$\frac{5 \times 5 + 15 \times 10 + 25 \times f_1 + 35 \times 4 + 45 \times 20 + 55 \times 3 + 65 \times f_2}{100} = 35$

$\frac{25 + 150 + 140 + 900 + 165 + 25f_1 + 65f_2}{100} = 35$

$1380 + 25f_1 + 65f_2 = 35\times 100$

$ 25f_1 + 65f_2 = 3500 - 1380$

$5 (5f_1 + 13f_2) = 2120$

$5f_1 + 13f_2 = \frac{2120}{5} = 424$

$5f_1 + 13f_2 =424$......................................(i)

$f_1 + f_2 = 58$

Multiply 5 on both sides,

$5f_1 + 5f_2 = 58 \times 5$

$5f_1 + 5f_2 = 290$......................................(ii)

Subtract (ii) from (i)

$5f_1 + 13f_2 =424$    $(-)$

$5f_1 + 5f_2 = 290$

________________

                $8f_2 = 134$

$f_2 = \frac{134}{8} \approxeq 17$

$f_1 + 17 = 58$

$f_1 = 58 - 17 = 41$

Therefore, the missing frequencies are 41 and 17 respectively.

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Updated on: 10-Oct-2022

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