The mean of the following distribution is 50.

Question:

The mean of the following distribution is 50.

Find the value of a and hence the frequencies of 30 and 70.

Solution:

We know that,

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

For the following data:

Mean $=\frac{(10 \times 17)+(30 \times(5 a+3))+(50 \times 32)+(70 \times(7 a-11))+(90 \times 19)}{17+5 a+3+32+7 a-11+19}$

$\Rightarrow 50=\frac{170+150 a+90+1600+490 a-770+1710}{60+12 a}$

$\Rightarrow 50(60+12 a)=2800+640 a$

$\Rightarrow 3000+600 a=2800+640 a$

$\Rightarrow 640 a-600 a=3000-2800$

$\Rightarrow 40 a=200$

$\Rightarrow a=\frac{200}{40}$

$\Rightarrow a=5$

Hence, the value of a is 5.

Also, the frequency of 30 is 28 and the frequency of 70 is 24.