The following data has a mean of 50. Determine the value of a and thus the frequencies 30 and 70.

$\begin{array}{cccccc}X& 10& 30& 50& 70& 90\\ F& 17& 5a+3& 32& 7a-11& 19\\ & & & & & \end{array}$

The sum of the values of all the observations $=f_1{x}_1+f_2{x}_2+\dots +f_n{x}_n$, and the number of observations $=f_1+f_2+\dots +f_n$.

So, the mean $\overline{x}$ of the data is given by

$\overline{x}=\frac{f_1{x}_1+f_2{x}_2+\cdots +f_n{x}_n}{f_1+f_2+\cdots +f_n}$

Mean $=\frac{\sum fixi}{\sum fi}=50$

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

$50(12\alpha +60)=640\alpha +2800$

$3000+600\alpha =640\alpha +2800$

$40\alpha =200$

$\alpha =5$

frequency of $30=5\alpha +3=25+3=28$

frequency of$70=7\alpha -11=35-11=24.$