Question

Question 8
The mean of the following frequency distribution is 50 but the frequencies $f_1$ and $f_2$ in classes 20-40 and 60-80, respectively are not known. Find these frequencies, if the sum of all the frequencies is 120.
$\begin{array}{cccccc}Class& 0-20& 20-40& 40-60& 60-80& 80-100\\ Frequency& 17& f_1& 32& f_2& 19\end{array}$

Answer 1

First, we calculate the class mark of given data
$\begin{array}{ccccc}Class& Frequency\left(f_i\right)& Classmarks\left(x_i\right)& u_i=\frac{x_i-a}{h}& f_i{u}_i\\ 0-20& 17& 10& -2& -34\\ 20-40& f_1& 30& -1& -f_1\\ 40-60& 32& a=50& 0& 0\\ 60-80& f_2& 70& 1& f_2\\ 80-100& 19& 90& 2& 38\\ & \sum f_i=68+f_1+f_2& & & \sum f_i{u}_i=4+f_2-f_1\end{array}$
Given that, sum of all frequencies=120
$\Rightarrow \sum f_i=68+f_1+f_2=120\Rightarrow f_1+f_2=52\left(i\right)$
Here, (assumed mean) a=50
and (class width) h=20
By step deviation method,
$Mean=a+\frac{\sum f_i{u}_i}{\sum f_i}\times h\Rightarrow 50=50+\frac{(4+f_2-f_1)}{120}\times 20\Rightarrow 4+f_2+f_1=0\Rightarrow -f_2+f_1=4....\left(ii\right)OnaddingEqs.\left(i\right)and\left(ii\right),weget2f_1=56\Rightarrow f_1=28$
Put the value of $f_1$ in Eq.(i), we get
$f_2=52-28\Rightarrow f_2=24$
Hence, $f_1=28and{f}_2=24.$