Question

The mean of the following distribution is 18. Find the frequency f of the class 19-21.

$\begin{array}{cccccccc}\text{Class}& 11-13& 13-15& 15-17& 17-19& 19-21& 21-23& 23-25\\ \text{Frequency}& 3& 6& 9& 13& f& 5& 4\end{array}$

OR

The following distribution gives the daily income of 50 workers of a factory:

$\begin{array}{cccccc}\text{Daily Income (in Rs)}& 100-120& 120-140& 140-160& 160-180& 180-200\\ \text{Number of workers}& 12& 14& 8& 6& 10\end{array}$

Convert the distribution above to less than type cumulative frequency distribution and draw its ogive.

Answer 1

$\begin{array}{cccc}\text{C.I.}& \underset{x_i}{\text{Mid value}}& f_i& f_i{x}_i\\ 11-13& 12& 3& 36\\ 13-15& 14& 6& 84\\ 15-17& 16& 9& 144\\ 17-19& 18& 13& 234\\ 19-21& 20& f& 20f\\ 21-23& 22& 5& 110\\ 23-25& 24& 4& 96\\ \text{Total}& & \sum f_i=40+f& \underset{704+20f}{\sum f_i{x}_i}\end{array}$

Now, Mean =18

$\Rightarrow \frac{\sum f_i{x}_i}{\sum f_i}=18$

$\therefore \frac{704+20f}{40+f}=18$

$\Rightarrow 704+20f=18(40+f)$

$\Rightarrow 704+20f=720+18f$

$\Rightarrow 20f-18f=720-704$

$\Rightarrow 2f=16$

$\Rightarrow f=8$

Less than type cumulative frequency distribution :

$\begin{array}{cc}\text{Daily Income}& \text{No.of workers}\\ \text{Less than 120}& 12\\ \text{Less than 140}& 26\\ \text{Less than 160}& 34\\ \text{Less than 180}& 40\\ \text{Less than 200}& 50\end{array}$