Question

If the arithmetic of the data given below is 28, find (a) the missing frequency and (b) the median of the series

$\begin{array}{ccccccc}\text{Profit per Retail shop (in Rs)}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ \text{Number of Retail shops}& 12& 18& 27& --& 17& 6\end{array}$

Answer 1

(a) Let the missing frequency be $f_1$

Arithmetic Mean =28

$\begin{array}{cccc}\text{Profit per Retail shop (in Rs)}& \text{Number of Retail shops}& \text{Mid Value}& \text{fm}\\ \text{(in Rs)Class Interval}& \text{Shops(f)}& \text{m}& \\ 0-10& 12& 5& 60\\ 10-20& 18& 15& 270\\ 20-30& 27& 25& 675\\ 30-40& f_1& 35& 35f_1\\ 40-50& 17& 45& 765\\ 50-60& 6& 55& 330\\ & \sum f=80+f_1& & \sum fm=2100+35f_1\end{array}$

$\overline{X}=\frac{\sum fm}{\sum f}$

or $28=\frac{2100+35f_1}{80+f_1}$

or $2240+28f_1=2100+35f_1$

or $2240-2100=35f_1-28f_1$

or $140=7f$

$f_1=20$

Hence, the missing frequency is 20

(b)

$\begin{array}{ccc}\text{Class Interval}& \text{Frequency}& \text{Cumulative Frequency}\\ 0-10& 12& 12\\ 10-20& 18& 30\\ 20-30& 27& 57\\ 30-40& 20& 77\\ 40-50& 17& 94\\ 50-60& 6& 100\\ \text{Total}& \sum f=100& \end{array}$

$\sum f=N=100$

So, the Median class = Size of $\frac{N}{2}$th item = 50th item

50 th item lies in the 57th cumulative frequency and the corresponding class interval is 20-30

Median = $L+\frac{\frac{N}{2}-CF}{f}\times i=20+\frac{\frac{100}{2}-30}{27}\times 10=20+\frac{50-30}{27}\times 10$

$=20+\frac{20}{27}\times 10=27.40$