Question

The distribution of the size of certain farms in a taluka (tehsil) is given below.

Size of farm (in acres)	5-15	15-25	25-35	35-45	45-55	55-65	65-75
No. of farms	7	12	17	25	31	5	3

Find the median size of the farms.

Answer 1

For the calculation of median, we need to prepare the cumulative frequency distribution (less than type)
table as shown.

Size of farm (in acres)	Frequency $\left(f_i\right)$	Cumulative frequency (less than type)
5-15	7	7
15-25	12	19
25-35	17	$36\to c.f$
35-45	$25\to f$	61
45-55	31	92
55-65	5	97
65-75	3	100

$$\begin{gathered} \mathrm{Total}\mathrm{frequency}=\sum _{}^{}{f}_i=N=100 \\ \therefore \frac{N}{2}=\frac{100}{2}=50 \\ \mathrm{So},\text{the}\mathrm{cumulative}\mathrm{frequency}\mathrm{should}\mathrm{be}\mathrm{just}\mathrm{greater}\mathrm{than}\mathrm{or}\mathrm{equal}\mathrm{to}50,\mathrm{i}.\mathrm{e}.61. \\ \mathrm{Therefore},\mathrm{the}\mathrm{median}\mathrm{class}\mathrm{will}\mathrm{be}35-45. \\ \mathrm{Median}=L+\left(\frac{N}{2}-c.f.\right)\frac{h}{f} \\ \mathrm{Here},L=\mathrm{lower}\mathrm{limit}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=35 \\ c.f.=\mathrm{cumulative}\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{class}\mathrm{preceeding}\mathrm{the}\mathrm{median}\mathrm{clas}s=36 \\ h=\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=10 \\ f=\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=25 \\ \therefore \mathrm{Median}=35+\left(50-36\right)\frac{10}{25} \\ =35+14\times 0.4 \\ =35+5.6 \\ =40.6 \\ \therefore \mathrm{Median}\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{farms}\mathrm{is}40.60\mathrm{acres}. \end{gathered}$$