Question

For the data of problem 14, find median number of packages received per day by a post office.

Answer 1

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

No. of packages	Frequency $\left(f_i\right)$	Cumulative frequency (less than type)
10-20	2	2
20-30	8	10
30-40	16	26
40-50	24	50
50-60	30	80
60-70	20	100

$$\begin{gathered} \mathrm{Total}\mathrm{frequency}=\sum _{}^{}{f}_i=N=100 \\ \therefore \frac{N}{2}=\frac{100}{2}=50 \\ \mathrm{So},\mathrm{cumulative}\mathrm{frequency}\mathrm{should}\mathrm{be}\mathrm{just}\mathrm{greater}\mathrm{than}\mathrm{or}\mathrm{equal}\mathrm{to}50,\mathrm{i}.\mathrm{e}.50. \\ \mathrm{Therefore},\mathrm{the}\mathrm{median}\mathrm{class}\mathrm{will}\mathrm{be}40-50. \\ \mathrm{Median}=L+\left(\frac{N}{2}-c.f.\right)\frac{h}{f} \\ \mathrm{Here},L=\mathrm{lower}\mathrm{limit}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=40 \\ c.f.=\mathrm{cumulative}\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{class}\mathrm{preceeding}\mathrm{the}\mathrm{median}\mathrm{clas}s=26 \\ h=\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=10 \\ f=\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=24 \\ \therefore \mathrm{Median}=40+\left(50-26\right)\frac{10}{24} \\ =40+24\times 0.41 \\ =40+9.84 \\ =49.84 \\ \approx 50 \\ \therefore \mathrm{Median}\mathrm{number}\mathrm{of}\mathrm{packages}\mathrm{received}\mathrm{per}\mathrm{day}\mathrm{by}\text{the}\mathrm{post}\mathrm{office}\mathrm{is}50. \end{gathered}$$