Question

The following table shows the ages of 300 patients getting medical treatment in a hospital on a particular day.

Age (in years)	10-20	20-30	30-40	40-50	50-60	60-70
No. of patients	60	42	55	70	53	20

Find the median age of the patients.

Answer 1

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

Age (in years)	Frequency $\left(f_i\right)$	Cumulative frequency (less than type)
10-20	60	60
20-30	42	$102\to c.f.$
30-40	$55\to f$	157
40-50	70	227
50-60	53	280
60-70	20	300

$$\begin{gathered} \mathrm{Total}\mathrm{frequency}=\sum _{}^{}{f}_i=N=300 \\ \therefore \frac{N}{2}=\frac{300}{2}=150 \\ \mathrm{So},\mathrm{cumulative}\mathrm{frequency}\mathrm{should}\mathrm{be}\mathrm{just}\mathrm{greater}\mathrm{than}\mathrm{or}\mathrm{equal}\mathrm{to}150,\mathrm{i}.\mathrm{e}.157. \\ \mathrm{Therefore},\mathrm{the}\mathrm{median}\mathrm{class}\mathrm{will}\mathrm{be}30-40. \\ \mathrm{Median}=L+\left(\frac{N}{2}-c.f.\right)\frac{h}{f} \\ \mathrm{Here},L=\mathrm{lower}\mathrm{limit}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=30 \\ c.f.=\mathrm{cumulative}\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{class}\mathrm{preceding}\mathrm{the}\mathrm{median}\mathrm{class}=102 \\ h=\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=10 \\ f=\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=55 \\ \therefore \mathrm{Median}=30+(150-102)\frac{10}{55} \\ =30+48\times \frac{10}{55} \\ =30+\frac{480}{55} \\ =30+8.73=38.73 \\ \therefore \mathrm{The}\mathrm{median}\mathrm{age}\mathrm{of}\text{the}\mathrm{patients}\mathrm{is}38.73\mathrm{years}. \end{gathered}$$