Question

The frequency distribution of IQ (Intelligent Quotient) of 80 candidates is given below.

IQ	70-80	80-90	90-100	100-110	110-120	120-130	130-140
No. of candidates	7	16	20	17	11	7	2

Find the median IQ of the candidates.

Answer 1

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

IQ	No. of candidates	Cumulative frequency (less than type)
70-80	7	7
80-90	16	23
90-100	20	43
100-110	17	60
110-120	11	71
120-130	7	78
130-140	2	80

$$\begin{gathered} \mathrm{Total}\mathrm{frequency}=\sum _{}^{}{f}_i=N=80 \\ \therefore \frac{N}{2}=\frac{80}{2}=40 \\ \mathrm{So},\mathrm{cumulative}\mathrm{frequency}\mathrm{should}\mathrm{be}\mathrm{just}\mathrm{greater}\mathrm{than}\mathrm{or}\mathrm{equal}\mathrm{to}40,\mathrm{i}.\mathrm{e}.43. \\ \mathrm{Therefore},\mathrm{the}\mathrm{median}\mathrm{class}\mathrm{will}\mathrm{be}90-100. \\ \mathrm{Median}=L+\left(\frac{N}{2}-c.f.\right)\frac{h}{f} \\ \mathrm{Here},L=\mathrm{lower}\mathrm{limit}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=90 \\ c.f.=\mathrm{cumulative}\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{class}\mathrm{prece}\mathrm{ding}\mathrm{the}\mathrm{median}\mathrm{clas}s=23 \\ h=\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=10 \\ f=\mathrm{frequency}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{class}=20 \\ \therefore \mathrm{Median}=90+\left(40-23\right)\frac{10}{20} \\ =90+8.5 \\ =98.5 \\ \therefore \mathrm{Median}\mathrm{IQ}\mathrm{of}\text{the}\mathrm{candidates}\mathrm{is}98.5. \end{gathered}$$