Question

Calculate the Interquartile Range for the data given below:

Class	:	35-40	30-35	25-30	20-25	15-20	10-15	5-10	0-5
Frequency	:	1	4	9	11	10	6	5	4

Answer 1

Class	Frequency (f)	c.f.
0−5	4	4
5−10	5	9
10−15	6	15
15−20	10	25
20−25	11	36
25−30	9	45
30−35	4	49
35−40	1	50
	N = 50

The first quartile class is given by the size of the ${\left(\frac{N}{4}\right)}^{\mathrm{th}}$item, i.e. the ${\left(\frac{50}{4}\right)}^{\mathrm{th}}$item, which is the 12.5^th item.
This corresponds to the class interval 10−15, so this is the first quartile class.
$$\begin{gathered} Q_1=l_1+\frac{{\displaystyle \frac{N}{2}-c.f}}{f}\times i \\ =10+\frac{12.5-9}{6}\times 5 \\ =10+\frac{3.5\times 5}{6} \\ =10+2.92 \\ =12.92 \end{gathered}$$
The third quartile class is given by the size of the 3${\left(\frac{N}{4}\right)}^{\mathrm{th}}$item, i.e. the 3${\left(\frac{50}{4}\right)}^{\mathrm{th}}$item, which is the 37.5^th item.
This corresponds to the class interval 25−30, so this is the third quartile class.

$$\begin{gathered} Q_3=l_1+\frac{3\left({\displaystyle \frac{N}{4}}\right)-c.f.}{f}\times i \\ =25+\frac{37.5-36}{9}\times 5 \\ =25+\frac{1.5\times 5}{9} \\ =25.83 \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{Interquartile}\mathrm{range}=Q_3-Q_1 \\ =25.83-12.92 \\ =12.91 \end{gathered}$$