Question

Calculate the Semi-interquartile Range and its Coefficient of the following data:

Marks	:	0-10	10-20	20-30	30-40	40-50	50-60	60-70
No. of Students	:	4	8	10	14	10	9	5

Answer 1

Marks	No. of Students (f)	c.f
0−10	4	4
10−20	8	12
20−30	10	22
30−40	14	36
40−50	10	46
50−60	9	55
60−70	5	60
	N = 60

The first quartile class is given by the size of the ${\left(\frac{N}{4}\right)}^{\mathrm{th}}$item, i.e. the ${\left(\frac{60}{4}\right)}^{\mathrm{th}}$item, which is the 15^th item.
This corresponds to the class interval 20−30, so this is the first quartile class.

$$\begin{gathered} Q_1=l_1+\frac{{\displaystyle \frac{N}{4}-c.f}}{f}\times i \\ =20+\frac{15-12}{10}\times 10 \\ =23 \end{gathered}$$
The third quartile class is given by the size of the 3${\left(\frac{N}{4}\right)}^{\mathrm{th}}$item, i.e. the 3${\left(\frac{60}{4}\right)}^{\mathrm{th}}$item, which is the 45^th item.
This corresponds to the class interval 40−50, so this is the third quartile class.

$$\begin{gathered} Q_3=l_1+\frac{{\displaystyle \frac{3N}{4}-c.f}}{f}\times i \\ =40+\frac{45-36}{10}\times 10 \\ =40+9 \\ =49 \end{gathered}$$

$$\begin{gathered} \mathrm{Quartile}\mathrm{Deviation}=\frac{Q_3-Q_1}{2}=\frac{49-23}{2}=13 \\ \mathrm{Coefficient}\mathrm{of}\mathit{}Q\mathit{.}D.=\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{49-23}{49+23}=\frac{26}{72}=0.361 \end{gathered}$$