Question

Calculate the Quartile Deviation and Coefficient of Quartile Deviation of the following data:

Marks	:	5-9	10-14	15-19	20-24	25-29	30-34	35-39
Students	:	1	3	8	5	4	2	2

Answer 1

Marks	Students (f)	Cumulative Frequency
5−9	1	1
10−14	3	1 + 3 = 4
15−19	8	4 + 8 = 12
20−24	5	12 + 5 = 17
25−29	4	17 + 4 = 21
30−34	2	21 + 2 = 23
35−39	2	23 + 2 = 25
	N = Ʃf = 25

$$\begin{gathered} Q_1=\mathrm{Size}\mathrm{of}{\left(\frac{N}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left(\frac{25}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =6.{25}^{\mathrm{th}}\mathrm{item} \end{gathered}$$
This corresponds to the class interval (15−19).
$$\begin{gathered} Q_1=l_1+\frac{{\displaystyle \frac{N}{4}-c.f}}{f}\times i \\ =15+\frac{6.25-4}{8}\times 4 \\ =15+\frac{2.25}{8}\times 4 \\ =15+\frac{9}{8} \\ =15+1.125 \\ =16.125 \end{gathered}$$
$\therefore Q_1=16.125$

$$\begin{gathered} Q_2=\mathrm{Size}\mathrm{of}{\left(\frac{N}{2}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left(\frac{25}{2}\right)}^{\mathrm{th}}\mathrm{item} \\ =12.5^{\mathrm{th}}\mathrm{item} \end{gathered}$$
This corresponds to the class interval (20−24).

$$\begin{gathered} Q_2=l_1+\frac{{\displaystyle \frac{N}{2}-c.f}}{f}\times i \\ =20+\frac{12.5-12}{5}\times 4 \\ =20+\frac{2}{5} \\ =20+0.4 \\ =20.04 \end{gathered}$$
$\therefore Q_2=20.04$

$$\begin{gathered} Q_{\mathit{3}}=\mathrm{Size}\mathrm{of}3{\left(\frac{N}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}18.{75}^{\mathrm{th}}\mathrm{item} \end{gathered}$$
This corresponds to the class interval (25−29).

$$\begin{gathered} Q_3=l_1+\frac{{\displaystyle \frac{3N}{4}-c.f}}{f}\times i \\ =25+\frac{18.75-17}{4}\times 4 \\ =25+1.75 \\ =26.75 \end{gathered}$$
$\therefore Q_3=26.75$
$$\begin{gathered} \mathrm{Coefficient}\mathrm{of}\mathrm{Quartile}\mathrm{Deviation}=\frac{Q_3-Q_1}{Q_3+Q_1} \\ =\frac{26.75-16.125}{26.75+16.125} \\ =\frac{10.625}{42.875} \\ =0.25 \end{gathered}$$