Question

Find out quartile deviation and coefficient of quartile deviation from the following data:

Class Interval	0−10	10−20	20−30	30−40	40−50	50−60
Frequency	4	8	5	4	9	10

Answer 1

Class interval	Frequency (f)	Cumulative Frequency (c.f.)
0 − 10 10 − 20 20 − 30 30 − 40 40 − 50 50 − 60	4 8 5 4 9 10	4 4 + 8 = 12 12 + 5 = 17 17 + 4 = 21 21 + 9 = 30 30 + 10 = 40
	$\Sigma f$=N = 40

Q₁ = Size of $\left(\frac{N}{4}\right)\mathrm{th}$ item
= Size of $\left(\frac{40}{4}\right)\mathrm{th}$ item
= Size of 10th item

10th item lies in group 10 − 20 and fall within 12th c.f. of the series.

${\mathrm{Q}}_1=l_1+\frac{{\displaystyle \frac{\mathrm{N}}{4}}-c.f.}{f}\times i$

Here,
l₁=Lower limit of the class interval
N= Sum total of the frequencies
c.f.=Cumulative frequency of the class preceding the first quartile class
f= Frequency of the quartile class
i= Class interval

$$\begin{gathered} Thus, \\ {Q}_1=10+\frac{10-4}{8}\times 10 \\ or,Q_1=10+\frac{60}{8} \\ or,Q_1=10+7.5 \\ \mathbf{\Rightarrow }\mathbf{}{\mathit{Q}}_{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{17}\mathbf{.}\mathbf{5} \end{gathered}$$

Likewise,
Q₃ = Size of $3\left(\frac{\mathrm{N}}{4}\right)\mathrm{th}$ item
= Size of $3\left(\frac{40}{4}\right)\mathrm{th}$ item
= Size of 30th item

30th item lies in group 40-50 and fall within the 30th c.f. of the series

$$\begin{gathered} \mathrm{Thus}, \\ {Q}_3=l_1+\frac{3\left({\displaystyle \frac{N}{4}}\right)-c.f.}{f}\times i \\ or,Q_3=40+\frac{30-21}{9}\times 10 \\ or,Q_3=40+\frac{9}{9}\times 10 \\ or,Q_3=40+10 \\ \mathbf{\Rightarrow }\mathbf{}\mathbf{}{\mathit{Q}}_{\mathbf{3}}\mathbf{=}\mathbf{50} \\ \mathrm{Quartile}\mathrm{deviation}\left(QD\right)=\frac{{\mathit{Q}}_{\mathit{3}}\mathit{-}{\mathit{Q}}_{\mathit{1}}}{\mathit{2}}=\frac{50-17.5}{2}=\frac{32.5}{2}=16.25 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{Q}.\mathrm{D}=\frac{{\mathit{Q}}_{\mathit{3}}\mathit{-}{\mathit{Q}}_{\mathit{1}}}{{\mathit{Q}}_{\mathit{3}}\mathit{+}{\mathit{Q}}_{\mathit{1}}}=\frac{50-17.5}{50+17.5}=\frac{32.5}{67.5}=0.48 \end{gathered}$$