Question

Given the following data, find out quartile deviation and the coefficient of quartile deviation:

Wages (₹)	0−5	5−10	10−15	15−20	20−25	25−30
Number of Workers	4	6	3	8	12	7

Answer 1

Wage	No. of worker (f)	Cumulative Frequency (c.f.)
0 − 5 5 − 10 10 − 15 15 − 20 20 −25 25 − 30	4 6 3 8 12 7	4 4 + 6 = 10 10 + 3 = 13 13 + 8 = 21 21 + 12 = 33 33 + 7 = 40
	$\mathrm{\Sigma }f=N=40$

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

10^th item lies in group 5 − 10 and falls within 10^th c.f. of the series.

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

Here,
l_₁=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

$$\begin{gathered} \mathrm{Thus}, \\ {Q}_1=5+\frac{10-4}{6}\times 5 \\ {Q}_1=5+\frac{6}{6}\times 5 \\ \mathbf{\Rightarrow }{\mathit{Q}}_{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{10} \end{gathered}$$

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

30^th item lies in the group 20-25 within the 33^rd c.f. of the series.

$$\begin{gathered} \mathrm{Thus}, \\ {Q}_3=l_1+\frac{3\left({\displaystyle \frac{N}{4}}\right)-c.f}{f}\times i \\ \mathrm{or},Q_3=20+\frac{30-21}{12}\times 5 \\ \mathrm{or},Q_3=20+\frac{9\times 5}{12} \\ \mathrm{or},Q_3=20+3.75 \\ \mathbf{\Rightarrow }\mathbf{}\mathbf{}{\mathit{Q}}_{\mathbf{3}}\mathbf{}\mathbf{=}\mathbf{23}\mathbf{.}\mathbf{75} \\ \mathrm{Quartile}\mathrm{deviation}(Q.D.)=\frac{Q_3\mathit{-}{Q}_1}{2}=\frac{23.75-10}{2}=\frac{13.75}{2}=6.87 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{quartile}\mathrm{deviation}=\frac{Q_3\mathit{-}{Q}_1}{Q_3\mathit{+}{Q}_1}=\frac{23.75-10}{23.75+10}=\frac{13.75}{33.75}=0.4 \end{gathered}$$