Question

Find out quartile deviation and coefficient of quartile deviation of the following series:
28, 18, 20, 24, 30, 15, 47, 27

Answer 1

Arranging the data in ascending order.

Sr. No.	1	2	3	4	5	6	7	8
Observation	15	18	20	24	27	28	30	47

In order to find the quartile deviation in case of individual series, we need to find out the values of third quartile and first quartile using the following equations:

Q₁ = Size of ${\left(\frac{N+1}{4}\right)}^{\mathrm{th}}$ item

or, Q₁= Size of ${\left(\frac{8+1}{4}\right)}^{\mathrm{th}}$ item

or, Q₁= Size of 2.25^th item

or, Q₁= Size of 2^nd item + $\frac{1}{4}\left[\mathrm{Size}\mathrm{of}{3}^{\mathrm{rd}}\mathrm{item}-\mathrm{Size}\mathrm{of}{2}^{\mathrm{nd}}\mathrm{item}\right]$
$\mathrm{or},Q_1=18+\frac{1}{4}\left[20-18\right]$

$\Rightarrow$ Q₁ = 18.5

Q₃ = Size of $3{\left(\frac{N+1}{4}\right)}^{\mathrm{th}}$

or, Q₃ = Size of 6.75^th item
or, Q₃ = Size of 6^th item + $\frac{3}{4}\left[\mathrm{Size}\mathrm{of}{7}^{\mathrm{th}}\mathrm{item}-\mathrm{Size}\mathrm{of}{6}^{\mathrm{th}}\mathrm{item}\right]$
$$\begin{gathered} \mathrm{or},\mathit{}{Q}_{\mathit{3}}=28+\frac{3}{4}\left[30-28\right] \\ \mathrm{or},Q_{\mathit{3}}=27+\frac{3}{4}\left[2\right] \\ \mathrm{or},Q_3=28+1.5 \\ \mathbf{\Rightarrow }\mathbf{}{\mathit{Q}}_{\mathbf{3}}\mathbf{=}\mathbf{29}\mathbf{.}\mathbf{5} \\ \mathrm{Quartile}\mathrm{deviation}(Q.D.)=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{\mathit{2}} \\ \mathrm{or},Q.D.=\frac{29.5-18.5}{2} \\ \mathrm{or},Q.D.=\frac{11}{2} \\ \Rightarrow Q.D.=5.5 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{Quartile}\mathrm{deviation}=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{Q_{\mathit{3}}\mathit{+}{Q}_{\mathit{1}}} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{29.5-18.5}{29.5+18.5} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{11}{48} \\ \Rightarrow \mathrm{Coefficient}\mathrm{of}Q.D.=0.229=0.23 \end{gathered}$$

Hence the Q.D. of the series is 5.5 and Coefficient of the Q.D. is 0.23