Question

Estimate quartile deviation and the coefficient of quartile deviation of the following data:
8, 9, 11, 12, 13, 17, 20, 21, 23, 25, 27
Show that QD is the average of the difference between two quartiles.

Answer 1

Sr. No.	1	2	3	4	5	6	7	8	9	10	11
Data	8	9	11	12	13	17	20	21	23	25	27

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

Q₁ = Size of ${\left(\frac{N+1}{4}\right)}^{\mathrm{th}}$ item
or, Q₁ = Size of ${\left(\frac{11+1}{4}\right)}^{\mathrm{th}}$ item
or, Q₁ = Size of 3^rd item
$\Rightarrow$ Q₁ = 11

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

or, Q₃ = Size of $3{\left(\frac{11+1}{4}\right)}^{\mathrm{th}}$ item
or, Q₃ = Size of 9^th item
$\Rightarrow$ Q₃ = 23

Calculating Quartile Deviation and Coefficient of Quartile Deviation

$$\begin{gathered} \mathrm{Quartile}\mathrm{deviation}(Q.D.)=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{2} \\ \mathrm{or},Q.D.=\frac{23-11}{2} \\ \mathrm{or},Q.D.=\frac{12}{2} \\ \Rightarrow Q.D.=6 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{quartile}\mathrm{deviation}=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{Q_{\mathit{3}}\mathit{+}{Q}_{\mathit{1}}} \\ or,\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{23-11}{23+11}=\frac{12}{34}=0.353 \end{gathered}$$

We know that the difference between the third quartile (Q₃₎ and first Quartile (Q₁) of a series is called the inter Quartile range i.e.
Inter Quartile Range= Q_3$-$Q_{1
Where as Quartile deviation is half of the Inter Quartile Range. i.e.
Quartile Deviation (Q.D.)=} $\frac{Q_{3-}{Q}_1}{2}$
Hence, from the above formula it is clear that quartile deviation is the average of the difference between the two quartiles.