Question

calculate inter quartile range, quartile deviation and coefficient of quartile deviation from the following data.
size of shoes(in inches) No of persons
4 4
5 6
6 10
7 15
8 9
9 2
10 1

Answer 1

Hey Monica,

Size	No. of Persons	Cumulative frequency
4	4	4
5	6	10
6	10	20
7	15	35
8	9	44
9	2	46
10	1	47

$$\begin{gathered} Q_1=\mathrm{Size}\mathrm{of}{\left(\frac{N+1}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left(\frac{47+1}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left(12\right)}^{\mathrm{th}}\mathrm{item} \end{gathered}$$

12^th item corresponds to size 6.
$\therefore Q_1=6$


$$\begin{gathered} Q_3=\mathrm{Size}\mathrm{of}3{\left(\frac{N+1}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}3{\left(\frac{47+1}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left(36\right)}^{\mathrm{th}}\mathrm{item} \end{gathered}$$

36^th item corresponds to size 8.
$\therefore Q_3=8$

$$\begin{gathered} \mathrm{Interquartile}\mathrm{range}=Q_3-Q_1 \\ =8-6 \\ =2 \end{gathered}$$

$$\begin{gathered} \mathrm{Quartile}\mathrm{deviation}\left(QD\right)=\frac{Q_3-Q_1}{2}=\frac{2}{2}=1 \\ \mathrm{Coefficient}\mathrm{of}\left(QD\right)=\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{2}{14}=0.14 \end{gathered}$$