Question

Calculate Quartile Deviation and its Coefficient from the following data:

Size	5−10	10−15	15−20	20−25	25−30	30−35	35−40	40−45	45−50
Frequency	6	10	18	30	15	12	10	6	4

Answer 1

Size	Frequency	Cumulative Frequency
5−10	6	6
10−15	10	16
15−20	18	34
20−25	30	64
25−30	15	79
30−35	12	91
35−40	10	101
40−45	6	107
45−50	4	111
	Σf=N = 111

Q₁ = Size of $\left(\frac{N}{4}\right)\mathrm{th}$ item
= Size of $\left(\frac{111}{4}\right)\mathrm{th}$ item
= Size of 27.75^th item


27.75 item lies in group 15−20 and falls within 34^th c.f. of the series

$$\begin{gathered} Q_1=l_1+\frac{{\displaystyle \frac{N}{4}}-c.f.}{f}\times i \\ \mathrm{Here,} \\ l₁=Lower\mathrm{limit}\mathrm{of}\mathrm{the}\mathrm{class}interval \\ N=\; Sum\mathrm{total}\mathrm{of}\mathrm{the}frequencies \\ c.f.=Cumulative\mathrm{frequency}ofthe\mathrm{class}\mathrm{preceding}\mathrm{the}\mathrm{first}\mathrm{quartile}class \\ f=\; Frequency\mathrm{of}\mathrm{the}\mathrm{quartile}class \\ i=\; Classinterval \\ Thus, \\ {Q}_1=15+\frac{27.75-16}{18}\times 5 \\ or,Q_1=15+\frac{11.75\times 5}{18} \\ or,Q_1=15+\frac{58.75}{18} \\ or,Q_1=15+3.26 \\ \mathbf{\Rightarrow }{\mathit{Q}}_{\mathbf{1}}\mathbf{=}\mathbf{18}\mathbf{.}\mathbf{26} \end{gathered}$$


Like wise,
Q₃ = Size of $3\left(\frac{\mathrm{N}}{4}\right)\mathrm{th}$ item
= Size of 3$\left(\frac{111}{4}\right)\mathrm{th}\mathrm{item}$
= Size of 83.25^th item

83.25^th item lies in the group 30-35 within the 33rd c.f. of the series.


$$\begin{gathered} \mathrm{Thus}, \\ {Q}_3=l_1+\frac{{\displaystyle \frac{3N}{4}}-c.f}{f}\times i \\ or,Q_3=30+\frac{83.25-79}{12}\times 5 \\ or,Q_3=30+\frac{4.25\times 5}{12} \\ or,Q_3=30+\frac{21.25}{12} \\ or,Q_3=30+1.77 \\ \Rightarrow Q_3=31.77 \end{gathered}$$

$\mathit{Q}\mathit{u}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{l}\mathit{e}\mathbf{}\mathit{D}\mathit{e}\mathit{v}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}=\frac{Q_3-Q_1}{2}=\frac{31.77-18.26}{2}=6.75$
$\mathit{C}\mathit{o}\mathit{e}\mathit{f}\mathit{f}\mathit{i}\mathit{c}\mathit{i}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{Q}\mathbf{.}\mathit{D}\mathbf{.}=\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{31.77-18.26}{31.77+18.26}=\frac{13.51}{50.03}=0.27$