Question

Find out quartile deviation and the coefficient of quartile deviation of the following data:

Age	20	30	40	50	60	70	80
Members	3	61	132	153	140	51	3

Answer 1

Age	Numbers (f)	Cumulative Frequency (c.f.)
20 30 (Q₁)40 50 (Q₃)60 70 80	3 61 132 153 140 51 3	3 3 + 61 = 64 64 + 132 = 196 196 + 153 = 349 349 + 140 = 489 489 + 51 = 540 540 + 3 = 543
	$\mathrm{\Sigma }f=N=543$

Q₁ = Size of ${\left(\frac{N+1}{4}\right)}^{\mathrm{th}}$ item
= Size of ${\left(\frac{543+1}{4}\right)}^{\mathrm{th}}$ item
= Size of 136^th item

136^th item lies in 196^th cumulative frequency of the series. Age corresponding to 196^th (c.f.) is 40.

Hence, Q₁ = 40

Q₃ = Size of $3\left(\frac{N+1}{4}\right)\mathrm{th}$ item
= Size of $3\left(\frac{543+1}{4}\right)\mathrm{th}$ item
= Size of 408th item

408th item lies in the 489th c.f. of the series. Age corresponding to 489th c.f. is 60.

Hence, Q₃ = 60.

Calculating Quartile Deviation and Coefficient of Quartile Deviation:

$$\begin{gathered} \mathrm{Quartile}\mathrm{deviation}(Q.D.)=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{2} \\ \mathrm{or},Q.D.=\frac{60-40}{2} \\ \mathrm{or},Q.D.=\frac{20}{2} \\ \Rightarrow Q.D.=10 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{quartile}\mathrm{deviation}=\frac{Q_3\mathit{-}{Q}_1}{Q_3+Q_1} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{60-40}{60+40} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{20}{100} \\ \Rightarrow \mathrm{Coefficient}\mathrm{of}Q.D.=0.2 \end{gathered}$$

Hence, Q.D. of the series is 10 and Coefficient of Q.D. is 0.2