Question

Estimate quartile deviation and the coefficient of quartile deviation of the following series:

Height (inches)	58	59	60	61	62	63	64	65	66
Number of Students	15	20	32	35	33	22	20	10	8

Answer 1

Height (inches)	No. of student (f)	Cumulative frequency (c.f.)
58 59 60 61 62 63 64 65 66	15 20 32 35 33 22 20 10 8	15 15 + 20 = 35 35 + 32 = 67 67 + 35 = 102 102 + 33 = 135 135 + 22 = 157 157 + 20 = 177 177 + 10 = 187 187 + 8 = 195
	$\mathrm{\Sigma }f=N=195$

Q₁ = Size of ${\left(\frac{N+1}{4}\right)}^{\mathrm{th}}$ item
= Size of ${\left(\frac{195+1}{4}\right)}^{\mathrm{th}}$ item
= Size of 49^th item

49^th item lies in 67^th c.f. of the series. Height corresponding to 67^th (c.f.) is 60.

Hence, Q₁ = 60 inches

Q₃ = Size of $3{\left(\frac{\mathrm{N}+1}{4}\right)}^{\mathrm{th}}$ item
= Size of $3{\left(\frac{195+1}{4}\right)}^{\mathrm{th}}$ item
= Size of 147^th item

147^th item lies in 157^th c.f. of the series. Height corresponding to 157^th c.f. is 63.

Hence, Q₃ = 63 inches

Calculating Quartile Deviation and Coefficient of Quartile Deviation


$$\begin{gathered} \mathrm{Quartile}\mathrm{deviation}(Q.D.)=\frac{Q_{\mathit{3}}\mathit{-}{Q}_{\mathit{1}}}{2} \\ \mathrm{or},Q.D.=\frac{63-60}{2} \\ \mathrm{or},Q.D.=\frac{3}{2} \\ \Rightarrow Q.D.=1.5 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{quartile}\mathrm{deviation}=\frac{Q_3\mathit{-}{Q}_1}{Q_3+Q_1} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{63-60}{63+60} \\ \mathrm{or},\mathrm{Coefficient}\mathrm{of}Q.D.=\frac{3}{123} \\ \Rightarrow \mathrm{Coefficient}\mathrm{of}Q.D.=0.024 \end{gathered}$$

Hence, Q.D. of the series is 1.5 and Coefficient of Q.D. is 0.024