Question

Form the following figures, find the quartile deviation and its coefficient:

Height (cm)	150	151	152	153	154	155	156	157	158
No. of students	15	20	32	35	33	22	20	12	10

Answer 1

Height (c.m)	No. of Students	Cumulative Frequency
150	15	15
151	20	15 + 20 = 35
152	32	35 + 32 = 67
153	35	67 + 35 = 102
154	33	102 + 33 = 135
155	22	135 + 22 = 157
156	20	157 + 20 = 177
157	12	177 + 12 = 189
158	10	189 + 10 = 199
	N = 199

$$\begin{gathered} Q_1=\mathrm{Size}\mathrm{of}{\left[\frac{N+1}{4}\right]}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{\left[\frac{199+1}{4}\right]}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{50}^{\mathrm{th}}\mathrm{item} \\ Now,weneedtolocatethisiteminthecolumnofCumulativeFrequency.The \\ itemjustexceeding{50}^{th}itemis67(inthec.f.column),whichiscorrespondingto152 \\ Hence,Q_1=152 \\ Similarly, \\ {Q}_3=\mathrm{Size}\mathrm{of}3{\left[\frac{N+1}{4}\right]}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}f3{\left[\frac{199+1}{4}\right]}^{\mathrm{th}}\mathrm{item} \\ =\mathrm{Size}\mathrm{of}{150}^{\mathrm{th}}\mathrm{item}=155 \\ \mathit{Q}\mathit{u}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{l}\mathit{e}\mathbf{}\mathit{d}\mathit{e}\mathit{v}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}=\frac{Q_3-Q_1}{2}=\frac{155-152}{2}=1.5 \end{gathered}$$

$\mathit{C}\mathit{o}\mathit{e}\mathit{f}\mathit{f}\mathit{i}\mathit{c}\mathit{i}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{Q}\mathbf{.}\mathit{D}\mathbf{.}=\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{155-152}{155+152}=\frac{3}{307}=0.0098$