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Measures of Dispersion — Economics STD 11 Commerce — Question

CBSE BoardEnglish MediumSTD 11 CommerceEconomicsMeasures of Dispersion6 Marks
Question
Find the Inter-Quartile range and coefficient of quartile deviation from the following data:
X 11-20 21-30 31-40 41-50 51-60
f 4 8 20 12 6
✓

Answer

Since inclusive class intervals are given, we will first convert them into exclusive class intervals.
X Exclusive C.I. f c.f.
11-20 10.5-20.5 4 4
21-30 20.5-30.5 8 12
31-40 30.5-40.5 20 32
41-50 40.5-50.5 12 44
51-60 50.5-60.5 6 50
    $\Sigma\text{f}=50$  
$\text{Q}_1=\Big(\frac{\text{N}}{4}\Big)^{\text{th}}\text{item}=\frac{50}{4}$
$= 12.5^{th}$ item, which lies in the class 30.5-40.5$\text{Q}_1=\text{l}_{\text{1}}+\frac{\frac{\text{N}}{4}-\text{c.f.}}{\text{f}}\times\text{i}$
$=30.5+\frac{12.5-12}{20}\times10=30.75$
$\text{Q}_3=3\Big(\frac{\text{N}}{4}\Big)^{\text{th}}\text{item}=3\times12.5^{\text{th}}\text{item}$
$= 37.5^{th}$ item, which lies in the class 40.5-50.5$\text{Q}_3=\text{l}_1+\frac{3\Big(\frac{\text{N}}{4}\Big)-\text{c.f.}}{\text{f}}\times\text{i}$
$=40.5+\frac{37.5-32}{12}\times10=45.08$
Inter-quartile range $=\text{Q}_3-\text{Q}_1=45.08-30.75=14.33$ Coefficient of Q.D. $=\frac{\text{Q}_3-\text{Q}_1}{\text{Q}_3+\text{Q}_1}$$=\frac{45.08-30.75}{45.08+30.75}=\frac{14.33}{75.83}=0.18$

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