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Measures of Dispersion — Economics STD 11 Commerce — Question

CBSE BoardEnglish MediumSTD 11 CommerceEconomicsMeasures of Dispersion6 Marks
Question
Calculate Inter-Quartile Range and Coefficient of Quartile Deviation from the following data:
Size: 0-5 5-10 10-15 15-20 20-25 25-30
Frequency: 3 9 15 23 30 20
✓

Answer

Size f c.f.
0-5 3 3
5-10 9 12
10-15 15 27
15-20 23 50
20-25 30 80
25-30 20 100
  $\Sigma\text{f}=100$  
$\text{Q}_1=\Big(\frac{\text{N}}{4}\Big)^{\text{th}}\text{item}=\Big(\frac{100}{4}\Big)^{\text{th}}\text{item}$
$= 25^{th}$ item. which lies in the class 10-15.
$\text{Q}_1=\text{l}_1+\frac{\big(\frac{\text{N}}{4}\big)-\text{c.f.}}{\text{f}}\times\text{i}$
$\text{Q}_1=10+\frac{25-12}{15}\times5$
$\text{Q}_1=10+\frac{13}{15}\times5$
$=10+\frac{13}{5}=10+4.3=14.3$
$\text{Q}_3=3\Big(\frac{\text{N}}{4}\Big)^{\text{th}}=3\times25$
$= 75^{th}$ item. which lies in the class 20-25.
$\text{Q}_3=\text{l}_1+\frac{3\big(\frac{\text{N}}{4}\big)-\text{c.f.}}{\text{f}}\times\text{i}$
$\text{Q}_3=20+\frac{75-50}{30}\times5$
$\text{Q}_3=20+\frac{25}{30}\times5$
$=20+4.15=24.15$
$\therefore$ Inter-Quartile Range $=\text{Q}_3-\text{Q}_1=24.15-14.3=9.8$
coefficient of Q.D. $=\frac{\text{Q}_3-\text{Q}_1}{\text{Q}_3+\text{Q}_1}$
$=\frac{24.15-14.3}{24.15+14.3}=\frac{9.8}{38.4}=0.25$

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