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Measures of Dispersion — Economics STD 11 Commerce — Question

CBSE BoardEnglish MediumSTD 11 CommerceEconomicsMeasures of Dispersion6 Marks
Question
Calculate standard deviation and coefficient of variation from the following data.
Matks: Below 20 Below 40 Below 60 Below 80 Below 100
Number of Students: 8 20 50 70 80
Hint: Since, less than distribution is given, we first convert into an exclusive distribution.
✓

Answer

Calculetion of Standard Deviation and Coefficient of Variation
Marks Number of Students(f) Mid-value(m) $d = m - A(A = 50)$ $\text{d}'=\frac{\text{d}}{\text{c}}(\text{c}=20)$ $fd'$ $fd'^2$
0-20 8 10 -40 -2 -16 32
20-40 12 30 -20 -1 -12 12
40-60 30 50 0 0 0 0
60-80 20 70 +20 +1 +20 20
80-100 10 90 +40 +2 +20 40
  $\Sigma\text{f}=80$       $\Sigma\text{fd}'=12$ $\Sigma\text{fd}'^2=104$
$\bar{\text{X}}=\text{A}+\frac{\Sigma\text{fd}'}{\Sigma\text{f}}\times\text{c}$
$=50+\frac{12}{80}\times20=53$
$\sigma=\sqrt{\frac{\Sigma\text{fd}'^2}{\Sigma\text{f}}-\Big(\frac{\Sigma\text{fd}'}{\Sigma\text{f}}\Big)^2}\times\text{c}$
$=\sqrt{\frac{104}{80}-\Big(\frac{12}{80}\Big)^2}\times20$
$=\sqrt{1.3-0.0225}\times20$
$=\sqrt{1.2775}\times20$
$=1.130\times20=22.60$
$\therefore$ Coefticient of Variation (CV) $=\frac{\sigma}{\bar{\text{X}}}\times100=\frac{22.60}{53}\times100=42.64\%$

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