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Home  >>  CBSE XI  >>  Math  >>  Statistics
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The sum and sum of squares corresponding to length x (in cm) and weight (y) in gm of 50 plant products are given below: $ \sum \limits _{i=1} ^{50} x_i=212, \qquad\sum \limits_{i=1}^{50}x_i^2 =902.8;\qquad \sum \limits _{i=1} ^{50} y_i=261, \qquad\sum \limits_{i=1}^{50}y_i^2 =1457.6$ Which is more varying , the length or weight?

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1 Answer

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Toolbox:
  • The formula required to solve this problem are : Mean $ \bar{x}= \large\frac{\sum x_i}{n}$
  • Standard deviation $\sigma= \sqrt {\large\frac{\sum x_i^2}{\sum n} - \bigg( \large\frac{\sum x_i}{n} \bigg)^2 }$
  • Coefficient variation $=\large\frac{\sigma}{\bar {x} }$$ \times 100$
For length,
Mean $ \bar{x}= \large\frac{\sum x_i}{n}$
$\qquad= \large\frac{212}{50}$
$\qquad= 4.24$
Standard deviation $\sigma= \sqrt {\large\frac{\sum x_i^2}{\sum n} - \bigg( \large\frac{\sum x_i}{n} \bigg)^2 }$
$\qquad= \sqrt { \large\frac{902.8}{50} -\bigg(\large\frac{212}{50}\bigg)^2}$
$\qquad= \sqrt { \large\frac{902.8 \times 50-(212)^2}{50 \times 50}}$
$\qquad= \large\frac{1}{50} $$ \sqrt { 45140-44944}$
$\qquad= \large\frac{14}{50} =\frac{7}{25} $$=0.28$
Step 3:
Coefficient of variation : $ =\large\frac{\sigma}{\bar {x}} $$\times 100$
$\qquad= \large\frac{0.28}{4.24} $$ \times 100$
$\qquad= \large\frac{28}{4.24}$
$\qquad= 6.604$
For weight,
Step 1:
Mean $ \bar{x}= \large\frac{\sum x_i}{n}$
$\qquad= \large\frac{261}{50}$
$\qquad= 5.22$
Step 2:
Standard deviation $\sigma= \sqrt {\large\frac{\sum x_i^2}{\sum n} - \bigg( \large\frac{\sum x_i}{n} \bigg)^2 }$
$\qquad= \sqrt{\large\frac{1457.6}{50} \bigg( \large\frac{261}{50}\bigg)^2 }$
$\qquad= \large\frac{1457.6 \times 50 -(261 \times 261)}{50 \times 50}$
$\qquad= \large\frac{1}{50} $$\sqrt {72880 -68121}$
$\qquad= \large\frac{\sqrt {4759}}{50}$$=1.37$
Step 3:
Coefficient of variation : $ =\large\frac{\sigma}{\bar {x}} $$\times 100$
$\qquad= \large\frac{1.37}{5.22} $$ \times 100$
$\qquad= \large\frac{137}{5.22}$
$\qquad= 26.24$
COMPARISON:
Variability depends upon coefficient of variation.
Higher the coefficient of variation, higher is the variability.
$\therefore $ coefficient of variation of weight is higher as compared to coefficient of variation of length.
$\therefore $ weight is more varying than length.
answered Jul 1, 2014 by meena.p
 

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