Question

For 10 observations if mean and variance of the series is 16 and 20 respectively then sum of square of all observation is ________. [A] 2760 [B] 4000 [C] 3600 [D] 4160​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

For 10 observations if mean and variance of the series is 16 and 20 respectively then sum of square of all observation is

[A] 2760

[B] 4000

[C] 3600

[D] 4160

EVALUATION

Let the given 10 observations are

$\sf{x_1,x_2,..,x_{10}}$

Here number of observations = n = 10

$\displaystyle \sf{Mean = \sum\limits_{i=1}^{10} \: \frac{x_i}{n} }$

$\displaystyle \sf{ \therefore \: \: \sum\limits_{i=1}^{10} \: \frac{x_i}{n} = 16 \: \: \: - - - (1) }$

Again

$\displaystyle \sf{ Variance = \: \frac{\sum\limits_{i=1}^{10} {x_i}^{2} }{n} - { \bigg( \sum\limits_{i=1}^{10} \: \frac{x_i}{n} \bigg)}^{2} }$

$\displaystyle \sf{ \therefore \: \: \: \frac{\sum\limits_{i=1}^{10} {x_i}^{2} }{10} - { \bigg( \sum\limits_{i=1}^{10} \: \frac{x_i}{10} \bigg)}^{2} = 20 }$

$\displaystyle \sf{ \implies \frac{\sum\limits_{i=1}^{10} {x_i}^{2} }{10} - {(16)}^{2} = 20 } \: \: \: \: \: \: (from \: equation \: 1) \:$

$\displaystyle \sf{ \implies \frac{\sum\limits_{i=1}^{10} {x_i}^{2} }{10} -256 = 20 }$

$\displaystyle \sf{ \implies \frac{\sum\limits_{i=1}^{10} {x_i}^{2} }{10} = 276 }$

$\displaystyle \sf{ \implies \sum\limits_{i=1}^{10} \: {x_i}^{2} = 2760 }$

Hence the sum of square of all observation = 2760

FINAL ANSWER

The correct option is

[A] 2760

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Answer 2

Answer :-

Mean is summation of all observations divided by number of observation and hence, would be multiplied by 2 due to linearity. Hence, the new mean is 8.

Variance is sum of squares of each observation subtracted by the mean. Hence, due to squared dependence, it will be quadrupled to get 8.

Hence, both mean and variance are 8.