Question

calculate sum and sum of squares if mean of 20 terms is 30 and SD is 3​

Answer 1

Given :-

• Mean of 20 terms is 30 and SD is 3 .

To Find :-

• calculate sum and sum of squares ?

Solution :-

→ Total terms are = 20

→ Mean = 30 .

So,

→ sum of all terms = (No. of terms) * Mean = 20 * 30 = 600 (Ans.)

Now, given that, Standard deviation of 20 terms is 3.

we know that,

• (SD)² = (sum of square of all terms / n) - (sum of all terms /n)² . { n = number of terms. }

Putting all values we get,

→ (sum of square of all terms / 20) - (30)² = 3²

→ (sum of square of all terms / 20) = 9 + 900

→ (sum of square of all terms / 20) = 909

→ sum of square of all terms = 909 * 20

→ sum of square of all terms = 18,180 . (Ans.)

Hence, sum of all terms is 600 and sum of squares of all terms is 18,180 .

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO DETERMINE

The sum and sum of squares if mean of 20 terms is 30 and SD is 3

CALCULATION

$\sf{Let \: \: x_1, x_2,.., x_{20} \: \: be \: the \: 20 \: terms}$

$\therefore \sf{Mean = \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ \: x_i} }{20} }$

$\implies \sf{ \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ \: x_i} }{20} } = 30$

$\implies \sf{ \displaystyle \sum\limits_{i=1}^{20} \sf{ \: x_i} } = 30 \times 20$

$\implies \sf{ \displaystyle \sum\limits_{i=1}^{20} \sf{ \: x_i} } = 600$

Hence the sum of the 20 terms is 600

Again

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

$\implies \sf{ \sqrt{ \: \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} }{20} - \bigg( { \frac{\sum\limits_{i=1}^{20} \sf{ \: x_i} }{20} \bigg)}^{2} } } = 3$

$\implies \sf{ \: \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} }{20} - \bigg( { \frac{\sum\limits_{i=1}^{20} \sf{ \: x_i} }{20} \bigg)}^{2} } = 9$

$\implies \sf{ \: \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} }{20} - {(30)}^{2} } = 9$

$\implies \sf{ \: \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} }{20} = 9 + 900}$

$\implies \sf{ \: \displaystyle \frac{\sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} }{20} = 909}$

$\implies \sf{ \: \displaystyle \sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} } = 909 \times 20$

$\implies \sf{ \: \displaystyle \sum\limits_{i=1}^{20} \sf{ {\: x_i}}^{2} } = 18180$

Hence the sum of squares of the 20 terms is 18180

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