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  • Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all items and the sum of the squares of the items.

Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all items and the sum of the squares of the items.

 

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Mean and standard deviation of 100 items are 50 and 4 respectively

We have to find the sum of all items and the sum of the squares of the items.

As per the question, number of items n =100

\\ \text{Mean of the given items, }\overline{x}=50~ \text{But we know}, \overline{x}=\frac{ \Sigma x_{i}}{n}~~ \\\\ \\ \text{Substituting the corresponding values we get} 50= \frac{ \Sigma x_{i}}{100}~~~~~ \Sigma x_{i}=50\ast100=5000 \\\\ \\ \text{Hence,~the sum of all the 100 items}=5000 \\\\ \\ \text{Also given that the standard deviation of the 100 items is 4} \\\\ \\ ~~ \sigma =4 \\\\


\\ \text{ But we know that } \\\\ \\ \sqrt {\frac{ \Sigma x_{i}^{2}}{n}- \left( \frac{ \Sigma x_{i}}{n} \right) ^{2}}~~~ \\\\ \\ \text{Substituting the corresponding values, we get} 4= \sqrt {\frac{ \Sigma x_{i}^{2}}{100}- \left( \frac{5000}{100} \right) ^{2}} \\\\ \\ \text{~~Now taking square on both sides, we get }4^{2}=\frac{ \Sigma x_{i}^{2}}{100}- \left( 50 \right) ^{2}~~~ \\\\
\\ ~ 16=\frac{ \Sigma x_{i}^{2}}{100}-2500~ \\\\ \\ ~~ 16+2500=\frac{ \Sigma x_{i}^{2}}{100}~~ \\\\ \\ ~~\frac{ \Sigma x_{i}^{2}}{100}=2516 \\\\ \\ ~ \Sigma x_{i}^{2}=2516\ast100=251600 \\\\

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