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  • The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is A. 50000 B. 250000 C. 252500 D. 255000

The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is
A. 50000
B. 250000
C. 252500
D. 255000

Answers (1)

As per the question

Number of observations: n =100

\\ \text{Mean of the given observations, }\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 * 100=5000 \\ \\ \text{It is also given that the standard deviation of the 100 observations is 5 } \\\\ \sigma =5 \\ \\ \sigma = \sqrt {\frac{ \Sigma x_{i}^{2}}{n}- \left( \frac{ \Sigma x_{i}}{n} \right) ^{2}} \\ \\ \text{Substituting the corresponding values, we get }5=\sqrt {\frac{ \Sigma x_{i}^{2}}{100}- \left( \frac{5000}{100} \right) ^{2}}~ \\


\\ \text{ Now taking square on both sides we get }5^{2}=\frac{ \Sigma x_{i}}{100}- \left( 50 \right) ^{2}~ \\ \\ 25=\frac{ \Sigma x_{i}}{100}-2500 \\ \\ ~ 25+2500=\frac{ \Sigma x_{i}^{2}}{100}~~~~ \\ \\ ~~ \Sigma x_{i}^{2}=2525 * 100 \\ \\ ~ \Sigma x_{i}^{2}=252500 \\

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