Let $(X_1, X_2 ,\dots X_{121})$ be identically distributed independent random variable with variance $V(X_i)=1$ for all $i$. What is the standard deviation of their average $\overline{X}_{121} = \frac{X_1 + X_2 +\dots + X_{121}}{121}$ ?
Standard deviation of a random sample's average
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probability
standard-deviation
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1$X$ is a random variable, and the question is asking what the standard deviation of $X$ is equal to. Which part of the question do you not understand? – 2017-02-22
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0What is the variance of $\sum_1^{121} X_i$? What is the variance of $\frac1{121}\sum_1^{121} X_i$? What is the standard deviation of $\frac1{121}\sum_1^{121} X_i$? – 2017-02-22
1 Answers
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The standard deviation of the average $\overline{X}_n$ is the square root of the variance (here, $n=121$). Using the scaling and summation properties of the variance of uncorrelated random variables, one writes \begin{equation} \sigma(\overline{X}_n) = \sqrt{V(\overline{X}_n)} = \sqrt{\frac{1}{n^2} \sum_{i=1}^n V(X_i)} = \sqrt{\frac{1}{n^2} n} = \frac{1}{\sqrt{n}} \approx 0.09 \, . \end{equation}
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0Thanks a lot. Now I understand it. – 2017-02-22