Standard deviation of a random sample's average

Question

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 $\overline{X}_{121} = \frac{X_1 + X_2 +\dots + X_{121}}{121}$ ?

Question as an image

Answer 1

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}