Consider a typical random walk problem, where the probability to go right is $R$ and the probability to go left is $L$, where $R+L=1$. The particle can move 1 unit in each step, and starts at zero.
Let the particle move $n$ steps and write down its location away from zero.
Now repeat this a very large number of times.
What is the mean location of the particle from zero, and what is the standard deviation of the particle's location?
You want the mean and variance of the sum of the random variables $X_1, X_2, ... , X_n$ which i.i.d with $P(X_i=1) = R$ and $P(X_i = -1) = 1-R$.
The moments can be calculated by direct summation:
\begin{equation} \begin{split} \langle\sum_{i=1}^n X_i\rangle &= \sum_{i=1}^n\langle X_i\rangle\\ &= \sum_{i=1}^n (2R-1)\\ &= n(2R-1) \end{split} \end{equation}
and
\begin{equation} \begin{split} \langle\Big(\sum_{i=1}^n X_i\Big)^2\rangle &= \langle\sum_{i=1}^n X_i\sum_{j=1}^n X_j\rangle\\ &= \sum_{i=1}^n\langle X_i^2\rangle + \sum_{i\neq j}\langle X_iX_j \rangle\\ &= \sum_{i=1}^n\langle X_i^2\rangle + \sum_{i\neq j}\langle X_i\rangle \langle X_j \rangle\\ &= n + n(n-1)(2R-1)^2 \end{split} \end{equation}
So the standard deviation is
$$\sqrt{\langle\Big(\sum_{i=1}^n X_i\Big)^2\rangle - \langle\sum_{i=1}^n X_i\rangle^2} = 2\sqrt{nR(1-R)}$$