How do you find the standard deviation of the weighted mean?
The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$
The weighted standard deviation (since it is not specified, I take it as of the distribution) is defined:
$$s_w = \sqrt{\frac{N'\sum_{i=1}^N {w_i(x_i-\bar{x}_w)^2}}{(N'-1)\sum_{i=1}^N{w_i}}},$$
where $N'$ is the number of nonzero weights, and $\bar x_w$ is the weighted mean of the sample (source)
For an unweighted sample, calculating the standard deviation of the mean from the standard deviation of the distribution is described on Wikipedia.
How do I calculate it for the weighted mean, and how is the expression derived?