Standard deviation of the weighted mean 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?
I think that your answer is 

$$\frac{\sum w_i(\theta_i-\bar{\theta})(\theta_i-\bar{\theta})'}{\sum w_i}$$
TO understand what the weights do, assume that the weights are all integers (if they are rational you can multiply out the LCM of the denominators). To keep it simple assume you have 3 data points with weight 2, 3 and 5.

Your mean is then
$$
\frac{2 x_1 + 3 x_2 + 5 x_3}{2 + 3 + 5}
$$

Now suppose you go back to your $3$ data points and want to make $x_3$ important (i.e. bias the mean towards $x_3$). One way to do this is to make up additional data by writing $x_3$ 5times.  You do the same for $x_2$ except you write it $3$ times and $x_1$  two times. 

To be specific, suppose that $x_1=1$, $x_2=0$, $x_3=8$.

$$
\text{Unweighted mean} = \frac{1+0+8}{3} = 3
$$
With weights, your data is now $\left[1,1, 0,0,0,8,8,8,8,8\right]$. So your weighted mean is
$$
\text{Weighted mean} = \frac{1+1+0+0+0+8+8+8+8+8}{10} = 4.2$$
See how the mean shifted from $3$ towards $8$ to become $4.2$.

Weights are a way to make some data more important than others.