# Standard deviation of the weighted mean

How do you find the standard deviation of the weighted mean?

The weighted mean is defined: $\bar{x} = \frac{\sum{wx}}{\sum{w}}$

The weighted standard deviation (since it is not specified, I take it as of the distribution) is defined: $\sqrt{\frac{\sum{w(x-\bar{x})^2}}{\frac{(N-1)\sum{w}}{N}}}$

For an unweighted sample, calculating the standard deviation of the mean from the standard deviation of the distribution is described on Wikipedia here: http://en.wikipedia.org/wiki/Standard_deviation#Standard_deviation_of_the_mean

How do I calculate it for the weighted mean, and how is the expression derived?

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$$\frac{\sum w_i(\theta_i-\bar{\theta})(\theta_i-\bar{\theta})'}{\sum w_i}$$