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I have some measured datapoints. Each of the points has attached an error, depending on how precise the measurement was done.

Now I'm using the squared weighted mean with the inverse of the errors as weights so that I emphasize more on the preciser values and less on the values where the error is large.

What I'm wondering is what can I say about the precision of my squared weighted mean. Is there a way to calculate the expected error range for it?

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Are these observations independent? Do you know the mean and variance of each individual observation? –  Macro Aug 30 '12 at 20:15
    
This gets interesting if the individual point errors were themselves estimated, such as from repeated measurements ;-). Might that be the case? –  whuber Aug 30 '12 at 20:38
    
As it turns out the expected errors for each measured value are the variances from the multiple measurements for that value. We therefore have $w_i=\frac{1}{\sigma_i^2}$, $w_i$ being the weights for each value. –  cdecker Aug 30 '12 at 21:06
    
@cdecker I think using 1/standard deviation is a more appropriate weighting than dividing by the variance. –  Michael Chernick Sep 4 '12 at 14:47

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Suppose you have k independent random variables X$_i$ i=1,2,..,k each having variance σ$_i$$^2$ and let X$_w$= Σ c$_i$ X$_i$. The variance of X$_w$= Σ c$_i$$^2$ σ$_i$$^2$. You can use that variance to get the standard error of that estimate.

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