How do I calculate the standard deviation between weighted measurements?

Question

I have several weighted values for which I am taking a weighted average. I want to calculate a weighted standard deviation using the weighted values and weighted average. How would I modify the typical standard deviation to include weights on each measurement?

This is the standard deviation formula I am using.

When I simply use each weighted value for 'x' and the weighted average for '\bar{x}', the result seems smaller than it should be.

Solution

I just found this wikipedia page discussing data of equal significance vs weighted data. The correct way to calculate the biased weighted estimator of variance is

,

though the following, on-the-fly implementation, is more efficient computationally as it does not require calculating the weighted average before looping over the sum on the weighted differences squared

.

Despite my skepticism, I tried both and got the exact same results.

Note, be sure to use the weighted average

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Comments

[Steven C. Howell]

@DavidArenburg This is something I am trying to program and likely others have/will also use in the future. I just added more to the answer I posted to identify how to efficiently code this.

[Robert Dodier]

Note that the mean is also calculated using the weights (i.e. be careful to use the weighted mean in that formula, not the usual unweighted mean).

[Steven C. Howell]

@RobertDodier good point. I did not make that explicit but I will for completeness sake.

[akavalar]

That last sentence should say "weighted mean", not "weighted error", right? I suppose the weighted error would be the difference between x_i and this value.