How to calculate standard deviation between weighted measurements?

Question

I have several weighted values ​​for which I take the weighted average. I want to calculate the weighted standard deviation using the weighted values ​​and the weighted average. How to change a typical standard deviation to include scales in each measurement?

This is the standard deviation formula that I use.

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

Answer 1

I just found this wikipedia page that discussed data on equal significance and weighted data. The correct way to calculate biased weighted variance estimates

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although the next on-the-fly implementation is more computationally efficient, since it does not require calculating a weighted average before the cycle over the sum of the squared weighted differences

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

Please note that use a weighted average

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