Proof of the corrected variance formula

By Kardi Teknomo, PhD.

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Traditional variance formula is
We can rewrite the variance formula so that we can separate the term of mean from the summation

To simplify the notation, let us give symbol to the sum


Using this new symbol, we can write the variance as

Now we have corrected measurement data, which is to add with a new data and to minus with the wrong data. The corrected variance is

We can expand into


Then rearrange to get

And we simplify

(proved)

See Also: proof of corrected mean formula

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These tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi. (2006) Recursive Average and Variance.
http://people.revoledu.com/kardi/tutorial/RecursiveStatistic/index.html