Proof Recursive Variance Formula

By Kardi Teknomo, PhD .

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The usual way to compute time-variance is given by equation (2). I put here again for your convenient:

Proof Recursive Variance Formula (2)

Changing Proof Recursive Variance Formula to Proof Recursive Variance Formula makes equation (2) becomes

Proof Recursive Variance Formula

or

Proof Recursive Variance Formula (a)

Putting out the last term from equation (2) gives

Proof Recursive Variance Formula

Since Proof Recursive Variance Formula then

Proof Recursive Variance Formula (b) ( why? )

Equation (3) can be rearranged into

Proof Recursive Variance Formula (c)


The second term of the right hand side of equation (b) can be manipulated as

Proof Recursive Variance Formula

Proof Recursive Variance Formula

Put back the second term into equation (b) and replace the third term with equation (a) becomes

Proof Recursive Variance Formula

Proof Recursive Variance Formula (proved)


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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