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:

                                                   (2)

Changing  to  makes equation (2) becomes

or

                                                                    (a)

Putting out the last term from equation (2) gives

Since then                  

        (b)

Equation (3) can be rearranged into

                                                           (c)

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

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

 (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