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Incremental average (estimate update)

Statement: $\begin{displaymath} A_n = A_{n-1} + \frac{1}{n}(V_n - A_{n-1}) \end{displaymath}$

Alterative form: $\begin{displaymath}NewEstimate \leftarrow OldEstimate + StepSize[NewData - OldEstimate]\end{displaymath}$

The second form describes updating our estimate of the average by multiplying an error term, NewData - OldEstimate, by a weighting factor, StepSize. StepSize is $\frac{1}{n}$ when all data points are weighted equally.

The average, $A_{n-1}$ is known for a sequence of n-1 values, $V_1, V_2, ..., V_{n-1}$. If the next value, $V_n$ is included in the sequence, the new average, $A_{n}$, can be computed by adding the new component of the average, $\frac{1}{n}V_{n}$, to the old average transfered from an $n-1$ denominator to an $n$ denominator, ( $A_{n-1} - \frac{1}{n}A_{n-1}$).

The ( $A_{n-1} - \frac{1}{n}A_{n-1}$) component is easy to understand from an example: the difference between 1/4 and 1/5 is 1/(4*5). If we want to find the total divided by 5, and already have the total divided by 4, we can obtain the former by subtracting the known difference, 1/(4*5), which is the average divided by 4, divided by 5 again.