Moving Average - Cumulative Moving Average

In a cumulative moving average, the data arrive in an ordered datum stream and the statistician would like to get the average of all of the data up until the current datum point. For example, an investor may want the average price of all of the stock transactions for a particular stock up until the current time. As each new transaction occurs, the average price at the time of the transaction can be calculated for all of the transactions up to that point using the cumulative average, typically an unweighted average of the sequence of i values x₁, ..., x_i up to the current time:

The brute-force method to calculate this would be to store all of the data and calculate the sum and divide by the number of datum points every time a new datum point arrived. However, it is possible to simply update cumulative average as a new value x_i+1 becomes available, using the formula:

where can be taken to be equal to 0.

Thus the current cumulative average for a new datum point is equal to the previous cumulative average plus the difference between the latest datum point and the previous average divided by the number of points received so far. When all of the datum points arrive (i = N), the cumulative average will equal the final average.

The derivation of the cumulative average formula is straightforward. Using

and similarly for i + 1, it is seen that

Solving this equation for CA_i+1 results in: