How to calculate moving average without keeping the count and data-total?

I am trying to find a way to calculate a moving cumulative average without storing the count and total data that is received so far.

I came up with two algorithms but both need to store the count:

• new average = ((old count * old data) + next data) / next count
• new average = old average + (next data - old average) / next count

The problem with these methods is that the count gets bigger and bigger resulting in losing precision in the resulting average.

The first method uses the old count and next count which are obviously 1 apart. This got me thinking that perhaps there is a way to remove the count but unfortunately I haven't found it yet. It did get me a bit further though, resulting in the second method but still count is present.

Is it possible, or am I just searching for the impossible?

Solution 1:

You can simply do:

double approxRollingAverage (double avg, double new_sample) {

    avg -= avg / N;
    avg += new_sample / N;

    return avg;
}

Where N is the number of samples where you want to average over. Note that this approximation is equivalent to an exponential moving average. See: Calculate rolling / moving average in C++

Solution 2:

New average = old average * (n-1)/n + new value /n

This is assuming the count only changed by one value. In case it is changed by M values then:

new average = old average * (n-len(M))/n + (sum of values in M)/n).

This is the mathematical formula (I believe the most efficient one), believe you can do further code by yourselves

Solution 3:

From a blog on running sample variance calculations, where the mean is also calculated using Welford's method:

Too bad we can't upload SVG images.