Preventing overflow is essential while calculating the average of a long series of data. One simplest method to avoid the problem is running average (or moving average, rolling average, incremental mean).

## Problem

Suppose we have data $x_1, x_2, \cdots, x_n$. The $sum_{n+1}$ will overflow when the new data $x_{n+1}$ is counted, where $sum_i = x_1 + x_2 + \cdots + x_i$. At that time, $\mu_{n+1} = \frac{sum_{n+1}}{n+1}$ will be wrong, where $\mu_i$ is the average of $x_1, x_2, \cdots, x_i$, since $sum_{n+1}$ is an overflowed value.

How could we solve the problem?

## Naive Approach

uint64_t count = 0;
uint64_t sum = 0;

void Add(uint64_t data)
{
if (sum + data < sum) { // Overflow!
sum = sum / 2;
count = count / 2;
}

sum += data;
++count;
}

double GetAverage()
{
return static_cast<double>(sum) / count;
}


The above approach to prevent overflow is extracted from Gecko MP3Demuxer, but it’s wrong in most cases (it’s only correct when every incoming data is same). For example, if we have $x_1 = 10, x_2 = 20$ and $x_3 = 90$ will lead to overflow to the sum of $x_i$ so the average computed by above approach will be $\frac{(10+20)/2 + 90}{2/2 + 1} = 52.5$, which is different from $\frac{10 + 20 + 90}{3} = 40$. I cannot believe this incorrect code lives in Firefox more than 3 years. Fortunately, the overflow happens once in a blue moon so it’s not too much trouble.

### Proof

We can formally prove the above approach is incorrect. Let $sum_{k}$ and $\mu_k$ be the sum and arithmetic mean of $x_1, x_2, \cdots, x_k$ respectively, and $E_k$ be the estimated average from above approach.

Since the new incoming data $x_{n+1}$ will cause overflow to $sum_{n+1}$, so

$E_{n+1} = \frac{p + x_{n+1}}{q + 1}$ , where $p = \frac{sum_{n}}{2} = \frac{x_1 + x_2 + \cdots + x_n}{2}$ and $q = \frac{n}{2}$

As a result,

\begin{align} E_{n+1} &= \frac{p + x_{n+1}}{q + 1} \\ &= \frac{\frac{x_1 + x_2 + \cdots + x_n}{2} + x_{n+1}}{\frac{n}{2} + 1} \\ &= \frac{\frac{x_1 + x_2 + \cdots + x_n + 2 \cdot x_{n+1}}{2}}{\frac{n + 2}{2}} \\ &= \frac{x_1 + x_2 + \cdots + x_n + 2 \cdot x_{n+1}}{n+2} \end{align}

We could compare $E_{n+1}$ and $\mu_{n+1}$ to see if they are equal:

\begin{align} \mu_{n+1} &= \frac{x_1 + x_2 + \cdots + x_n + x_{n+1}}{n+1} \\ &= \frac{(n+2) \cdot (x_1 + x_2 + \cdots + x_n + x_{n+1})}{(n+1) \cdot (n+2)} \\ &= \frac{(n+1) \cdot (x_1 + x_2 + \cdots + x_n + x_{n+1}) + (x_1 + x_2 + \cdots + x_n + x_{n+1})}{(n+1) \cdot (n+2)} \\ E_{n+1} &= \frac{x_1 + x_2 + \cdots + x_n + 2 \cdot x_{n+1}}{n+2} \\ &= \frac{(x_1 + x_2 + \cdots + x_n + x_{n+1}) + x_{n+1}}{n+2} \\ &= \frac{(n+1) \cdot ((x_1 + x_2 + \cdots + x_n + x_{n+1}) + x_{n+1})}{(n+1) \cdot (n+2)} \\ &= \frac{(n+1) \cdot (x_1 + x_2 + \cdots + x_n + x_{n+1}) + (n+1) \cdot x_{n+1}}{(n+1) \cdot (n+2)} \\ &= \frac{(n+1) \cdot (x_1 + x_2 + \cdots + x_n + x_{n+1}) + (\overbrace{x_{n+1} + \cdots + x_{n+1}}^{n+1}))}{(n+1) \cdot (n+2)} \\ \mu_{n+1} - E_{n+1} &= \frac{ (x_1 + x_2 + \cdots + x_n + x_{n+1}) - (\overbrace{x_{n+1} + \cdots + x_{n+1}}^{n+1}))}{(n+1) \cdot (n+2)} \\ &= \frac{ (x_1 - x_{n+1}) + (x_2 - x_{n+1}) + \cdots + (x_n - x_{n+1}) + (x_{n+1} - x_{n+1})}{(n+1) \cdot (n+2)} \\ &= \frac{ (x_1 - x_{n+1}) + (x_2 - x_{n+1}) + \cdots + (x_n - x_{n+1}) }{(n+1) \cdot (n+2)} \end{align}

Thus, we can clearly see that $\mu_{n+1}$ will be equal to $E_{n+1}$ only when $x_1 = x_2 = \cdots = x_n = x_{n+1}$.

## Running Average

In fact, the average can be calculated without using the overflowed $sum_{n+1}$. Let $\mu_n$ be the mean of $x_1, x_2, \cdots, x_n$, we can get $\mu_n$ by $\mu_{n-1}$:

\begin{align} \mu_n &= \frac{ \mu_{n-1} \cdot (n-1) + x_n }{ n } \\ &= \frac{ n \cdot \mu_{n-1} + x_n - \mu_{n-1}}{ n } \\ &= \mu_{n-1} + \frac{ x_n - \mu_{n-1} }{ n } \end{align}

### Sample code

On this ground, we could correct the code into:

double average = 0;
uint64_t count = 0;
void Add(uint64_t data)
{
average += (data - average) / ++count;
}

double GetAverage()
{
return average;
}


or

double UpdateAverage(uint64_t data)
{
static double average = 0;
static uint64_t count = 0;
average += (data - average) / ++count;
return average;
}


or

class Averager
{
public:
Averager()
: average(0)
, count(0)
{}

~Averager() {};

void Add(uint64_t data) { average += (data - average) / ++count; }

double GetAverage() { return average; }

private:
double average;
uint64_t count;
}


## References

Bug 1423834 has been filed for this problem when I found this code in MP3Demuxer. You could find more detail there.

The following links are some related resources: