The $avg pipeline group accumulator computes the average by summing all elements then divide by the count.

src/mongo/db/pipeline/accumulator_avg.cpp :

return Value(_nonDecimalTotal.getDouble() / static_cast<double>(_count));

This may lead to an overflow in the sum for large number of records or large numbers.

To be evaluated for performance reasons : using a stream-average to compute the average.

count++
avg += (x - avg) / count;