From my personal observation, empire-wide Stability seems to be calculated fairly simply.

(sum of all Stability of your Cities) / (the number of your Cities) = Civilization-wide Stability

This is generally acceptable, but in some cases it does not make any sense.

For example, if only two cities exist in your Civilization, 100% Stability Capital with the population of 100+ and 0% newly-conquered City with population of 0(they all died in previous siege), then suddenly the Civilization-wide Stability plunges to 50%. Of course, each population in the game does not represent the entirety of people, who reside within Civilization. It is more like Manpower. Still, I think the number of populations should have some weight on this formula. Why would general Stability be halved, merely because a very small city on the other side of the continent is having little troubles?