I'm not really sure where to put this.
I spend most of my life world-building, pointlessly, and am a member of various groups to assist in that process. In one of these groups, we focus a lot on demographics, and I constantly notice people asking how many people should be in their village or city, and, more often, how fast their population should *GROW*.
It's one of those irritating questions that can never be answered accurately, but I'm ever eager to spout both facts and opinions. So I did some math, then gave up after a couple hours and used an online calculator, and ended up with some statistics on population growth. Since many people here seem to enjoy world-building, and as I've noticed a few mapping threads which focus on the changes in a world/nation over time, I thought I would share a few of these facts and opinions with you:
The crude death rate, the total number of deaths per year per 1000 people, for the whole world is about 8.37 per 1000 per year according to the current CIA World Factbook. The crude birthrate for the entire world is 20.6 live births per 1000 people.
So each year, for every 1000 people on the earth, 20 people are born, and 8 die, increasing the total population by 12. (Actually 20.6 - 8.37 = 12.23, but you can't have a partial person.) That's 1.2 people added to the population each year, per 100, or .0012 per person.
That doesn't sound like a lot does it?
If you have a village with 20 families, (Let's call that Man, wife, no kids yet) that's 40 people. Plus the cranky old guy who lives just outside the bounds of sanity. Just for fun. How fast will your world population grow, if only that village is left alive?
1 year - 40.048 people
after 21 years you get a whole new person!
21 years - 41.037 people
20 years later you'll have a population of 42 people:
40 years - 42.002 people
60 years - 43 people
80 years - 44 people So far it seems like the population isn't going anywhere.. 1 person every 20 years isn't much. But this is not a flat rate, it compounds, so the population grows faster (even if it doesn't seem that way for a while.
After 98 years (only 18 years later) you have 45 people (Actually, it's 45.032371604663200050712789175541, in case you were wondering..)
184 years - 50 people.
200 years - 51 people
250 years - 92 people
300 years - 168 people
350 years - 305 people
400 years - 554 people (!)
500 years - 1,825 people
750 years - 36,045 people
1000 years - 711,181 people.
Now, of course, this is all based on averages, of a 1.2% growth rate. It doesn't take into account sudden fluctuations in growth, like wars, or plagues, culture, or quality of healthcare.
Your medieval society might grown at as little as half this rate. (0.6%)
That puts you at 53 people after 50 years, 72 after 100 years, 131 at 200 years, 791 after 500 years, and 15,755 after 1000 years. Quite a vast difference.
How do you decide at what rate your population should grow?
Well, in the very worst of times, your population will die out completely - war, famine, plagues, rampaging wizards...
In the best case scenario, well, the highest birth rate in the world right now is 50 per 1000
The lowest death rate is 1.4 per 1000
That leaves you with a growth rate of 48.6 per 1000, or 4.86%, and a population of 409 after 50 years. 4389 after 100 years, and in 500 years, your 40 villagers would have swelled to 769,798,085,747
A bit unrealistic? Oh yes. But in an advanced society, with no war, medicines to cure anything and ample food (in a large solar system with plenty of room to grow) it's entirely possible.
If you want to play with some numbers on your own, there's a Population Calculator which can help a great deal.
In short - Your population growth depends on your world, and on your people. In general, though, small populations grow slowly, large populations quickly. Low-tech societies grow slowly (or, rather, they die off faster) and advanced societies will grow rapidly without some sort of check in place (like the Two Child Policy imposed by some governments.
Most of the statistics you will find will be worthless, growth-wise, if your world is unique. (No one will die from infection if everyone can magically heal themselves!) The best you can do is to think of all the good and bad in your world and plan consistently.
Best of Luck~