Results of simulation:
To maximise variation I set things up so that no draws occur. I used 100 players playing in 100 double round robin tournaments, 990000 simulated games altogether.
The mean is always 1800, not surprising since any reduction in one players rating results in a symmetrical rise in the opponent's rating. The standard deviation is between 50 and 60, the maximum rating is between 1900 and 1950, and the minimum rating between 1650 and 1700.
I may run this with more rounds later on just to see what happens over longer time scales.
Edit: Missing zero in number of games.