I think it's an interesting question. Haven't seen any study ... an idea for chess.com. Would be a useful guide to planning, if a gen. trend appeared.
I know this thread is old, but it's an interesting one. I'm not a software guy, but I wonder if a chess.com API exists, and whether it can be used to do this kind of statistical analysis. Or perhaps an API isn't needed, just someone that knows how to create a bot that scrubs the site for the needed data.
In any case, I think chess.com should definitely start a stats blog. It would be very interesting.
In otb games it does make a difference, as the average ratings of players who play more games at slow time limits is higher than folk who play less games.
There is/was a graph showing all the data on the English Grading site.
They use a different rating system call grading which has only 3 figures.
But you can convert the numbers approx.
Involving OTB long games, we can look the number of games of the top players have played. Carlsen had approximately played 300 fide long games from 2000 to 2500, 300 more games to reach 2700, and 200 games to reach 2800. Then :
When 2000-2500, 1 game = + 1.7 Elo point
When 2500-2700, 1 game = + 0.7 Elo point
When 2700-2800, 1 game = + 0.5 Elo point
For one of the most rapidly improving players ever anyway.
Carlsen had approximately played 300 fide long games from 2000 to 2500, 300 more games to reach 2700, and 200 games to reach 2800.
True. If we look at Aronian's rating : 600 games from 2600 to 2800. Then : 1 game = 0.3 Elo point for him (half of Carlsen's progression speed, and he's the second best player in the world).
About Caruana and Karyakin (number 5 and number 9 in the world), 400 games from 2500 to 2700, then :
1 game = + 0.5 Elo point.
About Nakamura, 500 games from 2500 to 2700, then :
1 game = + 0.4 Elo point.