Online turn-based chess is not the same as regular over-the-board chess. Therefore, it is only natural that your online rating would be at least somewhat different than your OTB one. I mean, on Turn-Based chess you get days to make your moves, and you even get to see your moves before you make them! If you don't do at least somewhat better when compared to regular timed games then you're doing something wrong.
But everyone gets the same advantage from the additional time then. The games might be of higher quality, agreed, but this won't manifest itself as higher ratings overall.
If I make better use of that additional time than others (and given my slow time-per-move stat it's arguable that I do compared to someone with a very quick time-per-move) then perhaps my rataing will be slightly higher than it might be in another pool, but it stands to reason then, that the individual who moves very quickly will have a lower rating than he might in another pool that has less opportunity for this type of disparity in usage patterns.
At the end of the day it just underscores the fact that you can't reliably compare between pools.
No, I understood you. That's what I was saying is incorrect. As long as those two players have played a reasonable number of games, those ratings will accurately predict the outcome of the game regardless of the size of the pool.
I am well aware that many chess players believe this last statement to be true.
I am not one of them.
Moreover,I wonder how many of them would continue to believe it,if they understood that the very fact that they believe it, is a serious impediment to the improvement of their chess skills.
Beyond the fact that you haven't shown the statement to be false, how does believing the ratings to be accurate a serious impediment to the improvement of chess skills?