The number of levels in chess Consider a weak chess player who has just learned how the pieces move and who will lose to a Scholar's Mate. (We can say that he's male and call him "Mr. Newcomer".) Let's define Mr. Newcomer's level to be 0. Moreover, let's say that player A is a level above player B if A's expected score against B is 0.9 (for example, 9 wins and 1 losses in 10 games). Given these definitions, at which level would you find a player like Magnus Carlsen (who is currently the player with the rightest rating in the world)?

I have a feeling that you can give a quite exact answer to this question if you know Carlsen's rating (which is 2872 at the time of writng) and Mr. Newcomer's rating, and if understand the rating system better than I do...

The choice of 0.9 in the above definition of a "level" is pretty arbitrary; if you think a number like 0.8 or 0.7 makes more sense, just pick that number instead.
The rating system already kind of allows you to calculate this since you can convert any rating difference into an expected score using the formula: expected = 1/(1 + 10^(diff/400)) so a rating difference of 382 points should give you the .9 expected score.  

Therefore Magnus Carlsen at 2872 should score 90%  (which could be 9 wins and one loss out of 10 games or 8 wins and 2 draws) against a player rated 2490.  and that 2490 player should score the same against a player rated 2108 and so on.

So you could keep subtracting 382 from Carlsen's rating until you got to a rating of about 100 which is probably what the worst chess player could be.  if you subtract 7 times you get down to 198, so I guess you could say Carlsen is 7 of your levels above the weakes beginner.

Now a rating difference of 200 translates to about a .75 expected score  and that happens to be the rating difference that the USCF uses for different classes.  For example: Master = 2200 - 2399 /
Expert = 2000 - 2199 /
Class A =  1800 -1999 /
Class B - 1600 - 1799 /
Class c - 1400 - 1599 etc..

so you can kind of think of each class as a "level" like you describe where the middle of each class has about a .75 expected score against the middle of the next lower class. So you could say that using that criteria, Carlsen is about 14 "levels" or class above the weakest player.