Every so often, somebody asks about how ratings are calculated. This is a quick explanation of how the United States Chess Federation (USCF) does it.

First of all, a "rating" is a "Performance Average". USCF uses 2 formulas -- one for provisional ratings and one for established ratings. A provisional rating is given to a player who has played less than 20 rated games. If the player defeats an opponent, he gets the opponent's rating plus 400 points; if the player loses to an opponent, he gets the opponents rating minus 400 points; if the player draws with an opponent, he gets the opponents rating. (For calculation purposes, unrated opponents are considered by USCF to be rated 1000.) All of these numbers are added together and averaged, and that is the rating. Expressed mathematically, the formula is:

Where R is the rating, Y is the average rating of all opponents, W is wins, L is losses, G is the number of games played. [NOTE: Each draw is equal to half-a-win plus half-a-loss.]

Once a player has played his 20th rated game, his rating becomes "established", and is calculated on a logarithmic scale called an "expectancy curve". The basic formula looks simple enough:

R is the new (post-event) rating, P is the old (pre-event) rating, W is wins, W' is the "win expectancy", and K is a constant whose value depends on the player's rating (K=32 for 0-2099, K=24 for 2100-2399, K=16 for 2400-3000).

The formula for calculating the "win expectancy" is somewhat more complicated:

W' is the win expectancy, x is the player's rating, y is the opponent's rating. [NOTE: the little "^" indicates an exponent.]

For established ratings, the points gained (or lost) depends on the rating difference between the two players but cannot be less than 1 point, nor more than 32 for any given game. If a player rated 1600 plays against an opponent rated 1500, the "win expectancy" is 0.640 -- meaning, in the long run, the 1600 player should win about 64% of the time against the 1500 player. Let's say for the sake of argument that the 1600 player defeats the 1500 player. How many points would he gain? Plug the numbers into the formula:

So the answer would be 12 points gained by the 1600 player, 12 points lost by the 1500 player.

Please be advised that this is an extremely rough thumbnail sketch of the process. USCF uses some additional calculations and periodical adjustments for fine tuning purposes. (If you are curious about these, write to USCF.) Other chess organizations may utilize these basic formulas with or without modifications, and/or using a different base number for unrated players. Here at #ChessMasters Channel, our base number for unrated players is 1500.