The application - and even more the deduction - of the rating scheme requires a lot of mathematics. But don't worry: as a player you don't need to understand this scheme - PlayChess does all the math automatically for you.
The following "mathematical treatise" is given here just to
Rating Scheme for Dummies
If you are just a humble human being, not a god of math, here are the essential ideas of the rating system:
The ratings of experienced players (more than 10 games) are calculated by a more complicated formula. There, the simple "arithmetic mean" formula above is not sufficient, because the rating of these players would change VERY slowly (imagine players with more than 100 games!).
The Complete Rating Scheme
Principle. The rating scheme is a numerical scheme, in which percentage results can be exchanged into rating differences and rating differences into percentage performance probabilities. The basis of the scheme is the normal probability distribution.
Provisional and Established Ratings. A player has an established rating, if he/she has finished at least 10 games, otherwise his/her rating is called provisional.
Cut Off. To keep the influence of largely different ratings small for the preliminary (and established) ratings, a rating cut-off is used. Anytime the difference is larger than 400 points, the opponents rating is treated as being 400 points higher/lower than the players rating.
The performance probability is calculated by the formula
P(D) = 1/(1+10^(-D/400)) (1)
P(D) is the performance probability
D is the difference of the ratings of the two players.
The expected rating changes based on the percentage result is given by
D(p) = -400 * log10((1-p)/p) (2)
D(p) is the expected rating changeThe percentage result is calculated by
p is the percentage result of the player
and D(0) = -800 and D(1)= 800
p = (2*W+D)/(2*N) (3)
p is the percentage result
W is the number of wins
D is the number of draws
N is the number of finished games
For the calculation of the ratings, the opponents "Tournament Entry Ratings", which are valid on the day of the rating run, are used.
Calculation of Established Ratings
For each finished game the rating change is calculated by
dR = k*(W-We) (4)
dR Rating change
W True game result (win 1, draw 1/2)
We Expected result
k Development coefficient
The expected result We is calculated using formula (1) with the rating difference of the two opponents. The development coefficient is a stabilisation factor and is given by
k = r*p (5)
The next rating is calculated by
Rn = Ro + SUM(dR) (6)
Rn new rating of the player
Ro old rating of the player
SUM(dR) Sum of all rating changes as calculated in (4) for each game
Calculation of Provisional Ratings
The Provisional Rating is calculated by
Rp = Rc + D(p)*F (7)
Rp Provisional Rating
Rc Average of the opponents tournament entry ratings
D(p) Expected rating change based on (2)
F A Correction factor given by F = -2*p*p + 2*p + 0.5