The normal and logistic distributions are, in a way, arbitrary points in a spectrum of distributions which would work well. In practice, both of these distributions work very well for a number of different games. Method for calculating the relative skill levels of players in zero-sum games such as chess Arpad Elo, the inventor of the Elo rating system Pairwise comparisons form the basis of the Elo rating methodology. [20] Elo made references to the papers of Good, [21] David, [22] Trawinski and David, [23] and Buhlman and Huber. [24] Mathematical details [ edit ] A further assumption is necessary because chess performance in the above sense is still not measurable. One cannot look at a sequence of moves and derive a number to represent that player's skill. Performance can only be inferred from wins, draws and losses. Therefore, if a player wins a game, they are assumed to have performed at a higher level than their opponent for that game. Conversely, if the player loses, they are assumed to have performed at a lower level. If the game is a draw, the two players are assumed to have performed at nearly the same level. Elo's central assumption was that the chess performance of each player in each game is a normally distributed random variable. Although a player might perform significantly better or worse from one game to the next, Elo assumed that the mean value of the performances of any given player changes only slowly over time. Elo thought of a player's true skill as the mean of that player's performance random variable.

For top players, the most important rating is their FIDE rating. FIDE has issued the following lists: Performance rating = 1000 + 400 × ( 0 ) 1 = 1000 {\displaystyle {\text{Performance rating}}={\frac {1000+400\times (0)}{1}}=1000} The following analysis of the July 2015 FIDE rating list gives a rough impression of what a given FIDE rating means in terms of world ranking: w + 400 + x + 400 + y − 400 + z − 400 4 w + x + y + z + 400 ( 2 ) − 400 ( 2 ) 4 {\displaystyle {\begin{aligned}&{\frac {w+400+x+400+y-400+z-400}{4}}\\[6pt]&{\frac {w+x+y+z+400(2)-400(2)}{4}}\end{aligned}}} The highest ever FIDE rating was 2882, which Magnus Carlsen had on the May 2014 list. A list of the highest-rated players ever is at Comparison of top chess players throughout history.

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The K-factor, in the USCF rating system, can be estimated by dividing 800 by the effective number of games a player's rating is based on ( N e) plus the number of games the player completed in a tournament ( m). [19] K = 800 N e + m {\displaystyle K={\frac {800}{N_{e}+m}}\,} Rating floors [ edit ] players had an active rating between 2400 and 2499, most of whom had either the International Master or the International Grandmaster title. A F = min ⁡ { 100 + 4 N W + 2 N D + N R , 150 } {\displaystyle AF=\operatorname {min} \{100+4N_{W}+2N_{D}+N_{R},150\}} From July 2000 to July 2009, four lists a year were issued, at the start of January, April, July and October.

Performance rating = 2000 + 400 × ( 2 ) 2 = 1400 {\displaystyle {\text{Performance rating}}={\frac {2000+400\times (2)}{2}}=1400} Performance rating = 1000 + 400 × ( 1 ) 1 = 1400 {\displaystyle {\text{Performance rating}}={\frac {1000+400\times (1)}{1}}=1400} The development of the Percentage Expectancy Table (table 2.11) is described in more detail by Elo as follows: [13] players had an active rating between 2600 and 2699, all of whom had the International Grandmaster title. From July 2009 to July 2012, six lists a year were issued, at the start of January, March, May, July, September and November.where N W {\displaystyle N_{W}} is the number of rated games won, N D {\displaystyle N_{D}} is the number of rated games drawn, and N R {\displaystyle N_{R}} is the number of events in which the player completed three or more rated games.

players had an active rating between 2500 and 2599, most of whom had the International Grandmaster title. players had an active rating in the range 2300 to 2399, which is usually associated with the FIDE Master title. The Elo [a] rating system is a method for calculating the relative skill levels of players in zero-sum games such as chess. It is named after its creator Arpad Elo, a Hungarian-American physics professor.Performance rating or special rating is a hypothetical rating that would result from the games of a single event only. Some chess organizations [14] :p. 8 use the "algorithm of 400" to calculate performance rating. According to this algorithm, performance rating for an event is calculated in the following way: A statistical endeavor, by contrast, uses a model that relates the game results to underlying variables representing the ability of each player. The phrase "Elo rating" is often used to mean a player's chess rating as calculated by FIDE. However, this usage may be confusing or misleading because Elo's general ideas have been adopted by many organizations, including the USCF (before FIDE), many other national chess federations, the short-lived Professional Chess Association (PCA), and online chess servers including the Internet Chess Club (ICC), Free Internet Chess Server (FICS), Lichess, Chess.com, and Yahoo! Games. Each organization has a unique implementation, and none of them follows Elo's original suggestions precisely.