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Chess Ratings
The Elo rating is a relative measure of a player's skill. What does Elo stand for?

'I'm 1875. What's your rating?' is a typical greeting from one chessplayer meeting another for the first time. The other player, rated 1750, knows immediately that a game between them will be a tough battle. A draw will be a satisfactory result and a win will be a small upset.

What is a rating and how is it calculated? Perhaps most importantly, a rating is only meaningful relative to other ratings. In July 1989, when Garry Kasparov became the first player to break the 2800 barrier, his accomplishment was by comparison with all other ratings, current and historical. Vladimir Kramnik's FIDE rating chart, shown at the bottom of the page, follows the evolution of his rating after winning the Braingames World Championship from Kasparov in 2000 through his successful defense of the unified title against Veselin Topalov in 2006.

Vladimir Kramnik's Rating Chart (FIDE)

The rating system may seem mysterious, but it is grounded in statistical theory. Obviously, two players with the same rating should have an equal chance of winning against each other. Less obviously, the same rating difference implies the same chance of winning. A player rated 2400 playing against a player rated 2200 has the same chance of winning as a 1400 against a 1200. The rating difference is 200 points in both cases.

The most widely used rating system is known as the Elo system. Arpad E. Elo, born 1903 in Hungary, emigrated to the United States at age 10. From 1935 to 1965, he was professor of physics and astronomy at Marquette University.

From 1935 to 1937, Elo was administrator of the American Chess Federation which merged in 1939 with the National Chess Federation to become the USCF. He was nine times champion or co-champion of Wisconsin.

In the preface to his book The Rating of Chessplayers, Past and Present (Batsford 1978), Elo wrote

In 1959 the late Jerry Spann, then president of the United States Chess Federation (USCF) named a committee to review the federation's rating system and to revise and improve its technical and administrative features. It fell upon the writer, as chairman, to examine the basic theory and rationale of the rating systems of the chess world and the sports world in general.
The Elo system was adopted by the USCF in 1960, and by FIDE in 1970. Elo served as Chairman of the USCF Rating Committee from 1959 to 1976. He was inducted into U.S. Chess Hall of fame in 1988 and died in 1992.

From The Rating of Chessplayers:-

Few chessplayers are totally objective about their positions on the board, and even fewer can be objective about their personal capacities and ratings. Most of them believe they are playing "in form" only when far above normal form, and they tend to forget that an outstanding tournament success is just as likely the result of off form performances by opponents as superior play by themselves. There is truth in the paradox that "every chessplayer believes himself better than his equal".

The chart below shows a recent ratings distribution for USCF Non-Scholastic Members. (Source: USCF 2002 Ratings Distribution).


How are ratings calculated? Already in 1959, the USCF rating system arbitrarily used 2000 as the upper level for strong club players and 200 point divisions to assign players to classes. Elo kept these measures because they were 'steeped in tradition'.

The table at the bottom of this page relates expected game results to rating differences. The P column is the expected Percentage for the result of a single game. The D column is the rating Difference corresponding to that expected result.

For example, two players with the same rating (D=0) each have a 50% (P=.50) chance of winning a game. Similarly, a player with a rating 100 points greater than an opponent (D=102 is the closest value in the table) has a 64% (P=.64) chance of winning a game.

Let's say you score +3-2=1 (three wins, two losses and a draw) against opposition with an average rating of 1500. Your score is 3.5-2.5, for a percentage of 58% (P=0.58). The value for P=0.58 in the table corresponds to D=57. Your performance is calculated as 1500 + 57 = 1557. If you had achieved the same score against opponents with an average rating of 2000, your performance would be 2057.

This method is used to calculate an initial rating for a previously unrated player. The more games used in the calculation, the more accurate the initial rating will be. Established players are rated using the following formula.

Rn = Ro + K (W - We)
      Rn New rating
  Ro Old rating
  K Value of a single game
  W Score; 1.0 for a win, 0.5 for a draw
  We Expected score based on Ro

The formula says that after an event has finished, a player's new rating is calculated from the old rating adjusted by the result of the event. The adjustment is the difference between the player's actual result and the expected result, which is based on the old rating.

The difference is multiplied by a coefficient ('K'), which is a number between 10 and 40. A lower coefficient gives more weight to previous events and changes the rating at a slower rate. A higher coefficient gives more weight to the most recent events and changes the rating faster.

Ratings allow for more competitive subdivisions of players. For example, the USCF class system is structured as follows.

Range Class
2400- Senior Master
2200-2399 Master
2000-2199 Expert
1800-1999 Class A
1600-1799 Class B
1400-1599 Class C
1200-1399 Class D
1000-1199 Class E

This class system permits tournaments and prizes restricted to players of similar ability. A Class E player, competing in a Class E event, has just as much chance of winning the event as the world's best players have of winning Linares.

No system is perfect, and the there are some problems with the rating system. Rating deflation is a natural phenomenon caused by young improving players entering the rating pool and old stable players leaving the pool. Rating manipulation happens when unscrupulous organizers submit fraudulent reports to a rating agency.

These problems are a small price to pay for the great benefits that Elo's rating system has provided to the chess world. His induction into the Chess Hall of Fame was an appropriate expression of his great contribution to chess.


Rating expectancies vs. differences
    .99 677     .79 230     .59 65     .39 -80     .19 -251
    .98 589     .78 220     .58 57     .38 -87     .18 -262
    .97 538     .77 211     .57 50     .37 -95     .17 -273
    .96 501     .76 202     .56 43     .36 -102     .16 -284
    .95 470     .75 193     .55 36     .35 -110     .15 -296
    .94 444     .74 184     .54 29     .34 -117     .14 -309
    .93 422     .73 175     .53 21     .33 -125     .13 -322
    .92 401     .72 166     .52 14     .32 -133     .12 -335
    .91 383     .71 158     .51 7     .31 -141     .11 -351
    .90 366     .70 149     .50 0     .30 -149     .10 -366
    .89 351     .69 141     .49 -7     .29 -158     .09 -383
    .88 335     .68 133     .48 -14     .28 -166     .08 -401
    .87 322     .67 125     .47 -21     .27 -175     .07 -422
    .86 309     .66 117     .46 -29     .26 -184     .06 -444
    .85 296     .65 110     .45 -36     .25 -193     .05 -470
    .84 284     .64 102     .44 -43     .24 -202     .04 -501
    .83 273     .63 95     .43 -50     .23 -211     .03 -538
    .82 262     .62 87     .42 -57     .22 -220     .02 -589
    .81 251     .61 80     .41 -65     .21 -230     .01 -677
    .80 240     .60 72     .40 -72     .20 -240

 Related Resources
• Tools to Calculate Your Chess Rating
• Rating Calculator
• Step By Step Calculation