Elo Rating Calculator (expected score & new rating)

Understanding the Elo rating calculator — concept and principle

The Elo system rates players by comparing each game's actual result with the result expected from the two players' ratings. This calculator gives your expected score against an opponent and your new rating after a win, draw or loss. It began in chess and is now used in many games and sports.

How it is calculated

• Expected score E = 1 ÷ (1 + 10^((R_b − R_a) ÷ 400)), where R_a is your rating and R_b the opponent's.
• Change = K × (S − E), where S is 1 for a win, 0.5 for a draw and 0 for a loss.
• New rating = R_a + K × (S − E).

K is the development coefficient: the maximum points that one game can move a rating. Under FIDE rules it is 40 for new players, 20 for most rated players and 10 for players who have reached 2400.

The principle

Arpad Elo modelled performance as a random variable around a player's true strength, so a rating difference translates into an expected share of points. In the logistic form used today, every 400 points of difference multiplies the odds by 10: a player rated 400 above an opponent is expected to score about 91%. After each game the rating moves in proportion to the surprise S − E. Beating a stronger player gives a big gain, beating a weaker one a small gain, and if results match expectations the rating stays stable. With equal K, one player's gain is the other's loss.

Worked example

1. Inputs: your rating 1500, opponent 1600, result win, K = 20.
2. E = 1 ÷ (1 + 10^(100 ÷ 400)) = 1 ÷ (1 + 1.7783) ≈ 0.3599, i.e. 35.99%.
3. Change = 20 × (1 − 0.3599) ≈ +12.8013.
4. New rating = 1500 + 12.8013 ≈ 1,512.8013.
5. With a draw the change would be about +2.80, and with a loss about −7.20.

How to read the result

The expected score is not quite a win probability: it is the expected points per game, counting a draw as half a point. An expected score of 36% could mean a 25% chance to win and a 22% chance to draw, for example. A large positive change means you beat expectations; consistent over-performance will raise your rating until expected and actual results agree. Official ratings are often calculated per rating period, so they may differ slightly.

Common mistakes

• Swapping R_a and R_b in the exponent, which reverses the expected score.
• Using the wrong K-factor for your federation or rating level.
• Comparing ratings from different pools, such as a national rating and an online site's rating.

Frequently asked questions

Why does my rating go up only a little when I beat a weaker player?

Because you were already expected to win; the change is K times the difference between the result and the expectation.

Does a draw against a stronger player raise my rating?

Yes. A draw scores 0.5, which is above your expected score if the opponent is higher rated.

What rating do beginners start with?

It depends on the organisation; FIDE gives a first rating only after a player has scored against rated opponents.

Sources: Arpad E. Elo, The Rating of Chessplayers, Past and Present, Arco Publishing, 1978 · FIDE Handbook, FIDE Rating Regulations