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Elo rating calculator: expected score, K-factor and rating change for any sport

An Elo rating turns the gap between two ratings into an expected score — the average result the stronger side should achieve against the weaker — and then moves both ratings by how far the actual result beat or missed that expectation. Chess federations use it, and FIFA’s men’s world ranking is built on it.

Choose a method, enter the two ratings and the result, and the calculator shows each side’s expected score and new rating. Each method follows its publisher’s own rules: Elo’s formula as US Chess states it, FIDE’s table and K-factors, or FIFA’s weighting by the importance of the match.

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Calculate an Elo rating change

One game or match between two sides, rated before it was played.

Expected score 1 ÷ (1 + 10^(−difference ÷ 400)); the same K for both sides.

How far one result can move a rating. FIDE’s values are 40, 20 and 10.

Expected score, A0.240out of 1

Expected score, B
0.760
Change for A
+15.2
Change for B
−15.2
New rating, A
1515.2
New rating, B
1684.8

The formula

Expected score of A against B

EA = 1 ÷ (1 + 10−(RA − RB) ÷ 400)

New rating after the game

R′A = RA + K × (SA − EA)

S is the actual score — 1 for a win, ½ for a draw, 0 for a loss — and K sets how far a single result can move a rating. These are formulas (1) and (2) in Glickman and Jones’s account of the Elo system, and the “standard winning expectancy” and rating formula in US Chess’s published rating procedure. With one K for both sides, what one side gains the other loses.

The expected score is an average, not a win probability: a player expected to score 0.24 may get there through draws as well as wins. Glickman and Jones make the point that the formula is a probability only when every game has a winner.

Three published versions

How the three methods differ
MethodExpected scoreHow much a result countsRounding and limits
Elo’s formulaFormula, 400-point scaleK — US Chess works it out from the games a rating is based onNone in the formula itself
FIDE (chess)Table 8.1.2, in hundredthsK = 40, 20 or 10 by experience and ratingGap capped at 400 below 2650; change rounded, .5 away from zero
FIFA (SUM)Formula, 600-point scaleMatch importance I, from 5 to 60Shoot-out 0.75 / 0.5; no points lost in final-competition knockout rounds

The three convert the same rating gaps into similar, but not identical, expected scores:

Expected score of the higher-rated side, by rating gap
Rating gapElo’s formula (400)FIDE tableFIFA (600)
00.5000.500.500
500.5710.570.548
1000.6400.640.595
2000.7600.760.683
3000.8490.850.760
4000.9090.920.823
6000.9690.920.909

The FIDE column is for a player rated below 2650, so the 600-point gap counts as 400. On FIFA’s wider scale the same gap is worth less: 200 points is 0.683 rather than 0.760.

Worked examples

Elo’s formula. Glickman and Jones’s example pairs players rated 1500 and 1700: the 1500 player’s expected score is 0.240 — their “about 0.24”. If the 1500 player wins and K is 20, the change is 20 × (1 − 0.240) = +15.2, and the 1700 player loses the same.

FIDE. An illustrative game: a player rated 2100 with K = 20 draws with one rated 2300 with K = 10. The 200-point gap reads .24 and .76 from table 8.1.2, so the lower-rated player gains (0.5 − 0.24) × 20 = 5.2, rounded to +5, and the higher-rated player loses (0.5 − 0.76) × 10 = -2.6, rounded to −3. With different K values the two changes do not cancel.

FIFA. FIFA’s own example: a team on 1300 points beats a team on 1500 in a continental qualifier, where I = 25. Its expected result is 0.317, so it gains 25 × (1 − 0.317) = 17.07 points — the 17 FIFA quotes — and moves to 1317.07, while the other team drops to 1482.93.

How to read it, and its limits

  • Ratings are estimates built from past results. They trail behind changes in form, fitness or line-ups.
  • None of these formulas includes home advantage, and none says how likely a draw is — only the average score.
  • K is a choice. A high K reacts quickly to results and is noisy; a low K is stable and slow.
  • Each method here follows its publisher’s rules for one game. Real systems add more, such as FIDE’s rating periods and limits on K over many games.
  • An expected score describes what two ratings imply. It is not a forecast and not betting guidance.

For how simulators use ratings without letting them decide everything, read how sports simulation engines work.

40-0 MMA & Boxing Fight Draft weighs ratings, records, eras and weight classes when you draft, and its simulators settle each bout on style as well as ratings — so, just as an expected score below 1 implies, the higher-rated fighter does not always get the nod.

Reign Creative LLC makes 40-0 MMA & Boxing Fight Draft, so we have a commercial interest in it. It is not affiliated with, or endorsed by, any league, team, players’ association or athlete.

Good to know

Frequently asked questions

What is an expected score in Elo?
The average score a player is expected to make against an opponent, counting a win as 1, a draw as ½ and a loss as 0. Glickman and Jones’s example: a player rated 1500 against one rated 1700 has a long-run expected score of about 0.24. Where draws are possible it is not the same as a win probability.
What K-factor should I use?
The one your rating system sets. FIDE uses 40 for players new to its list and for juniors rated under 2300, 20 for players rated under 2400, and 10 once a player has reached 2400. US Chess works its K out from the number of games a rating is based on, and FIFA uses the importance of the match, from 5 to 60, in the same place. A larger K makes ratings move faster.
Why does FIDE use a table instead of the formula?
FIDE’s rating regulations convert the rating difference into an expected score with a published table, table 8.1.2, and count a difference of more than 400 points as 400 for players rated below 2650. The table and the formula give similar figures: a 200-point gap is .76 in the table and .760 by the formula.
How does FIFA’s ranking use Elo?
FIFA’s “SUM” method adds or subtracts I × (W − We) after each match, where I is the importance of the match, W the result and We the expected result on a 600-point scale. A penalty shoot-out counts as 0.75 for the winner and 0.5 for the loser, and teams do not lose points in the knockout rounds of a final competition. FIFA’s own example: a team on 1300 that beats a team on 1500 in a qualifier gains 17 points.
Is an expected score a prediction?
It is what two ratings imply about the average result, nothing more. Ratings lag behind form and injuries, and one game is a small sample. Use it to understand how a rating system responds to results — not to forecast a game, and not as betting guidance.

Sources

  1. Rating the chess rating system (Chance, 12(2), 21–28, 1999) — the Elo formula and its update (opens in a new tab)Mark E. Glickman and Albyn C. Jones — accessed
  2. The US Chess Rating System (2020) — standard winning expectancy and rating formula (opens in a new tab)US Chess Federation (Mark E. Glickman and Thomas Doan) — accessed
  3. FIDE Rating Regulations effective from 1 March 2024 — table 8.1.2 and rules 8.3.1–8.3.4 (opens in a new tab)FIDE Handbook — accessed
  4. Revision of the FIFA / Coca-Cola World Ranking — the SUM formula (opens in a new tab)FIFA — accessed
  5. Men’s ranking procedure (opens in a new tab)FIFA (Inside FIFA) — accessed

League, governing-body and team names appear here only to identify the formulas and rules being used. Reign Creative LLC's sports games and calculators are not affiliated with, endorsed by or licensed by any league, team, players' association or athlete, and all names and marks belong to their owners.

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