Calculate your new Elo rating after any game. Enter your rating, your opponent's rating, and the result to get expected score, rating change, performance rating, and percentile.
Game Game Details
New Rating
1500
Enter a game result above to calculate
Expected Score
0.500
Rating Change
+20
New Rating
1520
Perf. Rating
1900
K-Factor
40
Percentile
~50th
E = 1/(1+10^((Opp-My)/400))
ΔR = K × (S − E)
The expected score E ranges from 0 (certain loss) to 1 (certain win), with 0.5 at equal ratings. The S-curve below shows how your expected score changes as the rating gap widens. Values update automatically from Tab 1.
Rating Difference (Your rating - Opponent's)
0
Expected Score
0.500
Score %
50.0%
Assessment
Even Match
Rating Diff (My−Opp)
Expected Score
Score %
Assessment
Simulate a series of games and track your rating trajectory. Add each game result to see how your rating evolves across a session or tournament.
+ Add Game
Syncs with Calculator tab. Changing it resets all games.
Input your current Elo rating and your opponent's Elo rating. Select the game result: Win, Draw, or Loss. The calculator computes the expected score (your win probability based on the rating difference) and the rating change for each outcome.
2
Select or Auto-Detect the K-Factor
The K-factor controls how much ratings change per game. Auto mode applies FIDE rules: K=40 for new players (rating <2100), K=20 for established players (2100–2400), K=10 for elite players (2400+). Select Manual to override with a custom K for non-FIDE systems (USCF, online platforms, team sports).
3
Review the Rating Change
The hero display shows your new rating and the change (ΔR = K × (S − E), where S is your score and E is your expected score). Stat cards show the expected score probability, performance rating for this game, and your estimated percentile in the rated player population.
4
Track a Tournament in the Tournament Simulator
Switch to the Tournament tab to simulate a full tournament rating path. Enter a starting rating, add game results one by one (each with an opponent's rating), and track cumulative rating changes, win/loss/draw record, and final rating across the full event.
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Reference
Formula & Methodology
3 formulas▸
Expected Score
E = 1 / (1 + 10^((R_opp − R_me) / 400))
E is the expected score (win probability) for the player with rating R_me against an opponent rated R_opp. The formula uses base-10 logarithm with a scale factor of 400, chosen so that a 400-point rating difference corresponds to a 10:1 odds ratio (E = 0.909). A draw is worth 0.5 points. If E = 0.75, the player is expected to score 0.75 points (win 75% and draw 25%, or some combination).
Rating Change
ΔR = K × (S − E) · R_new = R_old + ΔR
K is the K-factor (sensitivity), S is the actual score (1 = win, 0.5 = draw, 0 = loss), and E is the expected score. When you beat a much higher-rated player (upset), S=1 and E is small, so ΔR is large and positive. When you lose to a lower-rated player, S=0 and E is large, so ΔR is large and negative. The K-factor scales the magnitude: K=40 produces larger swings (appropriate for new players whose true strength is unknown); K=10 produces smaller swings (appropriate for elite players whose ratings are well-established).
The single-game performance rating estimates the rating that would be consistent with the game result against this opponent. Beating a 1500-rated player implies a performance of 1900; drawing implies 1500; losing implies 1100. This approximation follows from the expected-score formula: if your performance rating exactly matched R_opp ± 400, you would have expected score 0.909 or 0.091. Over a multi-game tournament, performance rating is averaged across all games.
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Glossary
Key Terms Explained
6 terms▸
Elo RatingA numerical system for rating the relative skill of players in zero-sum games, developed by physicist Arpad Elo in the 1960s for chess. The expected score formula E = 1/(1+10^(ΔR/400)) maps rating differences to win probabilities using a logistic function. Ratings are updated after each game based on the difference between actual and expected score. Elo is used in chess, Go, checkers, and has been adapted to team sports (FIFA world rankings), video games (online multiplayer matchmaking), and academic peer review.
K-FactorThe maximum rating points that can be gained or lost in a single game. Higher K = more volatile ratings that respond quickly to performance; lower K = more stable ratings. FIDE uses K=40 for new players (fewer than 30 games or rating below 2300), K=20 for established players, K=10 for elite players (2400+). Online chess platforms like Chess.com and Lichess often use much higher K-factors (32–64) for rapid convergence.
Expected Score (E)The probability-weighted expected outcome of a game based on the rating difference: E = 1/(1+10^((R_opp−R)/400)). E ranges from 0 (certain loss) to 1 (certain win). E = 0.5 when ratings are equal. E = 0.909 when the player is rated 400 points above the opponent. E is a probability in a single game, but is also interpreted as the expected fraction of points in a multi-game match.
FIDE RatingThe international chess rating published by the Fédération Internationale des Échecs (FIDE), the global chess governing body. FIDE ratings use the standard Elo formula with K-factor rules. Approximate FIDE rating levels: Grandmaster (2500+), International Master (2400+), FIDE Master (2300+), National Master (~2200), Expert (~2000), Class A (~1800), Class B (~1600). The current World Chess Champion typically holds a rating of 2800+.
Performance RatingA retroactive estimate of the rating a player performed at during a specific event or game. Computed by solving for the rating that would have given an expected score equal to the actual result. Over a tournament, performance rating = average opponent rating + 400 × (wins − losses) / games. It measures how well the player actually played, independent of their starting rating.
Rating Deflation/InflationThe phenomenon where average Elo ratings in a pool trend lower (deflation) or higher (inflation) over time. Deflation occurs when new players enter the pool at an inflated rating, then their rating drops as the pool absorbs the points. Inflation occurs when strong players retire with high ratings they no longer earn. FIDE and major platforms periodically recalibrate initial ratings and K-factors to counteract drift.
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Scenarios
Real-World Examples
2 worked examples▸
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Upset Win: Low-Rated vs. High-Rated
Rating Upset
My Rating 1200Opponent Rating 1800Result WinK-Factor 40 (auto, <2100)
E = 1/(1+10^((1800−1200)/400)) = 1/(1+10^1.5) = 1/(1+31.62) = 0.0307. S = 1 (win). ΔR = 40×(1−0.0307) = 40×0.969 = +38.8 ≈ +39 points. New rating: 1200 + 39 = 1239. A 600-point rating difference gives only a 3% win probability — making this a huge upset. The winner gains nearly a full K (39/40 = 97.5% of max), while the opponent loses the same 39 points. Performance rating for this game ≈ 1800 + 400 = 2200.
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Equal Opponents: Draw
Equal Match
My Rating 1500Opponent Rating 1500Result DrawK-Factor 40 (auto)
E = 1/(1+10^0) = 1/2 = 0.500. S = 0.5 (draw). ΔR = 40×(0.5−0.5) = 0. Rating unchanged. When ratings are equal, the expected score is exactly 0.5 and a draw produces no rating change for either player — the most fair outcome. A win would yield +20 points; a loss −20 points. Equal-rated players in a 100-game series would see their ratings converge to their true skill level as wins and losses balance out.
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Deep Dive
Elo Rating: How Chess Rankings Work and Why the System Is Used Everywhere
The Elo rating system, developed by physics professor Arpad Elo for the US Chess Federation in 1960, solved a fundamental problem in competitive ranking: how to measure relative skill between players who do not all play each other. By modeling win probability as a logistic function of rating difference and updating ratings after every game based on actual vs. expected outcomes, Elo created a system that continuously adjusts to new information without requiring a single round-robin tournament. The elegance and accuracy of Elo's original design has made it the foundation for rating systems in chess, Go, online gaming, team sports, and even academic peer review.
The Mathematics of Expected Score
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The expected score formula E = 1/(1 + 10^((R_opp − R_me)/400)) maps rating differences to win probabilities using a logistic curve. The scale factor of 400 is a convention: Arpad Elo chose it so that a 400-point rating difference corresponds to a 10:1 expected score ratio — meaning the higher-rated player should score approximately 10 points for every 1 point the lower-rated player scores. At 0 rating difference, E = 0.5 (50% win probability). At +200 points, E ≈ 0.76; at +400, E ≈ 0.91; at +600, E ≈ 0.97. The logistic shape reflects the empirical observation that skill differences in chess produce approximately this win probability distribution. The original Elo formula used a normal distribution rather than logistic, but the logistic version (adopted by FIDE in 1978) is mathematically equivalent for most purposes and easier to compute.
K-Factor: Stability vs. Responsiveness
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The K-factor controls the volatility of the rating system. A high K allows ratings to change rapidly, which is appropriate when a player's true strength is uncertain (new players, rapidly improving juniors) — the rating system needs to 'find' the right level quickly. A low K produces stable ratings that change slowly, appropriate for established elite players whose true strength is known with high confidence. FIDE uses three K-factor tiers: K=40 for players in their first 30 games or below 2300, K=20 for established players (2100–2400), K=10 for elite players above 2400 with lifetime game experience. Online chess platforms like Chess.com and Lichess often use higher K-factors (32–64) because online games are faster and more numerous, allowing faster convergence to true strength. The consequence of a high K-factor is that a single upset can move ratings by 30–40 points; a low K means even a major upset moves ratings by only 8–10 points. Both are correct for their intended use cases.
Elo Beyond Chess: Team Sports and Online Games
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Elo's elegance — requiring only win/draw/loss outcomes and current ratings — made it ideal for adoption beyond chess. FIFA uses an Elo-variant for national team rankings that assigns match weights by tournament importance and adjusts for goal difference. The NFL has used Elo-based power ratings for forecasting since the 1980s, and FiveThirtyEight's NFL Elo model achieved notable prediction accuracy. Video game multiplayer matchmaking systems in League of Legends, Dota 2, and most competitive games use Elo or TrueSkill (a Bayesian extension that handles team games and accounts for uncertainty). The key insight that made Elo extensible: any zero-sum game with binary outcomes (win/loss) or ordered outcomes (win/draw/loss) can be modeled with the same expected-score framework, with the K-factor and scale constant tuned to the specific game's competitive environment. The main limitation of classic Elo for team sports is that it treats teams as single players — individual performance variance is averaged out, and roster changes require manual adjustment.
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Questions
Frequently Asked Questions
6 questions▸
What is the Elo rating system?+
A method for calculating the relative skill of players in zero-sum games, developed by Arpad Elo for chess in the 1960s. Each player has a numerical rating; after each game, ratings are updated based on the difference between actual score (1=win, 0.5=draw, 0=loss) and expected score (derived from the rating difference). Higher ratings = stronger players. FIDE Grandmasters typically hold ratings of 2500+.
How is the expected score calculated in Elo?+
E = 1 / (1 + 10^((R_opp − R_me) / 400)). At equal ratings: E = 0.5 (50/50). At 200-point advantage: E ≈ 0.76. At 400-point advantage: E ≈ 0.91. At 600-point advantage: E ≈ 0.97. The expected score is your probability of winning the game.
What is the K-factor and how does it affect rating changes?+
K controls how many points can be gained or lost per game. ΔR = K × (S − E), so maximum gain/loss per game = K (if you win when expected score = 0, or lose when expected score = 1). FIDE K-factors: K=40 for new players (<2100), K=20 for established players (2100–2400), K=10 for elite players (2400+). Higher K = faster rating change but more volatility.
How much do I gain or lose for beating a higher-rated player?+
ΔR = K × (1 − E). If you're rated 1200 and beat a 1800-rated player: E ≈ 0.031, ΔR = 40 × (1 − 0.031) = +38.8 points. The bigger the upset (lower E), the more points you gain. Losing to a much lower-rated player: ΔR = K × (0 − E) = −K × E, which is large and negative.
What Elo rating is considered good in chess?+
Approximate FIDE levels: Beginner: 400–800. Novice: 800–1200. Class B: 1400–1600. Class A: 1600–1800. Expert: 1800–2000. National Master: ~2000–2200. FIDE Master: 2300+. International Master: 2400+. Grandmaster: 2500+. World Champion: 2800+. Average club player is approximately 1400–1600. Online ratings (Chess.com, Lichess) are typically 200–400 points higher than FIDE ratings due to different player pools and K-factor settings.
Is Elo used in sports other than chess?+
Yes. FIFA uses an Elo-variant for national team rankings. NFL, NBA, and MLB sabermetric sites use Elo-based power ratings for win probability models. Online multiplayer games (League of Legends, Dota 2, Starcraft) use Elo or Elo-derived systems (Microsoft TrueSkill) for matchmaking. Even academic paper and grant review systems have used Elo for ranking competing proposals.
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