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Elo Rating Calculator

Computes the expected score of two players from their Elo ratings with the logistic formula E = 1/(1 + 10^((R_b − R_a)/400)) and the rating change of each after a win, draw or loss for a given K-factor.

When to use

You want the win expectancy between two rated players or the new ratings after a result, for chess or any Elo-based ranking (games, sports, matchmaking).

Do not use when: The rating system uses a different scale or rating deviations (Glicko, TrueSkill), or you need ratings after a whole tournament with FIDE's rounding and 400-point rule. Mathematics only; not gambling advice.

Formula

E_a = 1 / (1 + 10^((rating_b − rating_a) / 400)); E_b = 1 − E_a; new_rating_a = rating_a + k_factor × (S_a − E_a); new_rating_b = rating_b + k_factor × (S_b − E_b); S = 1 for a win, 0.5 for a draw, 0 for a loss

Standard logistic Elo formula (Elo 1978) in which a 400-point difference gives the stronger player an expected score of about 0.91. FIDE reads E from a table and counts differences above 400 points as 400; both differ from this formula by at most a few hundredths of a rating point per game. Probability mathematics only; not gambling advice.

Inputs

ParameterTypeUnitRequiredDescription
rating_anumberyesCurrent Elo rating of player A. Range: ≥ 0, ≤ 4000
rating_bnumberyesCurrent Elo rating of player B (the opponent). Range: ≥ 0, ≤ 4000
resultenum: a_wins | draw | b_winsyesOutcome of the game from player A's point of view.
k_factornumberdefault 32Maximum rating change per game. FIDE uses 40 (new players and juniors), 20 (established) and 10 (rated 2400+); 32 is common in club and online systems. Range: ≥ 1, ≤ 200

Outputs

OutputTypeUnitDescription
expected_score_anumberE_a = 1/(1 + 10^((rating_b − rating_a)/400)).
expected_score_bnumberE_b = 1 − E_a.
new_rating_anumberrating_a + k_factor × (S_a − E_a) with S_a = 1, 0.5 or 0.
new_rating_bnumberrating_b + k_factor × (S_b − E_b).
change_anumberk_factor × (S_a − E_a).
change_bnumberk_factor × (S_b − E_b) = −change_a.
win_probability_a_percentnumber%E_a × 100: A's expected score as a percentage (the win probability when draws are impossible).

Example

1500 beats 1600, K = 32: {"rating_a":1500,"rating_b":1600,"result":"a_wins","k_factor":32} → {"expected_score_a":0.3599,"expected_score_b":0.6401,"new_rating_a":1520.48,"new_rating_b":1579.52,"change_a":20.48,"change_b":-20.48,"win_probability_a_percent":35.99}

2000 draws with 1800, K = 20: {"rating_a":2000,"rating_b":1800,"result":"draw","k_factor":20} → {"expected_score_a":0.7597,"change_a":-5.19,"new_rating_a":1994.81,"new_rating_b":1805.19}

GET https://tttkmbb.com/api/v1/calculate/elo-rating?rating_a=1500&rating_b=1600&result=a_wins&k_factor=32

Machine access

Sources

FAQ

Which K-factor should I use?

The one your rating system prescribes: FIDE 40/20/10 by experience and rating, USCF a rating-dependent formula, many online chess and game servers 16–32. A larger K makes ratings move faster.

Does the expected score equal the win probability?

Only when draws cannot happen. In chess E_a = P(win) + ½·P(draw), so a 0.36 expected score can be, for example, 25 % wins and 22 % draws.

Why is FIDE's published change slightly different?

FIDE reads the expected score from a table rounded to two decimals, caps rating differences at 400 points and rounds the sum of a tournament's changes, so individual games can differ by a fraction of a point.

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