Elo ratings explained

What an Elo rating is

An Elo rating is a relative rating that is updated after every result. It was devised by physicist Arpad Elo for chess, replacing an earlier and cruder ranking method, and its simplicity has since seen it adapted far beyond the chessboard — table tennis, tennis ladders, online games, and informal power rankings across many sports. The core idea never changes: your rating only means something in comparison with the other players it was earned against. There is no absolute scale; a number is high or low only relative to the pool.

Two quantities do all the work. First, before a game, the system computes an expected score for each player from the gap in ratings. Then, after the game, it nudges each rating toward the actual result — by more when the result was surprising, by less when it was expected.

The formula

The expected score for player A against player B depends only on the difference between their ratings:

After the game, A’s rating is updated by comparing the actual score with the expected one:

R_A, R_B

the current ratings of players A and B

E_A

A’s expected score, a number between 0 and 1

S_A

the actual result for A: 1 for a win, 0.5 for a draw, 0 for a loss

K

the K-factor, which sets how much one result can move the rating

400

the scale constant: a 400-point gap makes the stronger player 10× more likely to score

Because the two expected scores always sum to 1, whatever points A gains, B loses — the system is zero-sum for a single game.

Worked example

Take player A rated 1600 and player B rated 1400, with a K-factor of 32. A wins the game.

1. The rating gap is R_B − R_A = 1400 − 1600 = −200.
2. Expected score for A: E_A = 1 / (1 + 10^(−200/400)) = 1 / (1 + 10^(−0.5)) = 1 / (1 + 0.3162) ≈ 0.76.
3. A won, so S_A = 1. New rating: R'_A = 1600 + 32 × (1 − 0.76) = 1600 + 32 × 0.24 ≈ 1608.
4. B lost the same swing: R'_B = 1400 + 32 × (0 − 0.24) ≈ 1400 − 8 = 1392.

A gained about 8 points and B lost about 8 points. The move is small because the favourite was expected to win — had the 1400 upset the 1600, the swing would have been much larger (roughly 32 × 0.76 ≈ 24 points).

Rating gap and expected score

The expected score depends only on the gap, so it is worth reading it off directly. The table shows the expected score of the higher-rated player for a range of rating gaps.

A gap of 0 gives an even 0.50; every 400 points multiplies the odds by ten, which is why the expected score climbs steeply and then flattens as it approaches 1.

The K-factor

The K-factor is the single dial that governs volatility. A bigger K means faster adjustment: each result moves the rating more, so ratings track recent form quickly but jump around. A smaller K makes ratings sluggish but stable. Different leagues and governing bodies choose different values — often a higher K for new or provisional players and a lower K once a rating is established — and they also set their own starting ratings (1000, 1200 and 1500 are all common). There is no universal correct value; it is a design choice for the pool.

Is an Elo expected score a prediction?

This is the point most worth being clear about. An Elo expected score is how the system weights a result so it knows how far to move each rating — it is a bookkeeping device internal to the algorithm. It is not a forecast of who will win a particular match, and you should not treat it as a betting probability. The number knows only the two ratings; it has never seen the players.

What an Elo rating does not tell you

Frequently asked questions

Related tools and reading

Sources

• Elo, A.E. (1978), The Rating of Chess Players, Past and Present — the original definition of the expected-score and update formulas. Checked 24 Aug 2026.