When people ask about the ratings on Chess.com, they get sent to an article with some really intimidating math formulas. I understand these formulas and their implications, but that's because I have a master's degree in statistics. I thought it might be helpful to try and give a description of chess ratings without intimidating formulas, so that you don't have to go back to graduate school to understand them. Mind you, there is still a lot of information here (and even one formula). But hopefully it's a bit more understandable.

Expected Score

The first part of the rating system is the expected score. The expected score is calculated based on the ratings of the two players, and modified by their ratings deviations (more on those later), using one of those intimidating formulas. A simplified version of the formula is:

1 / (1 + 10 ^ (-1 * ((RW - RB) / 400)))

That will give you a good idea of your expected score, but it doesn't take into account the ratings deviations.

The expected score is what the rating systems thinks is the most likely result of the game you are about to play. So your expected score is going to be between zero and one. If you are rated higher than your opponent, it will be above 0.5, and if you are rated lower than your opponent it will be below 0.5.

The trick is that it might be something like 0.7. Now, in Chess there are only three possible scores: 0 (a loss), 0.5 (a draw), and 1 (a win). There are two ways to think about expected scores like 0.7: chance and long term.

The chance view is that it is your chance of winning the game (70%). At lower levels of play that works pretty well. However, as you get better, more and more of your games will be draws. Then you don't have a simple win/loss result that can be translated into a single probability.

The long term view is that the expected score would be your average score over several games. For example, a 0.7 expected score would mean that over a 10 game match against that person you would expect to get 10 * 0.7 = 7 points. There are several different ways you could get those seven points:

• Seven wins and three losses.
• Six wins, two draws, and two lossses.
• Five wins, four draws, and one loss.
• Four wins and six draws.

Actual Score and Ratings Adjustment
 
Once you play the game, your actual score for the game is compared to the expected score. If your actual score is higher than the expected score your rating goes up, and if your actual score is lower than the expected score your rating goes down. How much your rating goes up or down depends on how close the actual score was to the expected score. The farther apart the scores are the more points you will gain or lose from your rating.

Take a practical example. Say you are playing someone 400 points above you. By the formula above, your expected score would be 0.09. That is, you are probably going to lose. If you lose, the actual score is 0. That is less than the expected score, so you will lose some ratings points (and your opponent will gain some). However, it is close to the expected score, so you won't lose very many. On the other hand, say you win. Your score (1) is higher than the expected score, so you will gain ratings points (and your opponent will lose some). Additionally, your score is a lot higher than the expected score, so you will gain a lot of points.

Basically, what the rating system does is guess your score, and then compare that guess to the actual result of the game. If it was wrong, it assumes it misjudged your chances of winning, and changes your rating to correct for that error. And since it doesn't know if it's mistake was in misjudging your chances or misjudging your opponent's chances, it adjusts both ratings at the same time.

Rating Deviation

That's generally how most rating systems work. But Chess.com uses the Glicko rating system, which adds in a rating deviation (RD). This is a measure of the uncertainty in your rating. If you are a new player you get a starting rating of 1200. However, the system really has no clue what your chances are against the other players at Chess.com. So it assigns you a high ratings deviation to indicate that there is a high level of uncertainty about your chances. As you play more and more games, the system gets more information about you, and becomes more certain about your chances against other players. The system then lowers your RD to indicate that there is a low level of uncertainty about your chances.

The system also assumes that your chances could change. Maybe you are intensely studying chess books and improving your skill. Or maybe you haven't played in a while and you've gotten rusty. So the longer it has been since you played your last game, the more uncertain the system becomes about your rating, and the more it increases your RD. Conversely, if you are playing a lot of games on a steady basis, your RD will go down.

How does RD affect the ratings? The higher your RD is, the more your rating will change. In other words, if the system is uncertain about your rating, it is willing to change your rating more based on new information. On the other hand, the higher your opponent's RD is, the less your rating will change. So if the system is unsure about your opponent's rating, it won't penalize you as much for losing to them (or reward you as much for beating them).

A Couple of Consequences

The first thing to understand about the rating system is that it is rating your performance, not your skill. There is hopefully a relationship between the two, but it is important to keep in mind what is actually being rated. As an illustration, say that you win a game. You might have won that game by using a deep tactical combination to create a decisive advantage. You might have won that game by seeing a subtle weakness in your opponent's position and creating a long term plan to exploit that weakness. Or you might have blundered away a piece but your opponent blundered away two pieces so you ended up ahead. The rating system has no idea how you won the game. All it knows is that you did win that game.

Not only does the rating system only know that one score, it has to use that score to adjust two player's ratings. This leads to the second thing to understand about the rating system: it's relative. The information it is getting is linked to the people it is getting the information from. Outside of that context, it doesn't really have any meaning. So if you have a rating from Chess.com, and you want to know what your rating might be if you played FIDE games, you really can't tell. This is because FIDE is a different group of people, so you lose the context that gives meaning to the Chess.com ratings.

This is a common misunderstanding of the ratings, so let me give you an example from basketball. One statistic for a basketball player is their free throw percentage. Free throws are done without interference, so it is as a pure a measure of skill as you get in basketball. Put that same player in a different division, a different league, a different country, and his free throw percentage will pretty much stay the same. But again, ratings are not measures of skill, they are measures of performance.

A better analogy for ratings would be a basketball team's win/loss record. That's a measure of their performance, and it depends on who they've been playing. Put them in a different division and you can expect that record to change. Move a college team into the NBA and you can expect it to change a lot. Don't fall into the trap of thinking that your rating is like a player's free throw percentage. It is more like an advanced win/loss ratio, that takes into account how strong your opponents were for each game.

Sure, you could take all the players at Chess.com who also play in FIDE, compare their ratings in the different systems, and come up with a statistical relationship between the two. It might tell you that an 1800 player on Chess.com is a 1500 player in FIDE (I don't really know, I just made that up as an example). But if you really take a look at that statistical relationship there is going to be a lot of variability in it. So it would be more honest to say that an 1800 player on Chess.com is a 1300 - 1700 player in FIDE. And that's a huge range in terms of chess ratings.