# Carlsen and Wei - out of control!

Is Magnus Carlsen really *that* much better than everyone else? Many people have asked that question, and in this post I provide an answer based on math. That's right, you that read right. Math! (Be afraid, be *very* afraid!)

In a previous post, I have presented an analysis of the rating progress of some of the top GMs. I was not entirely satisfied with the analysis I did, since it was based on the number of rating periods after a player reaches a 2200 rating. I feel that player age should be a better predictor of the players' star potential. Also, I wanted to redo the analysis with recent data.

Everyone knows that Magnus Carlsen has been sitting on the top of the rating lists for quite some time now. Some argue that he is in fact in a league of his own, and that no other player is in the same category. I wanted to see if this is just an empty argument or if it can be supported by ~~science~~ statistics.

Furthermore, I have heard many people talking about Wei Yi as a rising star, and that he may become a challenger to the top rating position and the world championship title.

So armed with statistical methods, I dug into FIDE's rating lists and selected a few of the top GMs: Aronian, Carlsen, Caruana, Giri, Karjakin, Nakamura, So, Vachier-Lagrave and Wei.

The first step of my analysis was to plot the rating progress versus the age of the players. This is what I found.

There are some interesting observations that can be made here. One is the extreme progress of Karjakin. His development at a younger age is higher than anyone else, but he has not maintained the progress and has receded into an "average" performance. Several players deviate from the general pattern at points, but not consistently. Carlsen's performance at young age is very good, but not as good as Karjakin's. However, he had a very strong development between ages 11 and 13, and after this period, his progress follows the overall pattern, but at a significantly higher level than anyone else. Furthermore, we can see that Nakamura seems to be a very average Super GM. His performance is more or less in the middle of the crowd.

The graph indicates that these top players usually have a very steep rating progress at a young age, and that the increase tapers off with time. I modeled this progress with a logarithmic equation.

Rating = -6197 - 5,840*Age + 1882,9*LN(Age)

For this equation, I use the player age measured in months. The dotted line in the graph above represents this function.

Based on the rating variation between the different players, I have established two limits that indicate the range within which we can expect a super GM to perform. This kind of analysis is often used to track measurements and evaluate the performance of various processes. When measurements fall outside the calculated limits, something extraordinary has probably happened, and the process is said to be out of control. Similarly, a when a player's rating falls outside the limits, this is a signal that something extraordinary has happened in the world of chess.

If a player exceeds any of these limits, that is a signal that the player deviates from the general pattern. In other words, a player whose rating falls below the lower limit is probably not Super GM material. On the other hand, a player whose rating goes above the upper limit is out of the ordinary, and most likely a Super Super GM - one of a kind.

So how do Carlsen and young Wei perform compared to a "normal" Super GM? Let's take a look.

As we see in this graph, neither player performs outside the calculated limits. However, we see that their performance of late has been consistently towards the upper limit. The probability of this happening at random is next to nothing, and with more advanced techniques we can establish beyond any doubt that both Carlsen and Wei are out of control (in statistical terms). The proof can be found when comparing the actual rating progress of the players to the expected progress from the mathematical model.

I am not going to go into details about these two graphs, but the principle is the same as before. Points above the upper limit suggests an extraordinary performance. These two players seem to have established themselves permanently outside the normal performance of the Super GMs, suggesting that they are in fact in a league of their own.

So the next time you want to know whether a rising star has the potential to compete among the very best in the future, just plot their rating progress in the chart above, and see how they compare.

If Wei Yi can maintain his level and continue to develop along the same lines as Carlsen has, we can expect him to become a dangerous competitor for the most prestigous titles in chess.