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Top 10 active players as of December 31, 1951

#	Player	Rating	+/-
1	Botvinnik, Mikhail M	2750	54
2	Keres, Paul	2736	54
3	Bronstein, David I	2722	51
4	Fine, Reuben	2717	67
5	Reshevsky, Samuel H	2713	59
6	Smyslov, Vassily V	2712	53
7	Najdorf, Miguel	2683	54
8	Boleslavsky, Isaak E	2669	55
9	Lipnitsky, Isaak O	2658	57
10	Geller, Efim P	2645	56

About these ratings

These ratings, calculated as of December 31, 1951, incorporate all games (against rated opponents) played in the time between January 1, 1951 and December 31, 1951, along with the ratings that were already calculated as of December 31, 1950. Players who have played an insufficient number of games in recent years will not appear on the list, even though you may see them on previous and/or subsequent lists. Any players who have been completely inactive for three years are removed from the list.

Definition of columns in the above list:

# refers to the player's world rank on this list.

Player is the full name of the player. Click on the player's name to see a career progression of the player's ratings.

Rating is the calculated rating for the player, as of December 31, 1951. This represents the player's estimated strength on January 1, 1952.

+/- is the standard deviation of the player's rating estimate. It is based on a geometric average of the number of rated games played in past years. Players who play very frequently will have a smaller +/- value, indicating a greater confidence in the accuracy of their rating. Players whose +/- value exceeds 76 are classified as "inactive" and removed from the ranking list due to their infrequent play against rated opponents. Mathematically, a player's rating is an estimate of their exact strength on January 1, 1952. The +/- value represents the standard deviation of that estimate; the player's true strength on January 1, 1952 should be within one standard deviation of their calculated rating approximately 68% of the time, and within two standard deviations of their calculated rating approximately 95% of the time.

Best? is the estimated likelihood that the player was actually the strongest player in the world on January 1, 1952. Since ratings are known to be inaccurate, it is possible that a lower-rated player is actually much stronger than their rating would indicate, and it is also possible that other players are weaker than their ratings would indicate. This number indicates the likelihood that the player is indeed the strongest player in the world. On a particular list, it will add up to 100% across the complete list of all players. This value is only calculated for the top 100 players on any rating list.

75%? gives a 75%-confidence world ranking for the player. Because ratings are known to be inaccurate, even if a particular player is rated third in the world, we cannot say with 100% certainty that they are truly one of the three strongest players in the world. This number indicates the world rank that we are 75% sure the player is "no worse than". Thus if a player has a value of #12 in this column, we are 75% sure that the player is one of the twelve strongest players in the world. This value is only calculated for the top 100 players on any rating list.

Age is the player's exact age in years, on December 31, 1951. If a player's age is displayed as a link, that means that this rating was active at the time of their next birthday. Click on the link to see the sorted list of all players' ratings on their birthdays when turning that same age.

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