user13744: For example in the first turn there are 20 possible legal moves (16 pawn moves and 4 knights moves). This number increases in the middlegame and then decreases in the Endgame.
Are there some calculations on the average? Google search seem to be focused only on "average number of moves per game". Would appreciate if somebody have some links to some scientific researches or also blog posts. 

user19383: The variability in number of legal moves is about the same for black & white... except for that odd little bump for white on move 7.  This phenomenon demands further research!



user12523: What you are looking for is called the branching factor, and I've always seen the number 35 mentioned, but I don't know what the original source is. I guess someone estimated it some 50 years ago by counting the number of moves in a number of random positions from games, and then it became "common knowledge". The number 35 is reasonable enough in practice, but of course is not exact.
In today's age of big data, it should be easy to take a game database and answer the question of what is the exact average number of moves per position in the database. I'm sure someone must have tried that already, but I haven't seen it.

user12951: Whilst acknowledging the comment by @SmallChess that this is pointless, it is also relatively straightforward to do. I analysed 2,539,871 games from a ChessBase mega database counting the number of moves for the next player to move before each move was played. I did not include the number of moves available after the final move of the game had been played. The total number of moves played was 194,389,820 (76.5 per game) and the total number of available moves was 6,039,013,721 giving an average of about 31.1 per move.
Here is an analysis of the average number of moves available for each colour at each move point for the first 100 moves of each. Move 0 is prior to the first move of each colour. It is interesting to observe that, on average, black has fewer options than white. I am not sure why this would be-hopefully, it isn't an error in the analysis (!) but I don't think it is.


13744: <p>For example in the first turn there are 20 possible legal moves (16 pawn moves and 4 knights moves). This number increases in the middlegame and then decreases in the Endgame.</p>

<p>Are there some calculations on the average? Google search seem to be focused only on "average number of moves per game". Would appreciate if somebody have some links to some scientific researches or also blog posts. </p>
 19383: <p>The variability in number of legal moves is about the same for black &amp; white... except for that odd little bump for white on move 7.  This phenomenon demands further research!</p>

<p><a href="https://i.stack.imgur.com/jqKEl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jqKEl.png" alt="mean number of legal moves after move number"></a>
<a href="https://i.stack.imgur.com/3FInO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3FInO.png" alt="std dev legal moves after move number"></a></p>
 12523: <p>What you are looking for is called the <a href="https://en.wikipedia.org/wiki/Branching_factor" rel="noreferrer">branching factor</a>, and I've always seen the number <strong>35</strong> mentioned, but I don't know what the original source is. I guess someone estimated it some 50 years ago by counting the number of moves in a number of random positions from games, and then it became "common knowledge". The number 35 is reasonable enough in practice, but of course is not exact.</p>

<p>In today's age of big data, it should be easy to take a game database and answer the question of what is the exact average number of moves per position in the database. I'm sure someone must have tried that already, but I haven't seen it.</p>
 12951: <p>Whilst acknowledging the comment by @SmallChess that this is pointless, it is also relatively straightforward to do. I analysed 2,539,871 games from a ChessBase mega database counting the number of moves for the next player to move before each move was played. I did not include the number of moves available after the final move of the game had been played. The total number of moves played was 194,389,820 (76.5 per game) and the total number of available moves was 6,039,013,721 giving an average of about 31.1 per move.</p>

<p>Here is an analysis of the average number of moves available for each colour at each move point for the first 100 moves of each. Move 0 is prior to the first move of each colour. It is interesting to observe that, on average, black has fewer options than white. I am not sure why this would be-hopefully, it isn't an error in the analysis (!) but I don't think it is.</p>

<p><a href="https://i.stack.imgur.com/7eqPQ.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/7eqPQ.gif" alt="Average number of moves available before each move"></a></p>
