I refereed a match where White had a rather large time advantage in the middlegame - on all boards. This isn't that remarkable, I expect that Black must invest more time just to equalize, and often must defend, which takes more time too. But already in the next match the n=8 stat flew out of the window.

Are there any large game databases (n=1000 surely is a minimum to decide about my hypothesis) which also record the time spent for each move? Note that I'm talking about tournament mode games, although the postulated effect might show up in rapid chess too.

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There are a lot of games in chess.com with clock info on every move. This clock info can be used to calculate the time spent per move. It can be tricky for time control with periods as additional time is added after say 40 moves. Also there are games with weird clk information in this case just ignore such games.

Go to chess.com events at https://www.chess.com/events, select events and download the event's pgn file

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