Have their been any statistics on how your knowledge on part of the game (opening, middle, end) corresponds to your rating or winning percentage? For example, I have found that if I play opening moves a good number of moves in, my chances of winning go way up. I am not sure how the middle game or end game affect this though. If player A was very strong in the opening, but semi-strong in the middle-game and player B was semi-strong in the opening, but very strong in the middle-game, does one player have an advantage? The same goes for the end-game? I have heard that the end-game is usually where the game is won. Do masters of the past and current have any direct correlation with these statistics?
No. There is a rating system based on wins, losses, and draws, not for proficiency in phases of the game. How would you quantify that? Knowing the main line on the MCO for your openings? People are always finding improvements to main and obscure opening lines. This is the TN (theoretical novelty) that people search feverishly for. So what might be sound play in the opening in January may be unsound by August.
Further, I have seen flawless openings and middlegames come to grief because of a tiny blunder in the endgame. As Tartakower once said, the winner is the player who makes the 2nd to last mistake. So there is not really an accepted method of tying knowledge of component phases of the game to winning percentage, because there are too many other factors at hand. If the game 's phases could be defined down to a formula of what is good play in the opening, middle game or ending, computers would always win. We still have imperfections of shortsightedness and obliging opponents, and time controls, and other distractions.
I can contribute a personal study, but it conflicts with your experience. While I was coaching a HS team, we started talking about how important the opening was (my position was "not very;" they weren't convinced).
So we looked at my last half-dozen or so tournaments. I had won 19 games during that stretch, and had been standing noticeably worse coming out of the opening in 14 of them. So, from that data, it would seem that opening knowledge wasn't contributing much at all to my winning percentage.
There's a complexity here in this that is not being accounted for, though. There are two players in a chess game, and the knowledge/ability of your opponent has an effect on this. The basic rule is the better your opponent, the less likely they are to make mistakes, therefore the more important your own opening knowledge becomes. A player with a reasonable tactical ability can recover in the middle game from an opening blunder against someone under 2000USCF, because the chances are they will make mistakes that will produce the opportunities. Against Carlsen? Yeah, good luck with that.
That's why the traditional advice has been to start improving from the back end of the game forward. You can steal back some half-points and full points later in the game that your opening ineptitude gives away.