The first task in understanding the game was to boil it down to its statistical foundations. What really makes one team better than another? There are 3 phases to the game: offense, defense, and special teams. Offense and defense can each be broken down into passing and running. Special teams consist of field goals, punting and kicking (and returning).
I theorized that offense and defense were particularly more important than special teams in winning. It's not that special teams don't matter in any one game, it's that they don't correlate strongly with winning over the course of a season. In the 2005 season, Field Goal Percentage correlated with wins at 0.054, which is very low and not statistically significant. Compared to the correlation between sacks and wins, which is 0.393, we see how much more sacks mean to winning than FG%.
Additionally, even if they correlated with winning, they are highly unpredictable from week to week. It's also difficult to measure the performance of a punting squad, for example. Are short punts bad if they pin the opposition inside the 10? What about FG%? It's hard to score because kickers with longer range are sent onto the field to try low-probability attempts.
So we're left with offensive running and passing, and defensive running and passing. There are many ways to measure these phases of the game but what is the best way to really measure how good a team is at each phase?
Total passing yards or total rushing yards would not be valid measurements. A team with a terrible defense is often playing from behind and will throw for large chunks of yardage in the 4th quarter "trash time." Teams with great defenses that carry leads into the 4th quarter will pound the ball on the ground, padding their total rushing yards. In each case, the "total" yards stat is a reflection on the team's defense as much as it is their offense. One might argue doesn't that count? Shouldn't these things factor in? Yes they should, but I want to isolate what really separates good teams from bad. Poor defense will factor in, but as you will see, the effect of a poor defense is isolated in the defensive stats.
The best measurement for each phase is yards per attempt. For passing, that means that a team is not rewarded for more attempts. All that matters is how many yards are gained with every drop-back. Incompletions, sacks, and interceptions count for zero. Running is simpler. There are no incomplete runs (though the Raiders try) so it is a straight average of yards per rush.
So here are the 4 primary "core" statistics for measuring how "good" a team is, and its correlation with team wins for the 03-05 seasons.
OFFENSE WIN CORRELATION
Yds Per Rush 0.415
Yds Per Pass Attempt 0.594
Yds Per Rush (given up) -0.351
Yds Per Pass Att (given up) -0.251
The 4 variables are statistically significant. Offensive stats appear to be more important than defensive, and passing appears to be more important than running.