I subscribe to the idea of "tempo-free" analysis, which ignores per-game statistics in favor of per-possession. For the first-time reader, this can make it a little difficult to understand the data behind my insights. For new subscribers, here is a glossary/primer on the data I like to use:

 

Possessions: Instead of calculating these, I track these using play-by-play data. I follow the standard definitions that a possession ends with a turnover, made shot, defensive rebound, or made free throws. Tracking this with data instead of calculating it is more time-consuming, and isn't completely error-free, but it should represent possessions in the most accurate way I can think of.

Plus/minus per possession: Instead of looking at absolute plus/minus, I look at it on a per-possession basis. So, I wouldn't say a team was +12 with Player X in; I'd say they were +0.20 per possession with Player X in. This can also be stated in terms of +/- per 100 possessions, just by multiplying by 100. Of course, the components of this are points scored and points allowed per possession.

Adjusted plus/minus per possession: Raw plus/minus numbers offer no context about the level of competition, and therefore aren't very useful when comparing game-to-game. To fix this, I adjust the raw plus/minus by using Ken Pomeroy's team ratings. This allows us to compare performances across the season, against a variety of opponents, and isn't influenced by opponent strength...at least, that's the idea.

Possessions and plus/minus are measured at the team level, although I'll also frequently discuss plus/minus for a team when a player is in the game...but I'm still measuring that for a team, not a player. There are also several stats I use that are measurable for teams or individual players:

Rebound rate: Instead of counting rebounds, or rebound margin, I prefer to look at rebound rate; that is, how often did a team get the available rebounds? I separate into offensive and defensive rebound rates. Obviously, a team's offensive rebound rate and their opponent's defensive rebound rate add to 100%, and vice versa.

Turnover rate: Again, instead of counting turnovers, I prefer to measure how frequently a team's possessions end in turnovers. I measure both a team's turnover rate and their opponent's turnover rate (which reflects how often the team in question forces turnovers). 

Foul rate: Continuing a theme, this measures how often a possession ends in a foul drawn (or committed, on defense). This is closely (but not perfectly) correlated to how often a team shoots or allows free throws.

Effective Field Goal Percentage (eFG%): To measure shooting, I like to use this metric which adjusts for the increased value of 3 pointers. The formula is (2PM +1.5* 3PM)/FGA. 

True Shooting Percentage (TS%): I don't use this a ton, since it incorporates free throw rate and accuracy and I prefer to look at field goal defense...but this is a great measure of individual player's offensive value. This counts total points scored, divided by 2* (Possessions used - Turnovers). 2 x TS% is basically how many points a player generates per shot or set of free throws drawn.

Assist rate: This measures how often field goals made are assisted. It's AST/FGM. I don't like using this one much because an assist doesn't make a field goal any more valuable, and good passing doesn't always mean assists. Assists are subjective at the whim of the scorekeeper.

Block rate: This captures how often field goal attempts are blocked. It's Blocks/FGA. I don't use this much, because a block doesn't mean a positive outcome. Someday I will go through my play-by-play data and determine how often a block results in a change of possession...until then, I'll stick to measuring more directly positive or negative things.

Finally, there are stats that are only measured for a player, not a team:

Usage rate: This stat measures how active a player is in the offense by measuring how many possessions they "use". A player "uses" a possessions by taking a shot, drawing free throws, or turning the ball over. Because I am counting possessions, my usage rate will vary somewhat from websites that estimate this through calculation. The difference shouldn't be huge, but is probably worth mentioning.

Points, rebounds, assists, etc. per 40 minutes: I really only talk about team performance using the per-possession or efficiency stats, but I will sometimes look at player stats in the context of a fixed amount of time. To approximate "per 40 minutes", I look at how many possessions the team averages per game and multiply per-possession stats by that number. It's not all that useful in isolation, but comparing the performance against different opponents, etc. can be interesting.

"Same Four" Plus/Minus: I have called this "Isolated Plus/Minus" as well, but I saw "Same Four" used online somewhere and like it. This stat is measuring the adjusted plus/minus per possession when a player is in the game, minus the adjusted plus/minus per possession when the rest of the lineup is in the game without him (hence the term, "same four"). This is attempting to isolate an individual player's effect on the game from the impact of his teammates. It's not perfect, but it does tend to identify which players bring unique skill sets, or are most indispensable.

I also have some other interesting ways to look at data that aren't specific stats, but add context to stats:

Performance by number of opponent starters: I track how many starters for the opposing team are in the game, and can look at most team or player stats according to this measure. Frequently I will control for this by only looking at performance when 3, 4, or 5 starters are in the game for the opponent. This usually filters out garbage time.

Shot quality: I also track shot attempts by the amount of time in a possession before the shot. I classify shots as Putbacks, 1-10 sec, 11-20 sec, or 21+ sec. I start the clock over on offensive rebounds, so the limit should be the limit on the shot clock. I track what percentage of a team's shots fall into each category, and their eFg% on these. This is an interesting way to measure the quality of possessions, and see how a team can force a style on their opponents.

This should prepare you for the way I look at basketball. If you have any questions, feel free to reach out...otherwise, I hope you enjoy reading my analysis.