In the previous post (Dec 26, 2010), I described a new basketball statistic -- the Successful Possession Rate (SPR), which I referred to as basketball's version of baseball's On-Base Percentage (OBP). Though the formula for SPR is described in detail in this previous post, the basic gist is that it is the total number of successful offensive possessions a player has had, divided by the total number of attempts he has made to achieve that success (i.e., score a basket, score a free throw, or make an assist).

But if SPR approximates the basketball equivalent of OBP, I wondered, what then would be the equivalent to baseball's Slugging Percentage (SLG%)? The SLG% is a batter's total bases divided by his total number of at-bats, and what it approximates is how close a batter comes to reaching home plate on any given plate appearance in which he does not walk (i.e., on any given "at-bat").

So what would be the basketball equivalent of SLG%? Schematically, such a statistic would have to approximate the player's likelihood of scoring a point on any given field goal attempt (FGA). So, just as baseball uses total bases (with four total bases awarded to a home run, three for a triple, and so on) in the numerator of its SLG% formula, the numerator of the basketball version would be just as straightforward: points scored (PTS). The trick here, however, is coming up with the denominator of the formula.

Traditionally, FGAs have been used -- like at-bats -- as the denominator in such basketball estimates as Points per Shot (PPS). The problems with the PPS formula -- PPS = PTS/FGA -- is that the denominator does not include all shot attempts, and thus credits the player with far too much efficiency. For example, points scored off of free throws are included in the numerator, but the free throw attempts (FTA) required to accumulate those points are not included in the denominator. This is a glaring inaccuracy, for FTA -- with the exception of "and-one" situations and technical fouls -- absolutely represent an attempt to put the ball into the hoop.

Free throws are usually rewarded when a player is fouled in the act of shooting -- and similar to a FGA, the ball becomes live after a missed FTA, or is awarded to the other team after a score, even in non-shooting situations (i.e., team bonus situations). In effect, a FTA is just as much a shot -- or, to be more accurate, half of a shot -- as a FGA. To not include some weighted version of the FTA in the denominator of a PPS gives far too much credit to players, as good as they may be, who practically live at the free throw line. Michael Jordan and LeBron James would be two standard examples of this inaccuracy, but perhaps the two best examples would be Shaquille O'Neal and Dwight Howard -- two players who have accumulated thousands of points at the free throw line, but have nevertheless squandered away thousands more without "paying for it" in their PPS.

And so I began to see that any accurate estimation of PPS would have to include a weighted sum of FTA in the denominator. Since a FTA is worth half as much as a standard FGA, I felt that FTA should be discounted by 50% (three-point field goal attempts, I reason, still only cost you two points worth of opportunity, and so I have left them alone). And so, in my basketball version of the SLG%, the denominator began to look to me like this:

[FGA + (FTA/2)]

At first, this seemed like a satisfactory estimate at shot attempts: FGA plus half of all FTA. Sure, there was some mild inaccuracy to it -- most notably, the fact that "and-one" situations and technical foul awards are counted as opportunites, when in fact they are "extra-credit bonuses" -- but it certainly was a more reflective approximation of shot attempts than just FGA alone. But something was still nagging at me.

That "something" was turnovers. Turnovers (TOV), for all intents and purposes, are failed shot attempts: they are a missed opportunity at a bucket, with possession changing hands into those of the opponent (they are, in fact, worse than a missed shot attempt, because with missed shots you at least have the opportunity to gather the rebound and start the possession anew). And turnovers are made, just like missed field goals and free throws, in an attempt to score a basket -- either taking the ball to the hoop, or advancing it up the court, or passing to a teammate that the player thinks is open. I suppose it can be debated as to whether or not a turnover is the equivalent of a missed shot -- I vote that it is worse -- but it is certainly a missed opportunity. And SLG%, I reasoned, doesn't care if you struck out, hit into a double play, or advanced the runner home on a ground-out -- they are all missed opportunities, and they are treated as statistical equals, even though their outcomes can have quite a different effect on your team's chances of winning.

And so, the more accurate denominator at shot attempts appeared to me as this:

[FGA + (FTA/2) + TOV]

On the other hand, a point is a point -- whether it comes from a two-point shot, a three-point shot, or a free throw, and so the numerator was left alone as total points scored (PTS). Taken together, the basketball version of SLG% -- which I refer to as the Turnover-Adjusted Points per Shot (TAPPS) -- has a formula that looks like this:

TAPPS = PTS/[FGA + (FTA/2) + TOV]

As usual, having developed a new statistical tool, I wanted to try it out on various players of different abilities, to challenge the statistic's credibility. From 1978-79 through 1981-82, for example, Kareem Abdul-Jabbar had an outstanding TAPPS of 1.04, without ever taking a single three-point shot. Shaq, despite his pedestrian free-throw shooting, has amassed a career TAPPS of 1.01, while Michael Jordan and Larry Bird have posted career TAPPS of 1.01 and 0.97, respectively.

Meanwhile, Charkes Oakley -- an old favorite of mine who nevertheless will never be stepping into anyone's Basketball Pantheon -- accumulated a career TAPPS of 0.84, which is a good 12% or more less than basketball's elite (and lest you ask, "How big a deal is 12%?", compare in your mind's eye the difference between a player with a SLG% of .500, and one with a .380). As another example, Brandon Jennings, who is an exciting player but one who I have always suspected of achieving high point tallies on poor shot selection, comes in with a career TAPPS of 0.83.

Meanwhile, my standard examples of basketball marginality -- Adam Morrison and Ken (The Animal) Bannister -- lend validity to the TAPPS estimate with near identity. Morrison, who can thank Mitch Kupchak for a championship ring, nevertheless has a career TAPPS of 0.79; Bannister, who has no such ring, edges out Morrison with a career TAPPS of 0.80.

And if you dare point out that The Animal is only 0.04 behind my man Oak in career TAPPS, let me provide you with a baseball analogy of such a comparison: Might your favorite team have use for a good defensive player who hits .270 (Oak), or would you be just as content with an O.K. fielder who hits .230 (The Animal)?

In future posts, TAPPS will be used to assess player efficiency, teamwork, and comparative trade value. For now, I invite you to ruminate on the Turnover-Adjusted Points per Shot statistic: basketball's version of Slugging Percentage.

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