# Relationship Between TS% and eFG%

In an effort to understand shooting efficiency, terms such as points-per-possession, effective field goal percentage, and true shooting percentage have come about as methods to quantify scoring efficiency. In fact, during my coaching days in Baltimore City (2013 – 2016), I developed a metric called points responsible for (PRF) that focused on distributing points to players based on their field goals, free throws, assists, and turnovers; relative to touches.

In this article, we focus on an advanced statistic called the True Shooting Percentage (TS%). To understand this metric, we focus on the simplistic field goal percentage (FG%) and build up to the more commonly used advanced analytic effective Field Goal Percentage (eFG%). We then will focus on the advancement of eFG% to TS% and develop an understanding of what TS% actually captures.

## Field Goal Percentage (FG%)

Field Goal Percentage is one of the oldest traditional statistics when it comes to understanding player performance. The equation is simple: take the number of field goal attempts and divide it into the number of made field goals. Similar to batting average in baseball, the analytic helps us understand the probability a player is able to make a field goal attempt. Primitive in its nature, field goal percentage is only able to capture a small portion of player capability.

Prior to 1979, FG% made complete sense as only two point field goal attempts existed. The idea of high probability shots referred to regions on the court that would lead to the highest probability of scoring two points. However, with the introduction of the three point line in 1979, field goal percentage did not make much sense anymore.

For instance, if I compared two shooters that took the same amount of field goal attempts and had the same field goal percentage; say both players took 2,000 field goal attempts and both made 40% from the field, we may not know which player is better today. If this were prior to 1979; then we know both players made 800 field goal attempts leading to 1600 points.

Post-1979, we need to carefully understand the number of three point attempts in this equation. For instance, suppose Player A is a traditional player: only two point field goal attempts. Then suppose Player B is a terrible three point shooter (say 20%) and takes 200 three point attempts. Then Player B has scored 1640 points. This means Player B is more efficient than Player A despite having the exact same field goal percentage.

This is where effective Field Goal Percentage comes into play.

## Effective Field Goal Percentage (eFG%)

Effective Field Goal Percentage (eFG%) is the slight alteration to capture the adjustments created by the three point line in 1979. Here, we compute

This means we weight the number of field goals made by 1.5 if the field goal made is a three point attempt. This way, if we multiply by two, we obtain the number of points scored from field goal attempts; which is no longer the case for FG%.

Now if we return to the comparison of Player A and Player B, we have a different understanding of the two players. While both players have the same number of field goal attempts and same field goal percentage, Player A has an eFG% of 40%. In comparison, Player B has an eFG% of 41%. This means that Player B is more efficient.

Furthermore, by multiplying each eFG% by 2 times the number of field goal attempts, we see that Player A has a value of 1600 while Player B has 1640. This is, in fact, the number of points scored by each player. This means eFG% captures scoring efficiency per field goal attempt in an unbiased manner in comparison to FG%.

## True Shooting Percentage (TS%)

True Shooting Percentage (TS%) attempts to take this idea a step further by introducing a “points per shooting possession” model. Here TS% attempts to capture the effect of shooting free throws. This type of model is effectively in response to Hack-A-Shaq type situations, where it becomes more favorable to foul terrible free throw-shooters in attempt to lower points per possession for a team.

As a new illustration, suppose we compare two players: Player B and Player C. Both players have identical FG% and eFG% of say . However, Player B is an 80% free throw shooter while Player C is a 60% free throw shooter. Then, obviously, it is important for a team to get Player C to the foul line more often than Player B. This means that the opponent recognizes that Player B is more efficient than Player C.

In light of this, TS% is calculated as

Think of this as eFG% on steroids. Here, we adjust field goal attempts with free throw attempts that terminate possessions. The factor 0.44 is merely the estimated percentage of free throws that terminate a possession. While this factor is not necessarily true anymore (we can actually fit it to be 0.43 this past season through linear models or count it to be closer to 0.426 using direct counting). Any which way, the goal is to capture equivalency of free throw attempts to field goal attempts.

From here, TS% includes the multiple of two in the denominator to account for using points in the denominator instead of FGM, 3PM, and FTM.

But what does TS% really mean?

## Rewriting TS%

While the genesis of TS% is easy to understand, let’s actually break up this quantity and see how it really relates to free throw shooting and eFG%. If we perform some straight-forward algebra, we see that:

Rewriting TS% in terms of eFG% and FT%.

We see that TS% is almost a mixture model of eFG% and FT%. Let’s quickly summarize the algebra done. First, we start with the definition of TS% in line one. In line two, we write points in terms of FGM, 3PM, and FTM.

In line three, we factor out the denominator and look at the two terms: Points Scored by FG and Points Scored by FT. We multiply each by 1 to obtain eFG% in the first term and FT% in the second term. In line four we explicitly swap in the definitions of eFG% and FT%.

Finally in line five, we distribute the denominator back in and write as a linear combination of eFG% and FT%.

However, if this were a pure mixture, the coefficients of eFG% and FT% would add to one. They don’t. This means either eFG% or FT% are being preferred; and this actually depends on the free throw shooting ability of player.

## TS% Inflates Free Throw Percentage

Using the linear combination form of TS%, we have that FT% is actually inflated by a relatively small amount. We see this by rewriting

Depending on how often a player gets to the line, we will see a player’s TS% change. For instance, a player who gets to the line frequently, like James Harden or Russ Westbrook (881 and 840 times, respectively) and has a high FT%, then obtaining more looks at the free throw line will help their TS%.

Let’s break down Russell Westbrook.

### TS% For Westbrook (Oklahoma City)

For the 2016-17 NBA season, Westbrook was 824-1941 (0.425) with 200-583 (.343) for three’s and 710-840 (.845) from the foul line. This resulted in an eFG% of 0.476. Since Westbrook’s FT% was higher than his eFG%, any extra free throw attempts would drive up Westbrook’s TS%.

In this case, Westbrook’s TS% should be 53.51%. However, since free throws are counted with that extra third term above and Westbrook has a higher FT% than eFG%, then Westbrook finishes with an extra 1.84%. This leads to Westbrook having a TS% of 55.35%.

### TS% For Andre Drummond (Detroit)

If we replicate this study for Andre Drummond, we find expected results: hurting players with poor free throw shooting abilities. For Drummond in the 2016-17 NBA season, we saw 483-911 (0.530) with 2-7 (0.286) for three’s and 137-355 (0.386) from the foul line. This resulted in an eFG% of 0.531.

Drummond’s TS% should be 50.98%. Instead, due to his terrible free throw shooting percentage, Drummond only manages to add on an extra 0.77% to obtain a TS% of 51.75%.

### (And it’s not the only one) but…

Despite the inflation of free throw shooting, we actually obtain an unbiased estimator for the number of points. This is inherently built in to the definition of TS%. That said, if we were to write 0.5 instead of 0.44; we would obtain an unbiased estimator for points with a perfect mixture. However, this assumes that half of all free throws result in completed possessions. Which it doesn’t. This begs the question…

### Why Not Use The Actual “0.44”?

Why are we not using the actual correction factor instead of an ill-fitted 0.44. We can actually walk through every free throw attempt and find that 42.6% of free throws resulted in terminated possessions. In this case, TS% is actually inflating over the years… unless you’re Andre Drummond. For instance, thanks to the 1.4% disparity, Harden’s TS% is inflated by 0.2%. This is similar for Westbrook and Jimmy Butler. Drummond, however, loses 0.1% of TS% due to using the wrong inflation factor.

While these percentages are not large in their discrepancies, we find that the expected number of points is actually far off. For example, Westbrook is expected to score 2545 points, despite scoring 2558 points. Similarly, Harden is expected to score 2340 points; despite scoring 2356. We see that TS% is indeed biased in the direction of the free throw inflation factor. To fully see this, consider Andre Drummond once again. Drummond is expected to score 1099 points, against an actual 1105. As free throw percentage decreases, this bias become negligible.

## So What Do We Do?

We find immediately here that TS% is biased in the direction of free throws and this bias actually under-estimates the number of points scored per possession for a player. Instead, we should either focus on developing a proper metric that incorporates the actual number of possessions or focus on an actual per possession statistics that TS% attempts to emulate.

A simple correction is to perform an in-season tally for the weight of free throws that terminate a possession. While this creates a fluid TS%, it will be the unbiased metric.

If this is not desirable, then we should opt to use a per possession statistic that is not as fluid. Here, we simply have the calculation of PTS / (Possession With A FG/FT attempt). To develop this statistic, we can form the correction to be the following:

## Per Possession

Suppose a player has the following break down: 7-10 in and-one attempts, 84-100 in two-free throw attempts, and 21-30 in three-free throw attempts. Despite having a 112-140 for 80% FT%, we can identify that the player averages 2.33 FT per possession: 140 free throws over 60 possessions. Note that this leads to 0.428 as the correction factor instead of 0.44.

Therefore we count only the 60 possessions as 120, similar to eFG%, and have a 112 over 120 contribution. This will result in an unbiased estimator; but once again favor free throws thanks to and-one situations.

So I guess the question is now this… how would you fix TS% to be an actual unbiased estimator?

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