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Estimating NBA Team Shot Selection Efficiency from
Aggregations of True, Continuous Shot Charts: A
Generalized Additive Model Approach
Basketball
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GAM dashboard: https://sportdataviz.syr.edu/TrueShotChart/
Open-Source Code Repository: https://github.com/Syracuse-University-SportAnalytics/continous_shot_selection
1. Introduction
We develop a novel type of basketball shot chart that uses a generalized additive model to estimate
total shot proficiency continuously in the half-court. This shot chart incorporates missed shots that
draw a shooting foul, and shot-pursuant free throw scoring, to determine total or true scoring yield
following a shot decision. A traditional discrete, or binned, shot chart is a size- and color-coded
spatial plot that tracks both location-dependent volume and points yielded from the floor,
respectively, on field goal attempts (FGAs). Shot charts provide distilled summaries of shotdistributional efficiency and are therefore a leading analytic tool for team game-planning (see, e.g.,
Papalexakis and Pelechrinis 2018; Jiao, Hu, and Yan 2021; Jieying, Guanyu, and Jun 2021; Franks,
Miller, Bornn, and Goldsberry 2015; Fichman and O’Brien 2019; Skinner and Goldman 2015;
Goldman and Rao 2011; Narayan 2019; Winston, Nestler, and Pelechrinis 2022). While descriptive,
traditional shot charts do not account for free throw scoring pursuant to shots from the field, nor
the locations of those shots from the field that lead to the free throw scoring opportunities. By
considering missed shots that result in a shooting foul, which are not considered FGAs, and made
shots that result in an “and-one” free throw opportunity, we correct for location-conditional scoring
yield estimate distortions in traditional shot charts.
Despite their limitations, traditional shot charts have been integral to improving shooting efficiency
in the NBA. In a 2020 NPR interview, Kirk Goldsberry, a basketball shot chart pioneer, discussed the
ability to infer disequilibria from shot chart aggregations: “It's that wild margin of inefficiency that’s
driven sort of these cartoonish trends and the rapid increase in 3-point shooting across the NBA.”
Shot charts, and their aggregations, have served as a key input in the NBA’s three-point revolution.
Figures 1a-b show the increased reliance on three-point shooting in the NBA during the past
decade, which has allowed teams to further leverage efficiency gains from the three-point premium.
The righthand side plot of Figure 1 shows the time trend in average number of three-point attempts
(3PAs) per NBA team-game. We observe a marked increase in the locally upward slope of the trend
beginning in 2013-14, coinciding with the ubiquitous adoption of SportVU player tracking
technology in all NBA arenas. The only precedent for such a steep rise in 3PA volume occurred from
1994 through 1997, when the three-point line was moved inward “above the break” from 23.75 feet
to 22 feet before being restored to its previous location. These negating leaguewide three-point line
position changes account for the locally steep rise-and-fall pattern of the mid-1990s. The lefthand
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side plot of Figure 1 features team-game level three-point attempt distributions for the 1980, 1985,
1990,…, and 2020 seasons (i.e., at five-year staggers).  This plot also demonstrates marked
rightward shifts in the team-game level 3PA density plots of 2015 and 2020 relative to earlier such
plots.
Figure 1a and 1b: NBA 3-Point Shots Taken Per Team-Game, Time Trend and Density Plots
2. Shot Charts with Spatial Continuity and True Estimates of
Point Yield
Shot charts help us understand trends in shot selection and the efficiency therefrom.  To obtain a
full picture of shot efficiency, however, we must consider all scoring pursuant to a given shot.
Herein, we overcome limitations of previous shot chart types by developing a generalized additive
model (GAM) continuous shot chart that accounts for all shot-pursuant free throw scoring. A GAM is
a type of generalized linear model, where the outcome variable depends linearly upon smooth
functions of the explanatory variables.  In fitting a generalized additive model, these smooth
functions are estimated. A GAM shot chart allows for the estimation of player or team shot
efficiency as a continuous, three-dimensional surface, where the third dimension is represented not
by the physical height of a surface at a particular (x,y) coordinate but via continuous color-coding.
Hastie and Tibshirani (1987) first developed GAMs to serve as a flexible and interpretable family of
non-linear models.
We call the GAM shot charts developed herein Continuous True Shot Charts and make them
available on a custom, searchable project dashboard at https://sportdataviz.syr.edu/TrueShotChart.
The project Github repository is available at: https://github.com/Syracuse-University-SportAnalytics/continous_shot_selection. Figure 2 shows a Continuous True Shot Chart for the 2021-22
Milwaukee Bucks. In a Continuous True Shot Chart, expected true point yield, from the field and line,
for each possible shot location of the half-court, is estimated, and the estimate is color-coded into
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the chart. Yellow ridge curves represent iso-yield curves or spatial level sets on yield such that
shots along a given set generate equal expected true points.
Figure 2: Continuous True Shot Chart, 2021-22 Milwaukee Bucks
The present work represents the first example of a continuous (GAM-based) true shot chart, to the
authors’ best knowledge. When examining the 2021-22 Milwaukee Bucks’ Continuous True Shot
Chart of Figure 2, we observe some surprises and some expected results. When including shotpursuant free throw scoring, we find the expected result that the Bucks scored proficiently at or
near the rim, as well as from the wing three-point region extended and the left corner three-point
area. However, we also find the surprising result that, for the Bucks of that season, long 2PAs were
not nearly as low yielding as the conventional wisdom would suggest. The black far-midrange
regions suggest that long 2PAs along the baseline extended were higher yielding than either close
midrange shots in the paint or midrange shots off the elbows. Even more surprisingly, long 2PAs
just inside the left three-point line break provided the Bucks with substantial yield (greater than 1.2
points per shot!). The 2021-22 Bucks obtained elite yield from a long 2PA region of the floor but
middling yield for much of the near-midrange paint region. Further, the Bucks obtained elite
scoring yield on 3PAs in some regions out to perhaps 26 feet, indicating that deep-threat 3PAs can
have substantial scoring benefits in addition to the spacing benefits they afford an offense.
However, any efficient scoring benefits of deep 3PAs fell off sharply beyond ~26 feet.
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At its limit, a modern shot chart reduces a team’s shooting regions down to 3PAs and 2PAs at or
near the rim. When considering shot-pursuant free-throw scoring, we observe that these distilled
areas are important but that there are potentially glaring exceptions to such reductionism. In Figure
3, we consider James Harden’s 2019-20 shot chart. We chose this player-season for several reasons.
This was Harden’s last year in Houston, where he was the on-court leader of a revolution in shot
chart reductionism, Daryl Morey having been the accompanying front office leader of said
revolution. Further, Harden won his third consecutive scoring title that season. Lastly, Harden has
an exceptionally high free throw rate such that we expect a great deal of additional information in
his true shot chart. In that season, Harden’s FT rate (FTA/FGA) was 0.557 or about 2.14 times the
league average of 0.26. We then consider the 2022-23 shot chart of Jimmy Butler in Figure 4. We
select Butler’s shot chart because he also has an all-time FT rate, especially among fellow Small
Forwards. In fact, the present authors cannot find a Small Forward with a higher seasonal FT rate
than Butler. His FT rate has been as high as 0.709 for a season and was 0.625 in 2022-23. As such,
we also expect Butler’s true shot charts to look substantially different than his traditional chart.
Figure 3: 2019-20 Continuous True Shot Chart of James Harden
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Figure 4: 2022-23 Continuous True Shot Chart of Jimmy Butler
Figures 3-4 have rich commonalities and differences. Buoyed by prolific free throw activity, both
charts feature the expected hot spots in parts of the three-point region, as well as at or near the rim.
However, some aspects of these shot charts reinforce surprises from the previous chart of Figure 2.
We observe areas of premium yield along the baseline extended long 2PA regions for Harden; these
same areas are of moderate yield for Butler, rather than low. As in the case of Figure 2, we observe
that premium long 2PA regions exist in the modern NBA shot chart when accounting for shotpursuant FT scoring.
For the 2021-22 Bucks again, Figure 5 presents a differential shot chart that visualizes the
difference, from each point in the half court, between the color coding, or yield, of the Continuous
True Shot Chart and that of a conventional shot chart that does not include shot-pursuant FT
scoring.
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Figure 5: Differential Shot Chart, Continuous True Shot Chart Yield minus Continuous
Conventional Shot Chart Yield, 2021-22 Milwaukee Bucks
We observe from Figure 5 that failing to include shot-pursuant FT scoring places a substantial, and
typically downward, bias on the estimated yield of shots near the basket. In general, the differential
is variable across the floor, suggesting a variable underlying shot position conditional FT rate. Thus,
traditional shot charts can substantially distort spatial characterizations of scoring yield. These
same results bear out for many other team seasons, as can be observed on our custom shot chart
dashboard at https://sportdataviz.syr.edu/TrueShotChart/. Figure 5 also explains the unexpected
result that, in a true shot chart, shots from the left baseline extended long 2PA region generated a
high yield for the 2021-22 Bucks. Shots from this region generated a high made FT rate. It can be
observed on the dashboard that James Harden’s 2019-20 yield premium from this region also
partly depended on a relatively high made FT rate for shots in that region.
Though largely reviled in the modern era of basketball analytics, long 2PAs in some cases generate
a high true scoring yield. In fact, we observe from Figure 2 that long 2PAs from this region for the
2021-22 Bucks returned a similar point yield, on average, than if the player had stepped back to the
adjacent three-point range! In this case, a comparison of Figures 2 and 5 suggests that the step-back
3PA garnered more points from the field but fewer points from the line. In consideration of the
latter effect, step-backs may discourage defenders from fouling because a foul on a missed 3PA
allows the offense 50% more points, in expectation, than fouling on a missed 2PA. Assuming league
average FT proficiency, a missed 3PA that draws a shooting foul would garner an expected
points, whereas a missed 2PA that draws a shooting foul would garner an expected
points. This premium is certainly not commensurate with any three-point premium
that exists in the absence of a shooting foul, as we will observe later in the paper. Therefore, fouling
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on a 3PA disproportionately punishes the defense. This calculus likely explains at least part of the
lower FT rate on 3PAs in Figure 5, as well as the lack of a sharp efficiency distinction between
certain 3PAs and their neighboring long 2PAs. In the subsequent section, we aggregate Continuous
True Shot Charts to determine whether results such as this suggest an erosion of the three-point
premium in recent NBA seasons.
3. Assessing Shot Selection Efficiency from Continuous True
Shot Charts
Another issue surrounding conventional shot chart analysis is the lack of a comprehensive measure
by which to summarize a shot chart in terms of shot selection efficiency. The present work
therefore further develops a shot selection measure called Shot Selection Efficiency that takes as
input a player’s or team’s expected true points and expected proportional volume from a simulated
grid of shots attempted somewhere on the half court and computes the player’s or team’s spatial
Pearson correlation between expected proportional volume and expected true points, from the field
and free throw line, across the half court.
This represents a measure of a player’s or team’s shot selection efficiency. The measure does not
depend on overall shooting ability but, rather, on how proficient a player or team is in shooting
from hot spots and avoiding cold spots on the floor, while taking into account shot location
conditional free throw scoring rate. A separate measure of exogenous shooting ability is developed
to estimate distinctly the average expected true points for a player (team) if they took a shot from
every possible location on the shot chart with equal likelihood. As such, this latter variable
estimates shooting ability but is exogenous of shot selection in that it weights each shot location
equally. With this control measure, we are subsequently able to validate the statistical contribution
or marginal effect of Shot Selection Efficiency to a basketball offense while controlling out the effect
of exogenous shooting ability. We validate the Shot Selection Efficiency measure by specifying an
XGBoost model, as well as linear, fixed effects regression models, at the team-season level that
estimate Pythagorean expected win percentage conditional on a set of variables that survived a
Variance Inflation Factor variable-paring model specification approach to limit multicollinearity.
These include exogenous shooting ability, defensive rating (DRtg), own turnover rate (TOV%),
offensive rebounding rate (ORB%), team fixed effects, Normalized Payroll, and Shot Selection
Efficiency. It stands to reason that these specific control variables survived VIF-paring. They
encompass the non-shooting offensive four factors (i.e., the offensive four factors not related to shot
selection efficiency or shooting ability) and the team’s overall defensive effectiveness in that
season.  Table 1 summarizes the dependent and independent variables of the study.
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Table 1: Feature Summary Statistics (n=210)
Characteristic     Median (SD) Range
Pyth.W.proportion 0.50 (0.14) 0.18 - 0.82
W.proportion 0.50 (0.14) 0.18 - 0.82
Net.Rating 0.40 (4.6) -10.5 - 11.6
Shot.selection.efficiency 0.27 (0.06) 0.01 - 0.44
Exogenous.shooting.ability 0.88 (0.05) 0.76 - 1.03
TOV% 12.60 (0.90) 9.90 - 14.90
ORB% 22.80 (2.21) 17.90 - 30.20
DRtg 111.2 (3.3) 102.9 - 120.0
Payroll (millions $) 123.57 (22.32) 79.18 - 192.91
Normalized.Payroll -0.02 (0.99) -2.93 - 2.59
Payroll normalized represents a team’s payroll in z-score terms, with respect to the distribution of
NBA team payrolls in that season. Net Rating is a team’s average scoring margin per 100
possessions, which approximates the average possessions per game in the 2022-23 NBA season. In
2022-23, the average points per NBA game was 114.7, and the average points per 100 NBA
possessions was 114.8. Therefore, average possessions per game in that season was (114.7/114.8)
or about 99.9.  This value has been converging upward throughout the course of the NBA playertracking movement, which has largely emphasized quick sets involving ball-movement, quicklydeveloping isolation plays, and pull-up 3PAs over slow sets involving foot-movement, slowlydeveloping isolation plays, and fewer pull-up 3PAs. Interestingly, rate stats for the NBA have been
measured in per 100 possession terms for decades, and the NBA’s analytic movement, which has
greatly propelled the prominence of these rate stats, has helped to converge these per 100
possession stats to the magnitude of per game stats on league average. At least for the time being,
this creates the advantage of less currency exchange in dealing with these two levels of statistics. In
analyzing league-wide data for 2022-23, the choice of per game or per 100 possession stats is
almost immaterial (material only at a margin of error less than 0.1%).
The median value for Exogenous Shooting Ability is 0.88, which matches the sample mean for this
variable. This is interpreted as median true points per shot for a team that randomly selects shot
locations, with equal likelihood, from across its shot chart. One might think of this team as one that
plays “hot potato” on offense or one programmed to get the ball to any point in the half court with
equal likelihood and then shoot. As mean and median are equal for this variable, such a team is
expected to score 0.88 true points per shot compared to the observed sample average true points
per shot of 1.13. We can meaningfully difference these two values to estimate that the net value of
shot selection, on league average, in our sample is approximately a quarter of a point per shot
taken.
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All teams in the sample exhibit a positive Shot Selection Efficiency value. Recall that this variable
represents the Pearson correlation between shot volume and true scoring from a grid of shot
locations across the floor. The median value is 0.27, suggesting that the typical NBA team is mildlyto-moderately effective in identifying hot spots and avoiding cold spots in shot selection.  The
minimum observed value is 0.01, suggesting there was a team in a season (the 2017-18 Portland
Trail Blazers) that had essentially no correlation between shot volume and true scoring in its shot
selection across the floor! Luckily for that team, it had a high exogenous shooting ability such that
their overall offense was effective. The maximum Shot Selection Efficiency was 0.44 and was set by
the 2021-22 Boston Celtics, a team well-known for analytically-driven sets and shot selection.
Another descriptive data summary question is whether the Shot Selection Efficiency has improved
as we progress deeper into the analytic era of basketball. Figure 6 addresses this question with a
time trend and embedded box plots for the variable by season.
Figure 6: Improvement of Shot Selection Efficiency over Time
In Figure 6, we observe a positive trend in expected team Shot Selection Efficiency over time. In
2016-17, the trend value was 0.23. By 2022-23, the trend value had risen to 0.32. With progressive
advances in analytics, NBA offenses have improved in Shot Selection Efficiency over time. This trend
result matches previous findings that offensive efficiency, and not defensive efficiency, is a leading
indicator of analytic advances (see, e.g., Ehrlich and Sanders 2021). Having summarized the key
model variables, a regression analysis will now estimate the respective marginal effects of Shot
Selection Efficiency, Exogenous Shooting Ability, and other variables upon team-season win
performance and net rating. Equations 1-3 provide the three primary model specifications.
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Equation 1-3 for Win Production Models
Model 1 estimates Pythagorean Expected Win Proportion for a team in a season as a function of the
team’s shot selection efficiency, exogenous shooting ability, and a vector of control variables in that
season. Our Pythagorean exponent for the model is 14, updating from similar earlier estimates, e.g.,
by Morey (1993), such that a team’s Pythagorean Expected Win Proportion in a season is given as
!"#$!"
!"#$!"%&"#$!".  Model 2 features the same righthand side but with each team’s (actual) Win
Proportion as the dependent variable, while Model 3 substitutes Net Rating as the dependent
variable. Results for the models are given In Table 2 as follows.
Table 2: Shot Selection Efficiency Effect on Win Performance Models
Pyth W Proportion W Proportion NetRtg
Predictors Estimates std.
Error p Estimates std.
Error P Estimates std.
Error p
(Intercept) 2.218 0.224 <0.001 2.097 0.249 <0.001 55.099 7.041 <0.001
DRtg -0.028 0.002 <0.001 -0.027 0.002 <0.001 -0.945 0.049 <0.001
gam shot
selection
efficiency
0.400 0.086 <0.001 0.334 0.095 0.001 14.149 2.691 <0.001
exogenous
shooting
ability
1.307 0.107 <0.001 1.393 0.119 <0.001 49.073 3.355 <0.001
off TOV% -0.025 0.006 <0.001 -0.021 0.007 0.001 -0.791 0.185 <0.001
off ORB% 0.018 0.002 <0.001 0.016 0.003 <0.001 0.535 0.073 <0.001
Observations 210 210 210
R2 / R2
adjusted
0.719 / 0.712 0.672 / 0.664 0.757 / 0.75
The regression data constitutes every NBA team-season from 2016-17 through 2022-23 (30 teams
multiplied by 7 years).  Shot Selection Efficiency has a positive and highly-significant conditional
effect on Pythagorean Expected Win Proportion, Win Proportion, and Net Rating. If an NBA team in a
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season can improve its weighted correlation between shot-volume and expected true points by 0.1
units, it gains an expected 0.04 units of win proportion or about 3.28 additional wins per 82-game
regular season, according to the Pythagorean Expected Win Proportion model. In other words, a
team’s expected regular season win count increases by 1 for roughly every 0.03 units of increase in
the spatial Pearson correlation between shot-volume and expected true points. To rephrase once
more, a standard deviation increase in Shot Selection Efficiency yields the expectation of a 0.17
standard deviation increase in Pythagorean Expected Win Proportion, which suggests that variation
in shot selection efficiency is a primary driver of winning in the NBA. From Model 3, we find that if
an NBA team in a season can improve Shot Selection Efficiency by 0.1 units, it gains an expected 1.41
points in terms of average game-level score margin (Net Rating).
Exogenous Shooting Ability is also significant and substantial in explaining team performance, as are
the other factor-type control variables. These results are both highly significant and substantial.
Further, shot selection gains do not necessarily require a costly roster overhaul. In principle at
least, shot selection efficiency is something that can be taught to players. We next examine whether
subsequent regressions will determine whether shot selection efficiency is a “moneyball,” or
inefficiently-priced source of wins. The three models feature strong explanatory power (𝑅'
between 0.672 and 0.757).
4. Is Shot Selection Efficiency Money(ball)?
Another model was estimated to consider the payroll-conditional effect of Shot Selection Efficiency
on team success by adding the features Normalized Payroll and Normalized Payroll Squared to
Model 1.  As stated, the Normalized Payroll feature is the raw payroll standardized (center and
scale) by season, which is calculated by subtracting the mean of payroll and then dividing by the
standard deviation. We also included a Normalized Payroll Squared feature to allow for a possible
quadratic payroll effect (due, e.g., to the possibility of some large-market teams being managerially
inefficient). Table 3 models represent payroll-augmented linear and quadratic versions of Model 1.
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Table 3: Moneyball Shot Selection Efficiency Opportunity
Pyth W Proportion Pyth W Proportion
Predictors Estimatesstd. Error P Estimatesstd. Error P
(Intercept) 2.218 0.224 <0.001 2.143 0.221 <0.001
DRtg -0.028 0.002 <0.001 -0.026 0.002 <0.001
gam shot selection
efficiency
0.400 0.086 <0.001 0.379 0.084 <0.001
exogenous shooting
ability
1.307 0.107 <0.001 1.195 0.111 <0.001
off TOV% -0.025 0.006 <0.001 -0.023 0.006 <0.001
off ORB% 0.018 0.002 <0.001 0.018 0.002 <0.001
normalized payroll 0.018 0.006 0.002
normalized payroll sq. -0.003 0.003 0.336
Observations 210 210
R2 / R2 adjusted 0.719 / 0.712 0.733 / 0.724
__________________
Payroll is significantly productive of expected wins in both models.  Further, the quadratic term on
Normalized Payroll is negative but insignificant in Model 2. The in-sample relationship between
payroll and expected wins is quadratic (inverted-U), but the coefficient on the quadratic payroll
term is insignificant. Even conditional on payroll, Shot Selection Efficiency is positive and significant,
indicating that the win productivity of Shot Selection Efficiency is not explained by variation in
team-season payroll. This may suggest either that players have different observed levels of Shot
Selection Efficiency but that the player labor market is not valuing higher levels of the attribute or
that team coaching and analytic staffs (i.e., team culture and values) are largely responsible for
variation in team Shot Selection Efficiency. In either case, Shot Selection Efficiency is a “moneyball” or
supra-payroll source of wins in the NBA. The magnitude of the coefficient for Shot Selection
Efficiency remains essentially the same when we control for team-season payroll.  This suggests
that the NBA player labor market is essentially not pricing in any variation in Shot Selection
Efficiency observed at the player level. Rather, this variation is driven by team coaching/analytics.
5. Robustness Check of Shot Selection Efficiency using XGBoost
To robustness check our linear, fixed effect model results, we fit an XGBoost model (Liu and Lust,
2020) to understand how Shot Selection Efficiency, along with other covariates, relate to
Pythagorean Expected Win Proportion. The covariates of interest include Normalized Payroll, DRtg,
ORB%, TOV%, Exogenous Shooting Ability, and Season. We partitioned 90% of the data into a
training set and 10% into a test set. The out-of-sample mean absolute error was 0.052 and the out
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of sample R-squared was 0.625, comparable to that observed in the linear, fixed effects models. For
the full model, the mean absolute error was 0.0004, and the R-squared was 0.999989. Also, Shapley
Additive exPlanations (SHAP) (Lundberg and Lee, 2017) values were calculated. Figure 7 shows a
beeswarm summary plot, which demonstrates the estimated effect of each parameter on expected
wins.
Figure 7: Beeswarm Summary Plot
Shot Selection Efficiency has a positive estimated Shapley Value such that the XGBoost results are
consistent with those of the linear, fixed effects models. Two SHAP dependency plots were also
rendered, where each shows estimated marginal effect that each feature has on Pythagorean
Expected Win Proportion. Figure 8 demonstrates the monotonically positive impact that Shot
Selection Efficiency has on Pythagorean Expected Win Proportion. This confirms that shot selection
efficiency is indeed important to team performance. For teams with a Shot Selection Efficiency
greater than 0.23, shot selection positively contributed toward winning, according to Figure 8.
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Figure 8: Shot Selection Efficiency Effect on Pythagorean Wins
Figure 9: Payroll Effect on Pythagorean Wins
6. Does an NBA 3PA Premium Remain according to
Aggregations of True Shot Charts?
By accounting for FT scoring, Continuous True Shot Charts can comprehensively address many
fundamental questions in basketball, such as: Does a three-point premium still exist in the NBA given
the escalation in three-point attempt volume? See, for example, Ehrlich and Sanders (2021), who
develop the first discrete, true shot chart to find evidence that the three-point premium eroded to
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some degree during the 2010s. While aggregations of conventional shot charts suggest the threepoint yield premium remains intact for the NBA, what happens when we consider shot-pursuant FT
scoring for recent seasons?
To consider this question, we aggregate both Continuous True Shot Charts and traditional shot
charts for all NBA shots taken in each season from 2016-17 through 2021-22. Doing so allows us to
estimate the three-point premium, with and without shot-pursuant free throw scoring in each
season. Figure 10 reports those results, where the red trend line is estimated from aggregations of
traditional shot charts and therefore reports the estimated difference between 3PA and 2PA yield
when not considering shot-pursuant FTAs. The blue line is estimated from aggregations of
Continuous True Shot Charts and therefore reports the estimated difference between 3PA and 2PA
yield when considering shot-pursuant FTAs.
Figures 10: NBA Three-Point Premium, 2016-17 – 2021-22: A Tale of Two Shot Charts
Observing the red trend line, we find a significant and fairly substantial leaguewide three-point
premium that existed from 2016-17 through 2020-21 when considering aggregations of traditional
shot charts. That is, NBA players scored significantly more proficiently from the field on three-point
attempts than on two-point attempts from 2016-17 through 2020-21.  In magnitude, this premium
ranged between 0.025 and 0.064 points per three-point shot over these years.  In 2021-22, there
was no significant leaguewide three-point premium or dispremium when considering aggregations
of traditional shot charts.  However, when considering aggregations of true shot charts, we find a
significant and deepening three-point dispremium at the league level since the 2018-19 season! In
2021-22, this dispremium swelled to -0.066 points per three-point shot or roughly a point less for
each 15 three-point shots.  Despite these findings, we observe that leaguewide three-point shot
volume has continued to rise dramatically.  It appears as though the typical team continued to
follow perceived three-point premia from aggregations of traditional shot charts rather than
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significant dispremia from aggregations of true shot charts. In the case of 3PA volume, there is
evidence that NBA teams overshot their market, assisted by a mis-specified analytic tool in the form
of traditional shot charts.
7. Conclusion
In this work, we have developed a set of novel GAM-estimated, Continuous True Shot Charts for
NBA teams and players and have demonstrated that they correct distortions in scoring yield
estimates found in traditional shot charts. They do so by including shot-pursuant FT scoring into
the color coding (scoring yield estimate) of the chart at all locations. We further construct a
measure of team shot selection efficiency as a means to summarize team and player shot charts in
terms of efficiency.  This measure represents the Pearson correlation between expected
proportional volume and expected true points, from the field and free throw line, across the half
court. Through linear, polynomial, and XGBoost modeling, we find that shot selection efficiency
contributes to team expected win proportion significantly and substantially.  Further, shot selection
efficiency is found to be a “moneyball” or supra-payroll source of wins in the NBA. Lastly, we use
Continuous True Shot Charts to find that there has been a widening NBA three-point dispremium
since 2018-19.  This dispremium result is not available if one aggregates from traditional shot
charts, which do not consider shot-pursuant FT scoring. Continuous True Shot Charts are both
novel and meaningful in basketball league settings.
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