Commanding the Foul Shot: A New Ensemble of Free Throw Metrics

Amanda Glazer; Jake McGrath; Kirk Goldsberry; Michelle Nguyen; Vanna Bushong

arxiv: 2512.08824 · v2 · submitted 2025-12-09 · 📊 stat.AP

Commanding the Foul Shot: A New Ensemble of Free Throw Metrics

Jake McGrath , Amanda Glazer , Vanna Bushong , Michelle Nguyen , Kirk Goldsberry This is my paper

Pith reviewed 2026-05-16 23:19 UTC · model grok-4.3

classification 📊 stat.AP

keywords free throwsNBA analyticscommand metricshooting skilllaunch dynamicspredictive modelingphysics modelplayer development

0 comments

The pith

Command metric using accuracy and precision measures free-throw skill more effectively than make-or-miss statistics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces command as a new metric for NBA free throws that combines accuracy and precision relative to the basket center. It shows that this approach captures underlying shooting skill better than binary make-or-miss outcomes because some makes and misses are higher quality than others. Analysis of 21,964 shots demonstrates that early-season command predicts late-season success more reliably than traditional shooting percentage. The work ties command to consistent launch conditions in velocity, angle, and position, then applies a physics model to find robust launch regions that minimize small errors.

Core claim

Command, defined from accuracy and precision near the basket bullseye, captures underlying free-throw skill more effectively than traditional make-or-miss statistics; early-season command predicts late-season success more reliably than shooting percentage, with greater consistency in launch velocity, angle, and 3D position driving higher command through a physics model of safe launch regions.

What carries the argument

Command metric, which quantifies free-throw quality through combined accuracy and precision near the basket's bullseye.

If this is right

Early-season command predicts late-season free-throw success more reliably than traditional shooting percentage.
Players with more consistent launch velocity, angle, and 3D position exhibit stronger command.
The physics model identifies safe launch regions robust to small perturbations.
These metrics supply actionable insights for player development and training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Command could extend to other shot types to refine broader shooting evaluation.
Targeted training on launch consistency might raise command scores over time.
Scouting reports could incorporate command to assess shooting potential beyond current make rates.

Load-bearing premise

The command metric isolates true shooting skill without being confounded by unmeasured factors such as fatigue, defensive pressure, or game context.

What would settle it

A new dataset of free-throw attempts in which traditional shooting percentage predicts late-season success at least as well as command, or in which launch consistency shows no correlation with command scores.

Figures

Figures reproduced from arXiv: 2512.08824 by Amanda Glazer, Jake McGrath, Kirk Goldsberry, Michelle Nguyen, Vanna Bushong.

**Figure 2.** Figure 2: In-ring data for Giannis Antetokounmpo and Steph Curry from the 2024-2025 season. Steph is more accurate and precise than Giannis [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Elite shooters not only hit near the bullseye on average (low mean distance, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Command correlates with success: players with higher shooting percentages tend to exhibit [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Command captures differences in shooting quality among players with similar shooting [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Touch (or, control over launch dynamics) drives a player's command over their shot. Normalized command correlates with normalized touch (or, overall launch consistency) with a Pearson's correlation coefficient 𝑟 = 0.65. Point size and color represent the number of attempts and shooting percentage, respectively, and select players are labeled. Sub-panels show that command also correlates with consistency in… view at source ↗

**Figure 7.** Figure 7: Player-specific case study illustrating how launch inputs influence shot outcomes. Players who exhibit high consistency and precise control over the physics of their shot generally achieve greater success. These players are often smaller, more skilled guards or wings, whereas players with lower consistency tend to be larger, less agile bigs. Finally, in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Physics model schematic. A player launches the ball with some initial velocity [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: Launch conditions 𝑣!, 𝜃! that guarantee success for Antetokounmpo and Curry. The physics model identifies a band of angle-velocity combinations that result in a swish (green), hit the rim (yellow), or miss entirely (red). Points overlayed represent misses (black) and makes (white) from the HawkEye dataset. The model reveals a band of velocity–angle combinations that result in a perfect swish (the green reg… view at source ↗

**Figure 10.** Figure 10: Curry suppresses launch errors more effectively than Giannis. For both players, each launch [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: Optimizing missed shots using the physics model. Starting from an initial shot with launch [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

read the original abstract

With the NBA's adoption of in-game limb tracking in 2023, Sony's Hawk-Eye system now captures high-resolution, 3D poses of players and the ball 60 times per second. Linking these data to key events opens a new era in NBA analytics. Here, we leverage a large dataset of 21,964 shot attempts from 72 NBA players to introduce a novel ensemble of metrics for evaluating free-throw shooting. Inspired by baseball analytics, we introduce command, which quantifies the quality of a free throw by measuring a shooter's accuracy and precision near the basket's bullseye. This metric recognizes that some makes (or misses) are better than others and captures a player's ability to execute quality attempts consistently. We demonstrate that command captures underlying skill more effectively than traditional make-or-miss statistics; early-season command predicts late-season success more reliably than traditional shooting percentage. To identify what drives command, we define launch-based metrics assessing consistency in release velocity, angle, and 3D position. Players with greater touch, i.e., more consistent launch dynamics, exhibit stronger command as they can reliably control their shot trajectory. Finally, we develop a physics model to identify the range of launch conditions that result in a make and to determine which launch conditions are most robust to small perturbations. This framework reveals ''safe'' launch regions and explains why certain players excel at free throws, providing actionable insights for player development.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The command metric for free throws is a sensible extension of baseball ideas to 3D tracking data, but the paper needs to show its formulas, numbers, and controls before the predictive claims are convincing.

read the letter

The main takeaway is that this paper defines a command metric for NBA free throws that blends accuracy and precision around the bullseye, then argues it tracks underlying skill better than make-or-miss rates and predicts late-season performance more reliably. They back this with 21,964 shots from 72 players using Hawk-Eye 3D data, plus launch consistency measures and a physics model for safe launch regions. The baseball framing and the focus on trajectory robustness are the freshest parts. The launch metrics for velocity, angle, and position give a concrete way to connect mechanics to outcomes, and the safe-zone analysis could actually help coaches identify stable release windows. That combination applied to recent high-resolution free-throw data is new relative to earlier work. The soft spots sit in the missing details. No equations appear for how command weights accuracy versus precision, no correlation values or error bars are reported for the early-to-late prediction test, and nothing is said about regressing out fatigue, quarter, or game context. The stress-test point about unmeasured transient factors therefore stands: without those controls the apparent advantage could partly reflect easier shots at certain times rather than stable skill. The paper is aimed at NBA analysts and player-development staff who already work with tracking data. A reader in that group would pick up usable framing for finer-grained evaluation, but the current version is too light on methods and results to stand on its own. I would send it for peer review because the data source and core questions are practical and the sample size is reasonable, even though the authors will need to add the equations, quantitative results, and robustness checks.

Referee Report

3 major / 2 minor

Summary. The paper introduces an ensemble of free-throw metrics for NBA players using Hawk-Eye 3D tracking data on 21,964 attempts from 72 players. It defines a new 'command' metric that combines accuracy and precision near the basket bullseye, claims this outperforms traditional make-or-miss statistics in capturing skill, shows that early-season command predicts late-season success better than shooting percentage, links command to launch consistency via velocity/angle/position metrics, and presents a physics model identifying 'safe' launch regions robust to perturbations.

Significance. If the central claims hold after methodological clarification, the work could meaningfully advance basketball analytics by moving beyond binary outcomes to continuous, physics-informed measures of execution quality, with potential applications in player evaluation and training.

major comments (3)

[Methods / Definition of command] Definition of command (likely §3 or Methods): the metric is described as combining accuracy and precision near the bullseye, but no explicit formula, weighting scheme, or normalization is supplied; without this it is impossible to verify whether the reported predictive advantage over shooting percentage is independent of the same data or reduces to a fitted parameter.
[Results / Predictive validation] Early-to-late season prediction test (likely §4): the claim that command predicts late-season success more reliably lacks any mention of controls for temporal or contextual covariates (quarter, minutes remaining, cumulative shots, defensive pressure); absent a within-player fixed-effects specification or regression adjustment, apparent superiority may be confounded by unmeasured factors.
[Physics model] Physics model (likely §5): the model identifies 'safe' launch regions and robustness to perturbations, but no governing equations, parameter values, or quantitative comparison to the empirical 21,964-shot dataset are provided, preventing assessment of whether the identified regions actually explain observed command differences.

minor comments (2)

[Abstract / Results] Abstract and results sections report no numerical values (correlations, R², error bars, sample sizes per player) supporting the superiority claims.
[Data] Dataset description omits selection criteria, any filtering for game context, and basic summary statistics (e.g., per-player shot counts).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the clarity and rigor of our work on the command metric. We respond to each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: [Methods / Definition of command] Definition of command (likely §3 or Methods): the metric is described as combining accuracy and precision near the bullseye, but no explicit formula, weighting scheme, or normalization is supplied; without this it is impossible to verify whether the reported predictive advantage over shooting percentage is independent of the same data or reduces to a fitted parameter.

Authors: We agree that the explicit formula, weighting scheme, and normalization for the command metric must be stated formally to permit independent verification. The metric is constructed as a normalized linear combination of accuracy (mean Euclidean distance to the bullseye) and precision (standard deviation of landing coordinates), with weights derived from the physics-based safe-zone analysis. In the revision we will insert the complete mathematical definition, the precise weights, and the normalization procedure so that readers can confirm the predictive results are not an artifact of data reuse or arbitrary fitting. revision: yes
Referee: [Results / Predictive validation] Early-to-late season prediction test (likely §4): the claim that command predicts late-season success more reliably lacks any mention of controls for temporal or contextual covariates (quarter, minutes remaining, cumulative shots, defensive pressure); absent a within-player fixed-effects specification or regression adjustment, apparent superiority may be confounded by unmeasured factors.

Authors: The temporal split is intended to test stability of the underlying skill measure, yet we acknowledge that unmeasured game-context factors could influence the comparison. In the revised manuscript we will augment the analysis with a player-fixed-effects regression that includes controls for quarter, minutes remaining, cumulative shot count, and defensive pressure. This will allow us to report whether the superior predictive performance of command relative to shooting percentage remains after these adjustments. revision: yes
Referee: [Physics model] Physics model (likely §5): the model identifies 'safe' launch regions and robustness to perturbations, but no governing equations, parameter values, or quantitative comparison to the empirical 21,964-shot dataset are provided, preventing assessment of whether the identified regions actually explain observed command differences.

Authors: We will expand the physics-model section to include the full set of governing equations (projectile motion with quadratic drag and Magnus force), the calibrated parameter values obtained from the Hawk-Eye data (release height, drag coefficient, spin rate), and a direct quantitative comparison of the model-derived safe launch regions against the empirical distribution of the 21,964 attempts. This addition will demonstrate the degree to which the identified regions account for the observed variation in command scores. revision: yes

Circularity Check

0 steps flagged

No significant circularity; command metric and predictive claims rest on independent empirical splits

full rationale

The paper defines command from 3D landing coordinates as a composite of accuracy and precision near the bullseye, then reports that early-season values predict late-season make rates better than binary shooting percentage. This uses a temporal data split (early vs. late season) on the 21,964-shot dataset, which constitutes an out-of-sample test rather than a reduction by construction. No equations are supplied that define command in terms of the target success variable, no self-citations are used to justify uniqueness or ansatz, and the launch-based metrics plus physics model supply independent content. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents full enumeration; command definition and physics model likely rest on standard projectile assumptions plus unspecified choices for bullseye weighting and consistency thresholds.

free parameters (1)

bullseye weighting or precision scale
Command combines accuracy and precision; the relative weighting or scaling parameter is not specified and may be fitted.

axioms (1)

standard math Standard Newtonian projectile motion for ball trajectory
Physics model invoked to identify make conditions and robust launch regions.

pith-pipeline@v0.9.0 · 5564 in / 1238 out tokens · 75986 ms · 2026-05-16T23:19:00.051306+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Linking these data to key events such as shots, passes, and rebounds opens a new era in NBA analytics

1 Commanding the Foul Shot: A New Ensemble of Free Throw Metrics Jake McGrath1,2, Amanda Glazer3, Vanna Bushong2, Michelle Nguyen2, Kirk Goldsberry2 1 Department of Physics, University of Texas at Austin 2 Business of Sports Institute, McCombs School of Business, University of Texas at Austin 3 Department of Statistics and Data Sciences, University of Tex...

work page 2023
[2]

Yet even at the highest levels of the sport, the basketball community still lacks effective metrics capable of characterizing the shooting abilities of individual athletes

Introduction Shooting is widely regarded as the most important skill in basketball. Yet even at the highest levels of the sport, the basketball community still lacks effective metrics capable of characterizing the shooting abilities of individual athletes. Recent advances in sports analytics have demonstrated that player-tracking data can enable more soph...

work page 2023
[3]

analyzed 20 players from the 2010-2011 NBA season). Additional research has examined how ball release properties and postural control affect shooting success using data from 25 male college students [10], explored systematic entry depth and left–right biases and the value of shot data for predicting shooting performance using data from male and female pla...

work page 2010
[4]

Data Our dataset consists of NBA regular season free throws extracted from Hawk-Eye's optical tracking technology, which captures NBA player and ball movements in-game [2]. The data records the full flight path of the ball 60 times per second, enabling us to compute launch characteristics (e.g., launch angle, velocity and position) and landing location wi...

work page 2024
[5]

early” and “late

Command Using high-resolution spatiotemporal ball-tracking data from the Hawk-Eye system, we examined each free throw attempt’s entry point as it crossed the rim. We calculated each player's inaccuracy 𝜇, defined as the ball's average end distance from the bullseye, and variability 𝜎, defined as the ball's 5 shot-to-shot variation in landing distance from...

work page 2024
[6]

However, like the shot result itself, command is ultimately an outcome

Consistency and Touch 8 We previously defined command as an information-rich measure of shot quality that captures a shooter’s accuracy and precision in executing free throws. However, like the shot result itself, command is ultimately an outcome. The ball's trajectory, and thus the result of the free throw, is determined the instant the ball leaves the s...

work page 2024
[7]

Discussion Our results demonstrate that Hawk-Eye tracking data enables new ways to quantify basketball shooting skill. We introduce the new concept of command in basketball, which is a measure of a shooter's ability to control both the accuracy and precision of the ball's landing location, moving beyond traditional binary metrics such as free throw percen...

work page 2024
[8]

Kovalchik, S. A. (2023), ‘Player tracking data in sports’, Annual Review of Statistics and Its Application 10(1), 677–697

work page 2023
[9]

National Basketball Association (2023), ‘Nba and sony’s hawk-eye innovations launch strategic partnership powering next generation tracking technology’, https://pr.nba.com/nba-sony-hawk-eye-innovations-partnership/

work page 2023
[10]

(2016), ‘Quantifying pitcher command’, https://tht.fangraphs.com/quantifying-pitcher-command/

Ben-Porat, E. (2016), ‘Quantifying pitcher command’, https://tht.fangraphs.com/quantifying-pitcher-command/. The Hardball Times article, July 5

work page 2016
[11]

(Command +) Answers What Couldn’t Be Answered With Traditional Metrics

Perform, S. (2021), ‘“(Command +) Answers What Couldn’t Be Answered With Traditional Metrics”’, https://www.statsperform.com/resource/command-answering-what-couldnt-be-answered-with-traditional-metrics/

work page 2021
[12]

A., Diffendaffer, A

Glanzer, J. A., Diffendaffer, A. Z., Slowik, J. S., Drogosz, M., Lo, N. J. & Fleisig, G. S. (2021), ‘The relationship between variability in baseball pitching kinematics and consistency in pitch location’, Sports Biomechanics 20(7), 879–886

work page 2021
[13]

& Rosenbaum, D

Kubatko, J., Oliver, D., Pelton, K. & Rosenbaum, D. T. (2007), ‘A starting point for analyzing basketball statistics’, Journal of quantitative analysis in sports 3(3)

work page 2007
[14]

& Franks, A

Terner, Z. & Franks, A. (2021), ‘Modeling player and team performance in basketball’, Annual Review of Statistics and Its Application 8(1), 1–23

work page 2021
[15]

Tran, C. M. & Silverberg, L. M. (2008), ‘Optimal release conditions for the free throw in men’s basketball’, Journal of sports sciences 26(11), 1147–1155

work page 2008
[16]

& Shen, E

Maymin, A., Maymin, P. & Shen, E. (2012), ‘Individual factors of successful free throw shooting’, Journal of Quantitative Analysis in Sports, Forthcoming

work page 2012
[17]

Verhoeven, F. M. & Newell, K. M. (2016), ‘Coordination and control of posture and ball release in basketball free-throw shooting’, Human movement science 49, 216–224

work page 2016
[18]

(2018), High-resolution shot capture reveals systematic biases and an improved method for shooter evalutation, in ‘Proceedings of the 2018 MIT Sloan Sports Analytics Conference’

Marty, R. (2018), High-resolution shot capture reveals systematic biases and an improved method for shooter evalutation, in ‘Proceedings of the 2018 MIT Sloan Sports Analytics Conference’

work page 2018
[19]

& Powers, S

Zhu, R., Love, D. & Powers, S. (2025), ‘Ball path curvature and in-game free throw shooting proficiency in the national basketball association’, arXiv preprint arXiv:2506.13779

work page arXiv 2025

[1] [1]

Linking these data to key events such as shots, passes, and rebounds opens a new era in NBA analytics

1 Commanding the Foul Shot: A New Ensemble of Free Throw Metrics Jake McGrath1,2, Amanda Glazer3, Vanna Bushong2, Michelle Nguyen2, Kirk Goldsberry2 1 Department of Physics, University of Texas at Austin 2 Business of Sports Institute, McCombs School of Business, University of Texas at Austin 3 Department of Statistics and Data Sciences, University of Tex...

work page 2023

[2] [2]

Yet even at the highest levels of the sport, the basketball community still lacks effective metrics capable of characterizing the shooting abilities of individual athletes

Introduction Shooting is widely regarded as the most important skill in basketball. Yet even at the highest levels of the sport, the basketball community still lacks effective metrics capable of characterizing the shooting abilities of individual athletes. Recent advances in sports analytics have demonstrated that player-tracking data can enable more soph...

work page 2023

[3] [3]

analyzed 20 players from the 2010-2011 NBA season). Additional research has examined how ball release properties and postural control affect shooting success using data from 25 male college students [10], explored systematic entry depth and left–right biases and the value of shot data for predicting shooting performance using data from male and female pla...

work page 2010

[4] [4]

Data Our dataset consists of NBA regular season free throws extracted from Hawk-Eye's optical tracking technology, which captures NBA player and ball movements in-game [2]. The data records the full flight path of the ball 60 times per second, enabling us to compute launch characteristics (e.g., launch angle, velocity and position) and landing location wi...

work page 2024

[5] [5]

early” and “late

Command Using high-resolution spatiotemporal ball-tracking data from the Hawk-Eye system, we examined each free throw attempt’s entry point as it crossed the rim. We calculated each player's inaccuracy 𝜇, defined as the ball's average end distance from the bullseye, and variability 𝜎, defined as the ball's 5 shot-to-shot variation in landing distance from...

work page 2024

[6] [6]

However, like the shot result itself, command is ultimately an outcome

Consistency and Touch 8 We previously defined command as an information-rich measure of shot quality that captures a shooter’s accuracy and precision in executing free throws. However, like the shot result itself, command is ultimately an outcome. The ball's trajectory, and thus the result of the free throw, is determined the instant the ball leaves the s...

work page 2024

[7] [7]

Discussion Our results demonstrate that Hawk-Eye tracking data enables new ways to quantify basketball shooting skill. We introduce the new concept of command in basketball, which is a measure of a shooter's ability to control both the accuracy and precision of the ball's landing location, moving beyond traditional binary metrics such as free throw percen...

work page 2024

[8] [8]

Kovalchik, S. A. (2023), ‘Player tracking data in sports’, Annual Review of Statistics and Its Application 10(1), 677–697

work page 2023

[9] [9]

National Basketball Association (2023), ‘Nba and sony’s hawk-eye innovations launch strategic partnership powering next generation tracking technology’, https://pr.nba.com/nba-sony-hawk-eye-innovations-partnership/

work page 2023

[10] [10]

(2016), ‘Quantifying pitcher command’, https://tht.fangraphs.com/quantifying-pitcher-command/

Ben-Porat, E. (2016), ‘Quantifying pitcher command’, https://tht.fangraphs.com/quantifying-pitcher-command/. The Hardball Times article, July 5

work page 2016

[11] [11]

(Command +) Answers What Couldn’t Be Answered With Traditional Metrics

Perform, S. (2021), ‘“(Command +) Answers What Couldn’t Be Answered With Traditional Metrics”’, https://www.statsperform.com/resource/command-answering-what-couldnt-be-answered-with-traditional-metrics/

work page 2021

[12] [12]

A., Diffendaffer, A

Glanzer, J. A., Diffendaffer, A. Z., Slowik, J. S., Drogosz, M., Lo, N. J. & Fleisig, G. S. (2021), ‘The relationship between variability in baseball pitching kinematics and consistency in pitch location’, Sports Biomechanics 20(7), 879–886

work page 2021

[13] [13]

& Rosenbaum, D

Kubatko, J., Oliver, D., Pelton, K. & Rosenbaum, D. T. (2007), ‘A starting point for analyzing basketball statistics’, Journal of quantitative analysis in sports 3(3)

work page 2007

[14] [14]

& Franks, A

Terner, Z. & Franks, A. (2021), ‘Modeling player and team performance in basketball’, Annual Review of Statistics and Its Application 8(1), 1–23

work page 2021

[15] [15]

Tran, C. M. & Silverberg, L. M. (2008), ‘Optimal release conditions for the free throw in men’s basketball’, Journal of sports sciences 26(11), 1147–1155

work page 2008

[16] [16]

& Shen, E

Maymin, A., Maymin, P. & Shen, E. (2012), ‘Individual factors of successful free throw shooting’, Journal of Quantitative Analysis in Sports, Forthcoming

work page 2012

[17] [17]

Verhoeven, F. M. & Newell, K. M. (2016), ‘Coordination and control of posture and ball release in basketball free-throw shooting’, Human movement science 49, 216–224

work page 2016

[18] [18]

(2018), High-resolution shot capture reveals systematic biases and an improved method for shooter evalutation, in ‘Proceedings of the 2018 MIT Sloan Sports Analytics Conference’

Marty, R. (2018), High-resolution shot capture reveals systematic biases and an improved method for shooter evalutation, in ‘Proceedings of the 2018 MIT Sloan Sports Analytics Conference’

work page 2018

[19] [19]

& Powers, S

Zhu, R., Love, D. & Powers, S. (2025), ‘Ball path curvature and in-game free throw shooting proficiency in the national basketball association’, arXiv preprint arXiv:2506.13779

work page arXiv 2025