Basketball Free Throw Form

Summary

This experiment investigates basketball free throw form. Central composite design to maximize accuracy and arc consistency by tuning release angle, release height, and backspin rate.

The design varies 3 factors: release angle deg (deg), ranging from 45 to 55, release height m (m), ranging from 2.0 to 2.5, and backspin rpm (rpm), ranging from 100 to 300. The goal is to optimize 2 responses: accuracy pct (%) (maximize) and arc consistency (pts) (maximize). Fixed conditions held constant across all runs include distance = 4.6m, ball = size_7.

A Central Composite Design (CCD) was selected to fit a full quadratic response surface model, including curvature and interaction effects. With 3 factors this produces 22 runs including center points and axial (star) points that extend beyond the factorial range.

Quadratic response surface models were fitted to capture potential curvature and factor interactions. The RSM contour plots below visualize how pairs of factors jointly affect each response.

Key Findings

For accuracy pct, the most influential factors were release angle deg (48.1%), release height m (30.9%), backspin rpm (20.9%). The best observed value was 76.0 (at release angle deg = 50, release height m = 2.25, backspin rpm = 200).

For arc consistency, the most influential factors were release angle deg (48.7%), release height m (35.7%), backspin rpm (15.6%). The best observed value was 7.4 (at release angle deg = 50, release height m = 2.25, backspin rpm = 200).

Recommended Next Steps

Run confirmation experiments at the predicted optimal settings to validate the model.
Consider whether any fixed factors should be varied in a future study.

Experimental Setup

Factors

Factor	Low	High	Unit
`release_angle_deg`	45	55	deg
`release_height_m`	2.0	2.5	m
`backspin_rpm`	100	300	rpm

Fixed: distance = 4.6m, ball = size_7

Responses

Response	Direction	Unit
`accuracy_pct`	↑ maximize	%
`arc_consistency`	↑ maximize	pts

Configuration

use_cases/212_basketball_shooting/config.json

{
  "metadata": {
    "name": "Basketball Free Throw Form",
    "description": "Central composite design to maximize accuracy and arc consistency by tuning release angle, release height, and backspin rate"
  },
  "factors": [
    {
      "name": "release_angle_deg",
      "levels": [
        "45",
        "55"
      ],
      "type": "continuous",
      "unit": "deg"
    },
    {
      "name": "release_height_m",
      "levels": [
        "2.0",
        "2.5"
      ],
      "type": "continuous",
      "unit": "m"
    },
    {
      "name": "backspin_rpm",
      "levels": [
        "100",
        "300"
      ],
      "type": "continuous",
      "unit": "rpm"
    }
  ],
  "fixed_factors": {
    "distance": "4.6m",
    "ball": "size_7"
  },
  "responses": [
    {
      "name": "accuracy_pct",
      "optimize": "maximize",
      "unit": "%"
    },
    {
      "name": "arc_consistency",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "central_composite",
    "test_script": "use_cases/212_basketball_shooting/sim.sh"
  }
}

Experimental Matrix

The Central Composite Design produces 22 runs. Each row is one experiment with specific factor settings.

Run	`release_angle_deg`	`release_height_m`	`backspin_rpm`
1	50	2.25	200
2	55	2	300
3	45	2.5	100
4	50	2.70644	200
5	50	2.25	200
6	40.8713	2.25	200
7	50	2.25	17.4258
8	50	2.25	200
9	55	2.5	100
10	59.1287	2.25	200
11	50	2.25	200
12	50	1.79356	200
13	50	2.25	200
14	45	2	300
15	50	2.25	200
16	55	2	100
17	50	2.25	382.574
18	55	2.5	300
19	50	2.25	200
20	45	2	100
21	45	2.5	300
22	50	2.25	200

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/212_basketball_shooting/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/212_basketball_shooting/config.json \
    --output use_cases/212_basketball_shooting/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/212_basketball_shooting/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/212_basketball_shooting/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/212_basketball_shooting/config.json

6

Multi-objective optimization

With 2 competing responses, use --multi to find the best compromise via Derringer–Suich desirability.

Terminal

$ doe optimize --config use_cases/212_basketball_shooting/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/212_basketball_shooting/config.json \
    --output use_cases/212_basketball_shooting/results/report.html

Features Exercised

Feature	Value
Design type	`central_composite`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	2 (accuracy_pct ↑, arc_consistency ↑)
Total runs	22

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: accuracy_pct

Top factors: release_angle_deg (48.1%), release_height_m (30.9%), backspin_rpm (20.9%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
release_angle_deg	4	342.8409	85.7102	1.644	0.2456
release_height_m	4	186.6742	46.6686	0.895	0.5052
backspin_rpm	4	99.3409	24.8352	0.476	0.7527
Lack	of	Fit	2	176.8598	88.4299
Pure	Error	7	364.8750
Error	9	541.7348	52.1250
Total	21	1170.5909	55.7424

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for accuracy_pct

Half-Normal Plot of Effects

Model Diagnostics

Response: arc_consistency

Top factors: release_angle_deg (48.7%), release_height_m (35.7%), backspin_rpm (15.6%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
release_angle_deg	4	8.5115	2.1279	2.082	0.1659
release_height_m	4	4.8082	1.2020	1.176	0.3836
backspin_rpm	4	1.6140	0.4035	0.395	0.8077
Lack	of	Fit	2	3.1295	1.5647
Pure	Error	7	7.1550
Error	9	10.2845	1.0221
Total	21	25.2182	1.2009

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for arc_consistency

Half-Normal Plot of Effects

Model Diagnostics

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

accuracy pct release angle deg vs backspin rpm

accuracy pct release angle deg vs release height m

accuracy pct release height m vs backspin rpm

arc consistency release angle deg vs backspin rpm

arc consistency release angle deg vs release height m

arc consistency release height m vs backspin rpm

Multi-Objective Optimization

When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.

Overall Desirability

D = 0.9545

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`accuracy_pct`	1.5	0.9545	76.00 0.9545 76.00 %	↑
`arc_consistency`	1.0	0.9545	7.40 0.9545 7.40 pts	↑

Recommended Settings

Factor	Value
`release_angle_deg`	50 deg
`release_height_m`	2.25 m
`backspin_rpm`	200 rpm

Source: from observed run #18

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`arc_consistency`	7.40	7.40	+0.00

Top 3 Runs by Desirability

Run	D	Factor Settings
#11	0.8633	release_angle_deg=50, release_height_m=2.25, backspin_rpm=17.4258
#22	0.8528	release_angle_deg=45, release_height_m=2.5, backspin_rpm=300

Model Quality

Response	R²	Type
`arc_consistency`	0.3761	quadratic

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.9545

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
accuracy_pct                 1.5       0.9545       76.00 %   ↑
arc_consistency              1.0       0.9545        7.40 pts   ↑

Recommended settings:
  release_angle_deg = 50 deg
  release_height_m = 2.25 m
  backspin_rpm = 200 rpm
  (from observed run #18)

Trade-off summary:
  accuracy_pct: 76.00 (best observed: 76.00, sacrifice: +0.00)
  arc_consistency: 7.40 (best observed: 7.40, sacrifice: +0.00)

Model quality:
  accuracy_pct: R² = 0.3812 (quadratic)
  arc_consistency: R² = 0.3761 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #18 (D=0.9545): release_angle_deg=50, release_height_m=2.25, backspin_rpm=200
  2. Run #11 (D=0.8633): release_angle_deg=50, release_height_m=2.25, backspin_rpm=17.4258
  3. Run #22 (D=0.8528): release_angle_deg=45, release_height_m=2.5, backspin_rpm=300

Full Analysis Output

doe analyze
=== Main Effects: accuracy_pct ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
release_angle_deg       17.2500       1.5918            48.1%
release_height_m        11.0833       1.5918            30.9%
backspin_rpm             7.5000       1.5918            20.9%

=== ANOVA Table: accuracy_pct ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
release_angle_deg            4     342.8409      85.7102      1.644     0.2456
release_height_m             4     186.6742      46.6686      0.895     0.5052
backspin_rpm                 4      99.3409      24.8352      0.476     0.7527
Lack of Fit                  2     176.8598      88.4299      1.696     0.2508
Pure Error                   7     364.8750      52.1250
Error                        9     541.7348      52.1250
Total                       21    1170.5909      55.7424

=== Summary Statistics: accuracy_pct ===

release_angle_deg:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  40.8713             1    51.0000     0.0000    51.0000    51.0000
  45                  4    67.7500     8.0984    56.0000    74.0000
  50                 12    68.2500     6.8639    54.0000    76.0000
  55                  4    67.7500     6.1305    60.0000    74.0000
  59.1287             1    59.0000     0.0000    59.0000    59.0000

release_height_m:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.79356             1    76.0000     0.0000    76.0000    76.0000
  2                   4    68.0000     5.4772    60.0000    72.0000
  2.25               12    64.9167     7.8330    51.0000    75.0000
  2.5                 4    67.5000     8.5440    56.0000    74.0000
  2.70644             1    74.0000     0.0000    74.0000    74.0000

backspin_rpm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 4    65.5000     6.6583    56.0000    71.0000
  17.4258             1    73.0000     0.0000    73.0000    73.0000
  200                12    65.7500     8.5400    51.0000    76.0000
  300                 4    70.0000     6.7330    60.0000    74.0000
  382.574             1    67.0000     0.0000    67.0000    67.0000

=== Main Effects: arc_consistency ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
release_angle_deg        2.4250       0.2336            48.7%
release_height_m         1.7750       0.2336            35.7%
backspin_rpm             0.7750       0.2336            15.6%

=== ANOVA Table: arc_consistency ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
release_angle_deg            4       8.5115       2.1279      2.082     0.1659
release_height_m             4       4.8082       1.2020      1.176     0.3836
backspin_rpm                 4       1.6140       0.4035      0.395     0.8077
Lack of Fit                  2       3.1295       1.5647      1.531     0.2809
Pure Error                   7       7.1550       1.0221
Error                        9      10.2845       1.0221
Total                       21      25.2182       1.2009

=== Summary Statistics: arc_consistency ===

release_angle_deg:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  40.8713             1     3.8000     0.0000     3.8000     3.8000
  45                  4     5.7500     1.4457     3.6000     6.6000
  50                 12     6.1417     0.9443     4.0000     7.4000
  55                  4     6.2250     0.4573     5.7000     6.7000
  59.1287             1     4.2000     0.0000     4.2000     4.2000

release_height_m:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.79356             1     7.4000     0.0000     7.4000     7.4000
  2                   4     6.3000     0.4546     5.7000     6.7000
  2.25               12     5.6250     1.1218     3.8000     7.0000
  2.5                 4     5.6750     1.4080     3.6000     6.6000
  2.70644             1     6.8000     0.0000     6.8000     6.8000

backspin_rpm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 4     5.6250     1.3817     3.6000     6.7000
  17.4258             1     6.4000     0.0000     6.4000     6.4000
  200                12     5.7667     1.2543     3.8000     7.4000
  300                 4     6.3500     0.4359     5.7000     6.6000
  382.574             1     6.1000     0.0000     6.1000     6.1000

Optimization Recommendations

doe optimize
=== Optimization: accuracy_pct ===
Direction: maximize

Best observed run: #18
  release_angle_deg = 50
  release_height_m = 2.25
  backspin_rpm = 200
  Value: 76.0

RSM Model (linear, R² = 0.0328, Adj R² = -0.1283):
  Coefficients:
    intercept                      +66.8636
    release_angle_deg              +0.5155
    release_height_m               +1.1679
    backspin_rpm                   -0.9959

RSM Model (quadratic, R² = 0.1008, Adj R² = -0.5736):
  Coefficients:
    intercept                      +66.7847
    release_angle_deg              +0.5155
    release_height_m               +1.1679
    backspin_rpm                   -0.9959
    release_angle_deg*release_height_m +0.5000
    release_angle_deg*backspin_rpm -0.0000
    release_height_m*backspin_rpm  +2.2500
    release_angle_deg^2            -1.0105
    release_height_m^2             +0.6395
    backspin_rpm^2                 +0.4895

  Curvature analysis:
    release_angle_deg              coef=-1.0105  concave (has a maximum)
    release_height_m               coef=+0.6395  convex (has a minimum)
    backspin_rpm                   coef=+0.4895  convex (has a minimum)

  Notable interactions:
    release_height_m*backspin_rpm  coef=+2.2500  (synergistic)
    release_angle_deg*release_height_m coef=+0.5000  (synergistic)

  Predicted optimum (from linear model, at observed points):
    release_angle_deg = 55
    release_height_m = 2.5
    backspin_rpm = 100
    Predicted value: 69.5428

  Surface optimum (via L-BFGS-B, linear model):
    release_angle_deg = 55
    release_height_m = 2.5
    backspin_rpm = 100
    Predicted value: 69.5428

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. release_angle_deg  (effect: 14.0, contribution: 40.6%)
  2. release_height_m  (effect: 10.2, contribution: 29.7%)
  3. backspin_rpm  (effect: 10.2, contribution: 29.7%)

=== Optimization: arc_consistency ===
Direction: maximize

Best observed run: #18
  release_angle_deg = 50
  release_height_m = 2.25
  backspin_rpm = 200
  Value: 7.4

RSM Model (linear, R² = 0.0158, Adj R² = -0.1482):
  Coefficients:
    intercept                      +5.8909
    release_angle_deg              +0.0898
    release_height_m               -0.0252
    backspin_rpm                   -0.1361

RSM Model (quadratic, R² = 0.1561, Adj R² = -0.4769):
  Coefficients:
    intercept                      +6.0896
    release_angle_deg              +0.0898
    release_height_m               -0.0252
    backspin_rpm                   -0.1361
    release_angle_deg*release_height_m +0.1625
    release_angle_deg*backspin_rpm +0.0375
    release_height_m*backspin_rpm  +0.3875
    release_angle_deg^2            -0.3193
    release_height_m^2             +0.0107
    backspin_rpm^2                 +0.0107

  Curvature analysis:
    release_angle_deg              coef=-0.3193  concave (has a maximum)
    release_height_m               coef=+0.0107  negligible curvature
    backspin_rpm                   coef=+0.0107  negligible curvature

  Notable interactions:
    release_height_m*backspin_rpm  coef=+0.3875  (synergistic)

  Predicted optimum (from linear model, at observed points):
    release_angle_deg = 55
    release_height_m = 2
    backspin_rpm = 100
    Predicted value: 6.1419

  Surface optimum (via L-BFGS-B, linear model):
    release_angle_deg = 55
    release_height_m = 2
    backspin_rpm = 100
    Predicted value: 6.1419

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. release_angle_deg  (effect: 2.2, contribution: 47.6%)
  2. backspin_rpm  (effect: 1.2, contribution: 27.0%)
  3. release_height_m  (effect: 1.2, contribution: 25.4%)