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Introduction
This report documents a comprehensive 2³ factorial Design of Experiments (DOE) using a paper helicopter as the experimental subject. The primary objective is to systematically evaluate how three key design factors affect the helicopter’s flight performance, measured as flight time in seconds.
Experimental Context and Rationale
A paper helicopter provides an excellent platform for learning DOE principles because:
Controllable factors: Physical dimensions and weight can be precisely modified
Measurable response: Flight time is easily quantifiable and repeatable
Physical interpretability: Results have clear engineering meaning
Cost-effective: Requires minimal materials and equipment
Research Questions
Main Effects: How does each individual factor (rotor length, rotor width, added mass) affect flight time?
Interaction Effects: Do the factors work together in ways that amplify or diminish each other’s effects?
Optimization: What combination of factor settings maximizes (or minimizes) flight time?
Model Adequacy: Can we build a reliable mathematical model to predict flight performance?
Experimental Factors and Levels
The experiment examines three factors, each at two levels:
Factor A - Rotor Length: 7.5 cm (low level) vs. 8.5 cm (high level)
Hypothesis: Longer rotors may provide more lift, increasing flight time
Factor B - Rotor Width: 3.5 cm (narrow) vs. 5.0 cm (wide)
Hypothesis: Wider rotors may create more air resistance, affecting flight dynamics
Factor C - Paper Clip Mass: 0 clips vs. 2 clips
Hypothesis: Added weight may increase stability but also increase descent rate
Design Strategy
A full 2³ factorial design was selected because:
Efficiency: Examines all possible factor combinations systematically
Interaction Detection: Can identify how factors work together
Statistical Power: Provides sufficient data for reliable conclusions
Educational Value: Demonstrates fundamental DOE principles
Paper helicopter design
Paper helicopter diagram
Figure 1: Schematic showing the paper helicopter design with labeled dimensions. The rotor length and width are the primary design variables, while paper clips are attached at the base to add mass.
Materials and Methods
Materials
Paper sheets (specify paper type, e.g., A4 office paper, 80 g/m²).
Ruler and scissors.
Paper clips (mass per clip should be measured; here “2” denotes two clips).
Measuring device (stopwatch or slow-motion video) and a fixed release height and method.
Build & Measurement protocol (step-by-step)
Cut and fold the paper helicopter rotor according to the schematic. Create rotors of lengths 7.5 cm and 8.5 cm; widths 3.5 cm and 5.0 cm as experimental factor levels.
Attach 0 or 2 paper clips to the hook point depending on the treatment.
Release the helicopter from a fixed height (e.g., 2.0 m) using the same release method each run to minimize bias. Start/stop timing from release to first contact with the ground.
Repeat runs according to the randomized run order and replicates listed in the dataset.
Record times in seconds in the Time_s column.
Experimental design A full factorial 2³ DOE was employed with replication (see data table below). Randomization was applied to run order to limit systematic error.
The following table provides a comprehensive overview of all variables collected during the experiment. Each variable serves a specific purpose in the experimental design and analysis.
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vars_table <-tibble(Variable =c("RunID", "RunOrder", "Replicate", "RotorLength_cm", "RotorWidth_cm", "PaperClip", "Time_s", "Treatment"),Description =c("Sequential ID for each run (1-24)","Randomized execution order used when conducting the experiment","Replicate number indicating repetition index (1, 2, or 3)","Rotor length in centimeters (levels: 7.5 = Short, 8.5 = Long)","Rotor width in centimeters (levels: 3.5 = Narrow, 5.0 = Wide)","Number of paper clips attached (0 = No clip, 2 = Two clips)","Measured flight time in seconds from release to landing","Treatment label summarizing factor high/low levels using standard DOE notation" ),Purpose =c("Data organization and tracking","Controls for time-based systematic errors","Enables estimation of experimental error","Primary design factor A","Primary design factor B", "Primary design factor C","Response variable (dependent variable)","Treatment identification for analysis" ))vars_table %>%gt() %>%tab_header(title =md("**Variable Definitions and Purposes**")) %>%cols_label(Variable ="Variable Name", Description ="Description", Purpose ="Experimental Purpose" ) %>%tab_style(style =cell_text(weight ="bold"),locations =cells_column_labels() )
Variable Definitions and Purposes
Variable Name
Description
Experimental Purpose
RunID
Sequential ID for each run (1-24)
Data organization and tracking
RunOrder
Randomized execution order used when conducting the experiment
Controls for time-based systematic errors
Replicate
Replicate number indicating repetition index (1, 2, or 3)
Number of paper clips attached (0 = No clip, 2 = Two clips)
Primary design factor C
Time_s
Measured flight time in seconds from release to landing
Response variable (dependent variable)
Treatment
Treatment label summarizing factor high/low levels using standard DOE notation
Treatment identification for analysis
Key Points for Table Interpretation:
RunID vs RunOrder: RunID is sequential (1,2,3…) while RunOrder shows the actual randomized sequence used during data collection
Treatment Notation: Uses standard DOE notation where lowercase letters indicate high levels (a=long rotor, b=wide rotor, c=with clips), and (1) represents all factors at low levels
Replication: Each of the 8 treatment combinations was repeated 3 times, providing 24 total observations
Experimental Design Summary
The table below shows how the 2³ factorial design is structured, displaying each unique treatment combination with its corresponding factor levels and summary statistics.
2 Treatment codes: (1) = all low, letters indicate high levels
How to Read This Design Table:
Factor Coding: The -1/+1 coding is standard in DOE:
Length: -1 = 7.5cm (short), +1 = 8.5cm (long)
Width: -1 = 3.5cm (narrow), +1 = 5.0cm (wide)
Clips: -1 = 0 clips, +1 = 2 clips
Treatment Codes: Follow standard factorial notation:
(1): All factors at low level
a: Only length at high level
b: Only width at high level
c: Only clips at high level
ab, ac, bc: Two factors at high level
abc: All factors at high level
Balance Check: Each treatment has exactly 3 replicates, confirming the design is balanced
Preliminary Observations:
Flight times range from ~2.3 to 4.2 seconds
Standard deviations are relatively small, suggesting good measurement precision
Treatment ab (long + wide, no clips) shows the highest mean flight time
Descriptive Statistics Analysis
Understanding the Response Variable Distribution
Before conducting formal statistical analysis, it’s crucial to examine the basic properties of our response variable (flight time). This section provides comprehensive descriptive statistics to understand the data’s central tendency, variability, and distribution characteristics.
Central Tendency: The mean flight time is approximately 3.2 seconds, with the median very close to the mean, suggesting a relatively symmetric distribution
Variability: The standard deviation of ~0.5 seconds indicates moderate variability in flight times
Coefficient of Variation: At ~15%, this shows reasonable precision in our measurements (values <20% are generally considered acceptable)
Range: Flight times span about 1.9 seconds from shortest to longest, indicating substantial differences between treatment conditions
Distribution Shape: Mean ≈ median suggests the data is approximately normally distributed, which is important for ANOVA validity
Treatment-Level Statistics (Ordered by Mean Flight Time)
Treatment
Replicates
Mean (s)
Std Dev (s)
Median (s)
IQR (s)
Min (s)
Max (s)
CV (%)
a
3
4.147
0.031
4.140
0.030
4.120
4.180
0.7
ab
3
3.610
0.171
3.660
0.165
3.420
3.750
4.7
(1)
3
3.407
0.100
3.400
0.100
3.310
3.510
2.9
ac
3
3.367
0.104
3.400
0.100
3.250
3.450
3.1
abc
3
3.110
0.069
3.070
0.060
3.070
3.190
2.2
c
3
3.007
0.040
3.030
0.035
2.960
3.030
1.3
bc
3
2.583
0.350
2.450
0.330
2.320
2.980
13.5
b
3
2.513
0.110
2.520
0.110
2.400
2.620
4.4
Key Insights from Treatment Comparisons:
Best Performing Treatment: The highest mean flight time shows which factor combination is optimal
Worst Performing Treatment: The lowest mean identifies the least effective configuration
Variability Assessment: Most treatments show similar standard deviations (~0.1-0.3s), suggesting consistent measurement precision across conditions
Coefficient of Variation: Most treatments have CV < 15%, indicating good experimental control
Treatment Differences: The range between best and worst treatments (~1.5+ seconds) is much larger than typical standard deviations, suggesting real treatment effects
** Length Factor Interpretation: - Effect Size: 0.68 seconds difference between levels - Better Level: Long (8.5cm) produces longer flight times - Practical Significance**: Large effect - practically significant
Width Factor Analysis
** Width Factor Summary Statistics**
Level
Observations
Mean Time (s)
Std Dev (s)
Median Time (s)
Narrow (3.5cm)
12
3.482
0.438
3.400
Wide (5.0cm)
12
2.954
0.495
3.025
** Width Factor Interpretation: - Effect Size: 0.528 seconds difference between levels - Better Level: Narrow (3.5cm) produces longer flight times - Practical Significance**: Large effect - practically significant
Clip Factor Analysis
** Clip Factor Summary Statistics**
Level
Observations
Mean Time (s)
Std Dev (s)
Median Time (s)
No Clip (0)
12
3.419
0.623
3.465
With Clip (2)
12
3.017
0.335
3.050
** Clip Factor Interpretation: - Effect Size: 0.402 seconds difference between levels - Better Level: No Clip (0) produces longer flight times - Practical Significance**: Large effect - practically significant
Summary of Factor-Level Findings:
This analysis reveals the individual contribution of each factor by comparing the average response when each factor is at its high vs. low level. The effect sizes help us understand which factors have the most substantial impact on flight performance, setting the stage for the formal ANOVA analysis that follows.
Data Visualization and Exploratory Analysis
Purpose of Visual Analysis
Visual exploration of experimental data serves several critical purposes in DOE:
Pattern Recognition: Identify trends and relationships before formal statistical testing
Assumption Checking: Assess normality, equal variance, and outlier detection
Effect Visualization: Understand the magnitude and direction of factor effects
Interaction Detection: Spot potential factor interactions through visual patterns
Communication: Present findings in an accessible, interpretable format
Main Effects Visualization
The following box plots display how each individual factor affects flight time. Box plots are ideal for showing both central tendency and variability simultaneously.
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library(patchwork)# Enhanced main effects plots with better aesthetics and informationp1 <- helicopter_coded %>%ggplot(aes(x = Length_Factor, y = Time_s, fill = Length_Factor)) +geom_boxplot(alpha =0.7, outlier.shape =21, outlier.size =2) +geom_jitter(width =0.2, alpha =0.6, size =2) +stat_summary(fun = mean, geom ="point", shape =23, size =3, fill ="red", color ="darkred") +labs(title ="Effect of Rotor Length on Flight Time",subtitle ="Red diamonds show treatment means",x ="Rotor Length", y ="Flight Time (seconds)" ) +scale_fill_manual(values =c("lightblue", "lightcoral")) +theme_minimal() +theme(legend.position ="none",plot.title =element_text(hjust =0.5, size =12, face ="bold"),plot.subtitle =element_text(hjust =0.5, size =10),panel.grid.minor =element_blank() )p2 <- helicopter_coded %>%ggplot(aes(x = Width_Factor, y = Time_s, fill = Width_Factor)) +geom_boxplot(alpha =0.7, outlier.shape =21, outlier.size =2) +geom_jitter(width =0.2, alpha =0.6, size =2) +stat_summary(fun = mean, geom ="point", shape =23, size =3, fill ="red", color ="darkred") +labs(title ="Effect of Rotor Width on Flight Time",subtitle ="Red diamonds show treatment means", x ="Rotor Width", y ="Flight Time (seconds)" ) +scale_fill_manual(values =c("lightgreen", "lightyellow")) +theme_minimal() +theme(legend.position ="none",plot.title =element_text(hjust =0.5, size =12, face ="bold"),plot.subtitle =element_text(hjust =0.5, size =10),panel.grid.minor =element_blank() )p3 <- helicopter_coded %>%ggplot(aes(x = Clip_Factor, y = Time_s, fill = Clip_Factor)) +geom_boxplot(alpha =0.7, outlier.shape =21, outlier.size =2) +geom_jitter(width =0.2, alpha =0.6, size =2) +stat_summary(fun = mean, geom ="point", shape =23, size =3, fill ="red", color ="darkred") +labs(title ="Effect of Paper Clips on Flight Time",subtitle ="Red diamonds show treatment means",x ="Paper Clips", y ="Flight Time (seconds)" ) +scale_fill_manual(values =c("lightpink", "lightsteelblue")) +theme_minimal() +theme(legend.position ="none",plot.title =element_text(hjust =0.5, size =12, face ="bold"),plot.subtitle =element_text(hjust =0.5, size =10),panel.grid.minor =element_blank() )# Combine plots with enhanced layout(p1 | p2 | p3) +plot_annotation(title ="Main Effects Analysis: Individual Factor Impacts",subtitle ="Box plots show distribution, jittered points show individual observations",theme =theme(plot.title =element_text(hjust =0.5, size =14, face ="bold"),plot.subtitle =element_text(hjust =0.5, size =11) ) )
How to Interpret Main Effects Plots:
Box Plot Elements:
Center line = median
Box boundaries = 25th and 75th percentiles (IQR)
Whiskers = extend to most extreme points within 1.5×IQR
Outliers = points beyond whiskers
Red diamonds = treatment means
Individual Points: Jittered to show actual data distribution and sample size
What to Look For:
Vertical separation between boxes indicates main effects
Box overlap suggests smaller effects
Similar spreads indicate equal variance assumption is reasonable
Outliers may indicate measurement errors or special causes
Interaction Effects Analysis
Interaction plots are crucial for understanding whether factors work together synergistically or antagonistically. These plots show if the effect of one factor depends on the level of another factor.
Parallel Lines: Indicate no interaction - the effect of one factor is consistent regardless of the other factor’s level
Crossing Lines: Suggest strong interaction - the effect of one factor reverses depending on the other factor
Diverging/Converging Lines: Indicate moderate interaction - one factor’s effect is amplified or diminished by the other
Error Bars: Show measurement uncertainty; overlapping bars suggest differences may not be significant
Preliminary Visual Insights:
Based on these interaction plots, we can make initial observations about: - Which factor combinations appear most/least effective - Whether factors work independently or synergistically
- The relative magnitude of main effects vs. interaction effects - Potential optimization strategies
These visual patterns will be formally tested in the subsequent ANOVA analysis.
ANOVA Analysis
Understanding Analysis of Variance in Factorial Experiments
Analysis of Variance (ANOVA) is the primary statistical method for analyzing factorial experiments. It allows us to:
Partition Total Variation: Separate the total variation in flight times into components attributable to each factor, their interactions, and random error
Test Statistical Significance: Determine which effects are larger than what we’d expect from random variation alone
Quantify Effect Sizes: Measure how much each factor contributes to the overall variation
Validate Model Assumptions: Check if our data meets the requirements for valid statistical inference
The ANOVA Model
For our 2³ factorial experiment, the statistical model is:
Practical significance: Large Effect (≥10% of variation)
Significant Interaction Effects
The following interaction effects are statistically significant:
** AC: Length × Clip **
Explains 5.2 % of total variation
Statistical significance: Very Significant (p < 0.01)
Interpretation: The effect of one factor depends on the level of the other factor(s)
Implication: Simple main effects analysis may be needed
** BC: Width × Clip **
Explains 3.3 % of total variation
Statistical significance: Very Significant (p < 0.01)
Interpretation: The effect of one factor depends on the level of the other factor(s)
Implication: Simple main effects analysis may be needed
ANOVA Assumptions Assessment
ANOVA requires three key assumptions:
1. Independence of Observations - Status:MET - Evidence: Randomized run order prevents systematic bias - Conclusion: Each helicopter was tested independently
2. Normality of Residuals - Test: Shapiro-Wilk test - p-value: 0.1117 - Status:PASSED (p > 0.05) - Conclusion: Residuals are approximately normal
3. Homogeneity of Variance (Equal Variances) - Test: Levene’s test
- p-value: 0.5286 - Status:PASSED (p > 0.05) - Conclusion: Variances are approximately equal across treatments
Overall ANOVA Validity: GOOD****
Stepwise Model Selection
Understanding Model Selection in DOE
After fitting the full factorial model, we often find that not all effects are statistically significant. Model selection helps us identify a simpler, more interpretable model that retains only the important effects. This process serves several purposes:
Parsimony: Simpler models are easier to interpret and communicate
Improved Precision: Removing non-significant terms can improve the precision of remaining effect estimates
Better Predictions: Simpler models often predict better on new data (avoid overfitting)
Focus on Key Factors: Identifies which factors truly matter for the response
Model Selection Strategies
There are several approaches to model selection:
Forward Selection: Start with no terms, add significant ones
Backward Elimination: Start with all terms, remove non-significant ones
Stepwise (Both Directions): Combines forward and backward at each step
All Subsets: Evaluate all possible model combinations
We’ll demonstrate both automated stepwise selection using statistical criteria and manual backward elimination using p-values.
Automated Stepwise Selection Using AIC
The Akaike Information Criterion (AIC) balances model fit with model complexity. Lower AIC values indicate better models.
Interpretation of AIC Selection: The stepwise algorithm evaluated adding and removing terms at each step, selecting the combination that minimized AIC. The modest AIC improvement suggests some benefit to model simplification.
While AIC provides an objective criterion, many practitioners prefer p-value based elimination for its interpretability. This approach removes the least significant term at each step.
Elimination Process: 1. Start with the full model 2. Identify the term with the highest p-value > 0.05 3. Remove that term and refit the model 4. Repeat until all remaining terms have p ≤ 0.05
Final Model Formula: Time_s ~ A_Length + B_Width + C_Clip + A_Length:B_Width + A_Length:C_Clip + , B_Width:C_Clip
Selection Rationale:
Statistical Optimality: This model achieved the lowest AIC value (-14.34), indicating the best balance between model fit and complexity
Adjusted R-squared: 0.917 - retains 91.7% of explainable variation
Parsimony: Uses 6 terms compared to 7 in the full model
Interpretability: Includes only statistically meaningful effects, making results easier to understand and communicate
This recommended model will be used for final effect interpretation and optimization analysis in subsequent sections.
Results and Interpretation
Statistical Analysis Summary
This section synthesizes the key findings from our comprehensive DOE analysis, translating statistical results into practical engineering insights for paper helicopter optimization.
Effect Magnitudes and Rankings
Show Code
effects_summary %>%gt() %>%tab_header(title =md("**Factorial Effects Ranked by Magnitude**")) %>%cols_label(Effect ="Effect",Estimate ="Effect Size (s)",SE ="Standard Error",t_stat ="t-statistic", Abs_Effect ="Absolute Effect" ) %>%tab_footnote(footnote ="Positive effects indicate higher flight times at high factor levels",locations =cells_column_labels("Estimate") ) %>%fmt_number(columns =everything() &where(is.numeric), decimals =3) %>%tab_style(style =cell_text(weight ="bold"),locations =cells_column_labels() ) %>%# Highlight the largest effectstab_style(style =cell_fill(color ="lightblue", alpha =0.3),locations =cells_body(rows =1:3) )
Factorial Effects Ranked by Magnitude
Effect
Effect Size (s)1
Standard Error
t-statistic
Absolute Effect
A (Length)
0.681
0.045
15.236
0.681
B (Width)
−0.527
0.045
−11.805
0.527
C (Clip)
−0.403
0.045
−9.007
0.403
AC
−0.237
0.045
−5.315
0.237
BC
0.188
0.045
4.196
0.188
ABC
0.131
0.045
2.928
0.131
AB
0.131
0.045
2.928
0.131
1 Positive effects indicate higher flight times at high factor levels
Significant Effects Analysis
Statistical Significance Findings
Our ANOVA analysis identified 5 statistically significant effects at the α = 0.05 level:
This positive effect indicates that longer rotors (8.5 cm) increase flight time by approximately 0.681 seconds compared to shorter rotors on average.
Factor B - Rotor Width: -0.527 seconds
This negative effect shows that narrower rotors (3.5 cm) increase flight time by approximately 0.527 seconds compared to wider rotors on average.
Factor C - Paper Clips: -0.402 seconds
This negative effect demonstrates that removing paper clips increases flight time by approximately 0.402 seconds compared to the two-clip condition.
Physical Mechanisms and Design Insights
Aerodynamic Considerations:
The observed effects can be understood through fundamental aerodynamic principles:
Lift Generation: Rotor dimensions directly affect the lift-generating surface area, with larger rotors potentially creating more lift but also more drag.
Stability and Control: The paper clip effect primarily influences helicopter stability and descent rate. The no-clip configuration appears more favorable, possibly due to reduced overall weight.
Interaction Effects: Significant interactions indicate that optimal design requires considering factor combinations rather than individual factors in isolation.
Optimal Configuration Analysis
Maximum Flight Time Configuration
Optimal Treatment: a
Design Specifications: - Rotor Length: 8.5 cm (Long) - Rotor Width: 3.5 cm (Narrow)
- Paper Clips: No clips - Average Flight Time: 4.147 seconds
Minimum Flight Time Configuration
Treatment: b
Design Specifications: - Rotor Length: 7.5 cm (Short) - Rotor Width: 5.0 cm (Wide) - Paper Clips: No clips - Average Flight Time: 2.513 seconds
Performance Range Analysis
Total Performance Range: 1.633 seconds
This represents a 65% difference between the best and worst configurations, demonstrating that design choices significantly impact helicopter performance.
Primary Recommendation: Use treatment a configuration - This design maximizes flight duration for applications requiring extended airtime - The combination provides optimal balance of lift generation and flight stability
For Minimum Flight Time Applications
Alternative Configuration: Treatment b may be preferred when: - Rapid descent is desired - Minimal flight time is the objective - Compact flight patterns are required
Practical Implementation Guidelines
Manufacturing Tolerances: Maintain rotor dimensions within ±0.5mm for consistent performance
Paper Clip Consistency: Use standard paper clips with consistent mass (typically ~0.5g each)
Release Technique: Ensure consistent release height and method for reproducible results
Environmental Factors: Consider air currents and temperature effects in practical applications
The systematic DOE approach has successfully identified the key factors controlling paper helicopter flight performance and provided clear optimization guidance for different application requirements.
