The paper investigated how paper helicopter flight times are affected by various factors using both full and fractional factorial experiments. It found that paper thickness, number of paperclips, and wing length significantly influenced flight time, but not paper color. The linear models for full and fractional factorial were almost identical, indicating fractional factorial is sufficient for this experiment while requiring fewer runs.

1. 1
Title: Paper Helicopters
Name: Suhrob Shirinov
Date: December 7, 2014
Objective:
The objective of this experiment was to get more familiar with the software Statease and compare the
full array with fractional array. Another objective of this experiment was to see how different the
fractional factorial is from full factorial and is fractional factorial more efficient than full factorial. The
experiment was to throw a paper helicopter from a certain height and record the time it takes for it to
hit the ground.
Summary and Conclusion:
From this experiment it was found that paper thickness (B), paperclips (C), and wing length (D) were
significant factors that influenced the response variable the most. Factor A which was paper color was
not significant. Based on our linear model it was found that it would take about 5.5 seconds for the
helicopter to drop with the best settings selected.
We expected 3 interactions in our affinity diagram. The interactions that we expected were BC, BD, and
CD. The ANOVA table showed us that none of the interactions were significant. The factors were
significant but the interactions were not.
These are the linear models for both full (top) and fractional array (bottom):
Y = 3.98 – 0.44B - 0.31C + 0.85D
Y = 3.96 – 0.43B – 0.28C + 0.87D
This experiment showed us that there is not a significant difference between running the full array and
running the fractional array. The linear models are almost the same and the slopes are the same. The
significant terms are the same. For this experiment it would be more efficient and time-saving to do the
fractional factorial instead of full factorial. However, that is not always the case. There are some
experiments where it is necessary to run the full array instead of the fractional.

2. 2
Lab Procedure:
1. Get the supplies needed:
a. Paper helicopters
b. Paperclips
c. Scissors
d. Stopwatch
2. Form a team:
a. Stopwatch-man to record the time
b. Throwing-man to throw the helicopters
c. Delivery-man to return helicopters
d. Data recorder to record the time
3. Make Affinity Diagram
4. From affinity diagram find out how big the fractional factorial array is going to be
5. Create Full Factorial Table
6. Run Each combination four times
7. Record data
8. Copy the numbers necessary to fractional factorial
9. Use Statease to get ANOVA and other graphs for both full and fractional factorial array
10. Interpret the data and graphs
11. Compare full factorial and fractional factorial
12. Draw conclusion

3. 3
Data:
Table 1 Full Factorial Array
Std Run
A:Paper
Color
B:Paper
Thickness C:Paperclips
D:Wing
Length
Time Time Time Time
Time
Avg.
Inches Inches sec sec sec sec sec
1 10 Pink 0.0055 1 3 3.46 3.50 3.30 4.00 3.57
2 14 Yellow 0.0055 1 3 4.01 3.93 3.93 3.49 3.84
3 5 Pink 0.0095 1 3 2.77 2.78 2.90 3.13 2.90
4 11 Yellow 0.0095 1 3 2.93 2.93 3.12 2.83 2.95
5 13 Pink 0.0055 2 3 3.20 2.99 3.10 3.40 3.17
6 1 Yellow 0.0055 2 3 3.33 3.29 3.13 3.29 3.26
7 4 Pink 0.0095 2 3 2.79 2.39 2.42 2.77 2.59
8 16 Yellow 0.0095 2 3 2.87 2.73 2.64 2.54 2.70
9 7 Pink 0.0055 1 5 5.71 5.91 6.18 6.60 6.10
10 15 Yellow 0.0055 1 5 5.76 5.83 5.72 5.64 5.74
11 2 Pink 0.0095 1 5 4.82 4.76 4.41 4.87 4.72
12 12 Yellow 0.0095 1 5 4.61 4.62 4.40 4.22 4.46
13 9 Pink 0.0055 2 5 5.02 5.23 4.92 4.80 4.99
14 8 Yellow 0.0055 2 5 4.74 4.77 4.39 4.79 4.67
15 3 Pink 0.0095 2 5 3.97 4.02 4.16 4.16 4.08
16 6 Yellow 0.0095 2 5 3.97 3.97 3.93 3.65 3.88
Table 1 is the full factorial array. This table shows all the possible combinations for a 2 level 4 factor
experiment. It is in standard order. It shows all 4 factors and the recorded time for each trial and
average time of those trials. The average time is our response variable.
Affinity Diagram
A. Paper Thickness Y-intercept 1 +
B. Paper Color # Factors 4 +
C. Paperclips # Expected Interactions 3 =
D. Wing Length Sum 7
Above is the affinity diagram. This diagram is used to determine the size of the fractional factorial array.
The round dots on the affinity diagram show us the expected interactions. They are AC or BD, AD or BC,
and CD. On the right of the affinity diagram we have a calculation to determine how many runs are
going to be in our fractional factorial array. Number of runs is equal to the sum of the y-int, number of
factors, and number of the expected interactions. Our sum was 7 so we have to round it up to 8. Our
fractional factorial is going to have 8 runs.

4. 4
Table 2 Fractional Factorial Array
Std Run
A:Paper
Color
B:Paper
Thickness
C:Paper
Clips
D:Wing
Length
Time
Avg
inch inch sec
1 1 Pink 0.0055 1 3 3.57
2 5 Yellow 0.0055 1 5 5.74
3 2 Pink 0.0095 1 5 4.72
4 6 Yellow 0.0095 1 3 2.95
5 4 Pink 0.0055 2 5 4.99
6 3 Yellow 0.0055 2 3 3.26
7 7 Pink 0.0095 2 3 2.59
8 8 Yellow 0.0095 2 5 3.88
Above is the fractional factorial array. Our affinity diagram helped us to determine the size of the array.
The fractional factorial array is half the size of the full factorial array. Not much can be interpreted at
this point.
Figure 1 Half-Normal Plot for Full Factorial
The above graph is the half-normal plot for full array. This is a picture from statease where we choose
the data points to cast in. The data points that have been chosen are factors B, C and D. The data points
that have not been chosen are cast out meaning they are not significant. The next tab on the software is
the ANOVA table. We can select the points and check in the next tab to see if they are significant or not.
Design-Expert® Software
Time
Shapiro-Wilk test
W-value = 0.913
p-value = 0.234
A: Paper Color
B: Paper Thickness
C: Paperclips
D: Wing Length
Positive Effects
Negative Effects
0.00 0.43 0.85 1.28 1.71
0
10
20
30
50
70
80
90
95
99
Half-Normal Plot
|Standardized Effect|
Half-Normal%Probability
B-Paper Thickness
C-Paperclips
D-Wing Length

5. 5
Figure 2 Half-Normal Plot for Fractional Factorial
Above is the half-normal plot for fractional factorial array. It looks pretty similar to full array except
there are less data points because each point on the graph is a factor or interaction. The full array had
all of the interaction and this one does not. Most of the interactions in fractional factorial confound with
each other.
Table 3 Confounding
Expected
Interactions
Complete Confounding Pattern
BC A ≈ BCD BC ≈ AD
BD B ≈ ACD BD ≈ AC
CD C ≈ ABD CD ≈ AB
The table above shows the expected interactions and the complete confounding pattern. This shows
why we have so few data points in half-normal plot. It also shows us the factors that we expected which
are in red from our affinity diagram. Our expected interactions should not confound with each other and
they should be carefully confounded with another interaction. They are also not allowed to be
confounded with single factors.
Design-Expert® Software
Time
Shapiro-Wilk test
W-value = 0.823
p-value = 0.149
A: Paper Color
B: Paper Thickness
C: Paperclips
D: Wing Length
Positive Effects
Negative Effects
0.00 0.44 0.87 1.31 1.74
0
10
20
30
50
70
80
90
95
Half-Normal Plot
|Standardized Effect|
Half-Normal%Probability
B-Paper Thickness
C-Paperclips
D-Wing Length

6. 6
Table 4 Effects of Confounding
Linear Model Full Array Sum
Linear Model for
Fractional Array
3.976 -0.014 *ABCD 3.962 3.953
-0.039 *A 0.034 *BCD -0.005 -0.015 *A
-0.441 *B 0.014 *ACD -0.427 -0.438 *B
-0.309 *C 0.026 *ABD -0.283 -0.293 *C
0.854 *D 0.016 *ABC 0.87 0.86 *D
0.001 *AB -0.116 *CD -0.115 -0.125 *AB
-0.001 *AC -0.104 *BD -0.105 -0.115 *AC
-0.001 *AD 0.086 *BC 0.085 -0.028 *AD
The table above shows how the confounding works. On the left side of the table we have the slopes of
every factor and every interaction for the full factorial. The full factorial is also the confounding for the
fractional factorial. The numbers and terms that are side by side are confounded with each other. For
example, factor A is confounded with BCD interaction. The “Sum” column is the column where we add
the slopes of the confounded terms together. The section that is on the right side of the table is the
slopes we have from our fractional factorial array. We can see that the slopes of the fractional array is
equal for the most part with the sum column. This means that whether we do fractional or full array we
get the same slopes. It also means that if factors or interactions are confounded with each other their
slopes are added together.

7. 7
Table 5 ANOVA for Full Factorial Array
ANOVA for selected factorial model
Analysis of variance table [Partial sum of squares - Type III]
Source
Sum of
df
Mean F p-value
Squares Square Value Prob > F
Model 16.303 3 5.434 87.731 1.97804E-08
B-Paper Thickness 3.115 1 3.115 50.293 1.26208E-05
C-Paper Clips 1.525 1 1.525 24.624 0.000329546
D-Wing Length 11.662 1 11.662 188.278 1.07168E-08
Residual 0.743 12 0.062
Cor Total 17.046 15
Table 3 is the ANOVA table for full factorial array. It shows us sum of squares, df, Mean square, F-value,
and p-value. What matters to us is the remaining factors and interactions in the table. There is only 3
factors available in this table. That means that factors B, C and D are significant and the other factor and
interactions are not significant. To determine which factors are significant we need to look at p-value. If
the p-value is lower than our 0.05 alpha risk than the factor or interaction is significant.
Table 6 Fractional Factorial Array
ANOVA for selected factorial model
Analysis of variance table [Partial sum of squares - Type III]
Source
Sum of Mean F p-value
Squares df Square Value Prob > F
Model 8.156 3 2.719 55.298 1.03E-03
B-Paper Thickness 1.462 1 1.462 29.739 5.49E-03
C-Paperclips 0.638 1 0.638 12.987 2.27E-02
D-Wing Length 6.055 1 6.055 123.167 3.75E-04
Residual 0.197 4 0.049
Cor Total 8.352 7
Table 4 is the ANOVA table for the fractional factorial array. This table also shows that factors B, C, and D
are significant. Factor A and other interactions are not significant. The only difference between these
two tables are the numbers. The numbers on the full factorial are about double the numbers on the
fractional factorial. This could be due to the size of the fractional factorial array since the full factorial
was twice the size of the fractional factorial.

8. 8
Full Factorial Linear Model:
Y = 3.98 – 0.44B - 0.31C + 0.85D
Above is the linear model in terms of coded factors for the full factorial array. The number with no
variable behind it is our y-intercept. The letters that have coefficients are the significant terms. The
coefficients are the slopes of those factors. This linear model can be used to predict the best result for
this experiment.
Fractional Factorial Linear Model:
Y = 3.96 – 0.43B – 0.28C + 0.87D
Above is the linear model in terms of coded factors for the fractional factorial array. It has the same
significant terms as the full factorial which was determined by the ANOVA table. The numbers of these
models do not match each other exactly but they are fairly close. The reason they are not exactly equal
is probably due to the rounding. This shows us that there is not big difference between fractional and
factorial array. They are almost exact and for this particular experiment we could have just done
fractional factorial to save time and be more efficient.
Design-Expert® Software
Time
Lambda
Current = 1
Best = -0.44
Low C.I. = -1.01
High C.I. = 0.11
Recommend transform:
Inverse Sqrt
(Lambda = -0.5)
Lambda
Ln(ResidualSS)
Box-Cox Plot for Power Transforms
-2
-1
0
1
2
-3 -2 -1 0 1 2 3
Figure 3 Box-Cox Plot for Full Factorial (top) and Fractional Factorial (bottom)

9. 9
Above is the Box-Cox Plot for the full factorial array and fractional factorial array. The assumption that is
made by is graph is normally distributed. However, the data gathered is not always normally distributed.
They look very similar to each other and they also have the same transformation recommendation,
inverse square. We did not do the transformation for fractional factorial either because it did not
improve our normal plot of residuals.
Figure 4 Normal Plot of Residuals for Full Factorial
Design-Expert® Software
Time
Color points by value of
Time:
6.1
2.59
Externally Studentized Residuals
Normal%Probability
Normal Plot of Residuals
-2.00 -1.00 0.00 1.00 2.00 3.00 4.00
1
5
10
20
30
50
70
80
90
95
99
Design-Expert® Software
Time
Lambda
Current = 1
Best = -0.25
Low C.I. = -1.01
High C.I. = 0.52
Recommend transform:
Inverse Sqrt
(Lambda = -0.5)
Lambda
Ln(ResidualSS)
Box-Cox Plot for Power Transforms
-4
-3
-2
-1
0
1
-3 -2 -1 0 1 2 3

10. 10
The normal plot of residuals is a graph that considers the residuals as errors. This graph compares the
actual data to what the model predicts. The log scale is used to create a normal distribution. The data
points should fall close to the red line to indicate a good graph. The majority of the data points fall close
to the line which indicates a good graph. There are couple points that are farther away from the line.
Figure 5 Normal Plot of Residuals for Fractional Array
Figure 5 is the normal plot of residuals for fractional array. The first and biggest difference between full
and fractional graphs is the number of data points on the graph. This graph is much better compared to
full factorial because the data points are not far away from the line. They are fairly close to the red line
which indicates a good graph. Also the inverse square transformation was done to both of these
datasets to see if it improves these graphs. The graphs did not change significantly so we decided not to
do the transformation.
Design-Expert® Software
Time
Color points by value of
Time:
5.74
2.59
Externally Studentized Residuals
Normal%Probability
Normal Plot of Residuals
-3.00 -2.00 -1.00 0.00 1.00 2.00
1
5
10
20
30
50
70
80
90
95
99

11. 11
Figure 6 Perturbation Graphs for Full Factorial (top) and Fractional Factorial (bottom)
Above we have the perturbation graph for full and fractional factorial. As it can be seen factor A is not in
the graphs because it is categoric. The graphs themselves look exactly alike. Factors B and C both have
negative slope and factor D has positive slope. The graphs look alike because the linear model for both
of the data sets were almost equal to each other. The steeper the slope the more effect it has on the
response variable.
Design-Expert® Software
Factor Coding: Actual
Time (Seconds)
Actual Factors
A: Paper Color = Pink
B: Paper Thickness = 0.0075
C: Paperclips = 1.5
D: Wing Length = 4
Factors not in Model
A
Categoric Factors
A
-1.000 -0.500 0.000 0.500 1.000
2
3
4
5
6
7
B
B
C
C
D
D
Perturbation
Deviation from Reference Point (Coded Units)
Time(Seconds)
Design-Expert® Software
Factor Coding: Actual
Time (Seconds)
Actual Factors
A: Paper Color = Pink
B: Paper Thickness = 0.0075
C: Paperclips = 1.5
D: Wing Length = 4
Factors not in Model
A
Categoric Factors
A
-1.000 -0.500 0.000 0.500 1.000
2
3
4
5
6
B
B
C
C
D
D
Perturbation
Deviation from Reference Point (Coded Units)
Time(Seconds)

12. 12
Figure 7 BC Interaction Plots Full Array (top) Fractional Array (bottom)
Above are the BC interaction plots for full and fractional array. It was not necessary to put this graph
here because BC interaction is not significant. It is here because it is required by the assignment to be
here.
Design-Expert® Software
Factor Coding: Actual
Time (Seconds)
X1 = B: Paper Thickness
X2 = C: Paperclips
Actual Factors
A: Paper Color = Pink
D: Wing Length = 4
C- 1
C+ 2
B: Paper Thickness (inches)
C: Paperclips
0.0055 0.0065 0.0075 0.0085 0.0095
Time(Seconds)
2
3
4
5
6
7
Interaction
Design-Expert® Software
Factor Coding: Actual
Time (Seconds)
X1 = B: Paper Thickness
X2 = C: Paperclips
Actual Factors
A: Paper Color = Pink
D: Wing Length = 4
C- 1
C+ 2
B: Paper Thickness (inches)
C: Paperclips
0.0055 0.0065 0.0075 0.0085 0.0095
Time(Seconds)
2
3
4
5
6
Interaction

13. 13
Table 7 Best Selected Levels for Full Factorial (top) and Fractional Factorial (bottom)
Best Selected
Factor Level Significant Reasons
A:Paper Color Yellow No
Yellow is a beautiful color. It is bright and
easier to see in the air.
B-Paper
Thickness
0.0055" Yes
Perturbation graph indicates the negative
level gives maximum flight time
C-Paper Clips 1 Yes
Perturbation graph indicates the negative
level gives maximum flight time
D-Wing
Length
5 Yes
Perturbation graph indicates the positive
level gives maximum flight time
Best Selected
Factor Level Significant Reasons
A:Paper Color Yellow No
Yellow is a beautiful color. It is bright and
easier to see in the air.
B-Paper
Thickness
0.0055" Yes
Perturbation graph indicates the negative
level gives maximum flight time
C-Paper Clips 1 Yes
Perturbation graph indicates the negative
level gives maximum flight time
D-Wing
Length
5 Yes
Perturbation graph indicates the positive
level gives maximum flight time
Above are the best selected levels for full and fractional factorial. The tables show all the factors
whether significant or not. It also shows the best levels selected and the reasons why they were
selected. The tables are exactly alike. They are same because both of them had similar or almost same
linear models and perturbation graphs.

14. 14
Prediction for Full Array:
Y = 3.98 – 0.44(-1) - 0.31(-1) + 0.85(1)
= 5.58 seconds
Above is the longest flight time we get based on the levels we selected from full factorial array. So if we
were to do confirming run with those levels we should get numbers close to the prediction above.
Prediction for Fractional Array:
Y = 3.96 – 0.43(-1) - 0.28(-1) + 0.87(1)
= 5.54 seconds
The number predicted above is from our fractional linear model. The numbers are fairly close to being
equal to the full factorial prediction. There is only 0.04 seconds difference and for our application that
difference is not significant. So based on these selected levels it would give the paper helicopter about
5.5 second flight time.
Noise:
Noises are factors that we can’t control which it can have effect on our response variable. The noises for
this experiment could be human error or how the wings of the paper helicopter is bent. Sometimes
during our experiment the helicopter did not rotate because the wings were not bent enough. It just fell
like a piece of paper. But the biggest place the noise was apparent was from our factor A which was
paper color. The table below shows that factor A has a slope when it should not have any slope. The
slope should have been zero. That small slope could be considered noise because it effects our response
variable. Even if you look at our data gathered in our Full Array Table it can be seen that yellow color
was a bit faster than pink.
Linear Model Full Array
3.976 -0.014 *ABCD
-0.039 *A 0.034 *BCD
-0.441 *B 0.014 *ACD
-0.309 *C 0.026 *ABD
0.854 *D 0.016 *ABC
0.001 *AB -0.116 *CD
-0.001 *AC -0.104 *BD
-0.001 *AD 0.086 *BC