Your career today is a
Statistical engineer
Leaders’ notes
Do not give to the students
Red text in italics denotes comments for leaders and example answers
Equipment and preparation required for one group (2-4 students) to
complete the workshop
 One printed worksheet and one pen or pencil for each student
 One copy of helicopter templates printed on paper (not card) single-sided so that it can be cut
out. Note: the file includes multiple copies of the templates to ensure there are enough
helicopters for the experiment to be completed for one group of students. Please print on
paper, not card as card makes the helicopters too heavy. Best printed with option ‘Actual size’,
not with ‘Fit’ or ‘Shrink oversized pages’.
 Two scissors, four paper clips, one stopwatch (or stopwatch function on phone), one calculator
Leaders’ notes – do not give these to the students
Royal Statistical Society careers workshop
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Your career today is a
Statistical engineer
Today you’ll be taking on the role of an engineering statistician.
You will work through an exercise which shows how statistics is used in engineering and you’ll see
how valuable the use of statistics is.
What is engineering?
“Engineering is the profession in which a knowledge of the mathematical and natural sciences,
gained by study, experience, and practice, is applied with judgment to develop ways to utilize,
economically, the materials and forces of nature for the benefit of mankind” – this definition is due
to the Accreditation Board of Engineering and Technology (ABET) in the United States.
For more information see: http://www.engineeringuk.com
What is an engineer?
Most engineering projects start with a question or problem. Solutions can then be investigated by
designing experiments and drawing conclusions from the data under some modelling assumptions.
These assumptions are consistent with the theories studied in physics, geometry, the properties of
materials, applied mathematics, electrical engineering, bioinformatics and computer science. By
quantifying the information available, the consequences of each potential solution choice can be
predicted. This involves testing and evaluating the results, increasingly using computer models
rather than physical tests (such as scaled models) to generate the data.
NB Decide in your group who will write notes and who will report back at the end of the
session. Try to complete page 6 within 45 minutes.
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1. Name three areas of engineering where you think statistics can be useful.
Students can name any areas for example: Bridges, Rollercoasters, Buildings, Aeroplanes,
Aerodynamics, Aerospace, Ships, Wind turbines, Watermills/Hydrodynamics, Cycling,
materials/machine tolerance
2. Why do you think statisticians can help engineers?
Design of experiments, measurement error, modelling, analysis of data
Today’s objective
To use statistical knowledge to maximise the flight time of
a paper helicopter.
You can change design variables such as the paper
dimensions and the addition (or not) of weights. You will
design your own machine to achieve the flight time
possible.
When the helicopter is constructed, it should look something like the diagram to the right.
Note the indication on the diagram of some of the dimensions and other variables that can be
changed:

r = rotor length

w = rotor width

l = tail length

p=with or without a paper clip
on the tail
Cut out Template 1. Build it as
shown and ask your session leader
where the best place is to do some
test flights.
You will need to inform the students
of a safe method of flying the
helicopters (stairwell, mezzanine
floor, table or chair)
3
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The dimensions of the helicopters have been written below for you. Refer back to the diagram so
you know what part of the helicopter each measurement refers to. For now we will not use any
paperclips on the tail.
Template 1 Helicopter:
r = 120 mm, w = 40 mm, l = 60 mm
3. Assign each of the following tasks to a different member of your team:
(a) Dropping the helicopter
(b) Timing the helicopter
(c) Using the stop clock and writing down the results
Fly the helicopter twice and write the results down below.
First flight time for template 1:
___________________________________________________
Second flight time for template 1: _________________________________________________
Did you get the same flight time with both flights and, if not, how similar were the times?
Students should observe that each time they drop the helicopters they will get a different time and
this could be due to random or systematic variation
4. Write down five things which may have influenced the flight times.
Hints: Did you drop them from the same height? What happens when you first release the
helicopter and how long does it take to spin?
If the students observe large variation in their measurements above then they need to discuss how
to reduce this. Students need to ensure the helicopters are released from the same height & travel
the same distance. They need to improve the consistency & accuracy of the stop watch timing by
deciding how the student with the stop clock knows when to start & stop it. Were the wings pre
bent before flying & what happens if they are not bent?
Regarding what happens when you release the helicopters, students should notice that it take a
few seconds for the helicopters to reach terminal velocity and start to spin. If they are not dropping
the helicopters from a sufficient height, then the helicopter may hit the ground before the spinning
starts. Therefore they may need to increase the height.
4
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Royal Statistical Society careers workshop
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Good quality experimental results come from good quality experimental design. Details of the
design are often documented in a “protocol”. In the space below, spend no more than 10 minutes
creating your own experimental protocol. This is used to help reduce the variation you observe
between helicopter flights which is not due to the experimental factors under investigation (i.e. to
try to reduce the experimental error).
You want to find the combination of factors which leads to the longest flight time. Bear in mind that
real life statisticians are also faced with minimising financial costs and the time associated with
each experiment they undertake. For this reason, today you are limited to just 10 flights. You can
choose to investigate one or more of the rotor length (r), rotor width (w), tail length (l) or paperclip
(p). Multiple copies of Template 2 (corresponding to w=40) and Template 3 (corresponding to
w=60) will be provided.
For now we will just investigate two levels of r (60mm or 120mm), w (40mm or 60 mm) and l
(30mm or 60mm). p will also be set at 2 levels (with or without adding a single paperclip to the tail).
Make sure you discuss and agree the following issues, writing details below:
 How many factors will you investigate (Up to 4 factors: choose from r, w, l or p)
 What will be your flight release height?
 Do you need to repeat each helicopter flight? How many times?
 How many people will time the flight and who will make the helicopters?
 Will the wings (rotors) be folded out prior to release?
 What other information would it be useful to specify before starting your experiment?
5. Maximising the helicopter flight time experiment protocol.
Students can carry out any 10 flights they like. The key thing is to get them thinking about a design
rather than just changing all four factors randomly. They may come to the conclusion that they can
only investigate 2 or 3 factors. It is good if they identify that a limitation of their experiment is that
they don’t have enough flights to do any repetition.
They need to define their release height and how they will be consistent. They also need to define
how they will know when to start & stop the clock.
They should specify anything they will keep consistent.
5
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Royal Statistical Society careers workshop
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Once your protocol is agreed, begin your experiment. Cut out the helicopters you need and use the
following template to record your results.
Flight
number
Rotor length (r)
Rotor width (w)
Tail length (l)
Paperclip (p)
e.g.
60 mm
40 mm
30 mm
No
Result
(seconds)
1
2
3
4
5
6
7
8
9
10
6 a) What was your flight time and what factor combination did it use?
______________________________________________________________________________
b) Did you have enough flights to be confident of your results?
Students may feel that with 10 flights they were unable to fully confirm the best flight options.
c) Did you try to investigate too many/too few of the factors?
______________________________________________________________________________
______________________________________________________________________________
If you had unlimited resources, you could have tested all of the different combinations of the four
factors. Investigating 4 factors at 2 levels you would need 24 =16 flights to fly each combination of
6
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Royal Statistical Society careers workshop
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factors once (as shown in the table below) and that is without repeating any combinations. See
how the number of experimental flights increases, the more factors you want to investigate,
particularly if you are interested in testing three levels of each factor instead of the two we have
looked at so far.
Number of
factors
Number
of levels
Number of flights
to use each
combination once
Number of
factors
Number
of levels
Number of flights
to use each
combination once
3
2
23=8 flights
5
2
25=32 flights
3
3
33=27 flights
5
3
35=243 flights
4
2
24=16 flights
(shown below)
6
2
26 =64 flights
4
3
34=81 flights
6
3
36 =729 flights
This is where the statistical technique of Design of Experiments comes in. The following diagram
shows 2 levels (low and high) for 4 factors (X1, X2, X3 and X4). The data dots represent each of the
16 experiments required to test all combinations once. These 16 experiments can be seen
(labelled flight number 1 to 16 in the table of flight times on the next page). We use a plus (+) to
indicate that a factor is at a high level and a minus (-) to indicate that a factor is at a low level.
7
Leaders’ notes – do not give these to the students
Royal Statistical Society careers workshop
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We could test the four factors at 2 levels one at a time however this would take 16 flights as shown
above. Instead, we use a technique called “fractional factorial designs”. Shown in the cubes
below, data is now collected at half of the combinations of the four factors (8 flights). The required
combinations are specifically chosen so that we can still determine the average effect that each
factor is having on the flight time.
The 8 experiments identified by a dot as needing data collected at this combination of factors in the
cubes above can be seen in the table of flight times below. For example, flight number 1
corresponds to all of the factors at their low levels (labelled flight no 1 above). Flight number 16
corresponds to all of the factors at their high levels (labelled flight no 16 above).
Only the 8 experiments shown in BOLD with results are required estimate the average effect of
each factor on the flight time. This experiment has been carried out for you dropping each
helicopter twice and the average result in seconds has been recorded. The helicopters were
dropped in a random order to prevent bias (such as improving the method of dropping over time).
If students are interested in why these specific 8 experiments are selected, here is the explanation.
Each run gives you information. If you do all of the 16 possible runs then you can estimate
information about the “main” effects of A, B, C and D (the effect each factor has on its own) AND
you can also estimate the effects of interactions between the factors (where the effect of one factor
depends on the level of others). If fewer runs are performed then you cannot separate from the
main effect the effect of some interactions. However, we use a specific design which “aliases” the
3-way and 4-way (higher order) interactions (which are likely to have smaller effects) with the main
effects (which you hope have a bigger effect size). This ensures the minimum loss of information
and that we can estimate the main effects separately from the 2-way (2 factor) interactions.
This design is called a 24-1 fractional factorial. As described on the next page for Rotor length, to
estimate the main effect of factor A you take the average of its runs at its high level subtracted
8
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from the average of its runs at it low level: [(flight no 4 + flight no 6 + flight no 10 + flight no 16)/4] –
[(flight no 1 + flight no 7 + flight no 11 + flight no 13)/4]. However when using this design we have
aliased the effect of A with the BCD interaction. Therefore the effect of A is designated to be
intertwined with the B*C*D interaction. The reason for this is shown in column B*C*D below. By
multiplying the positives and negatives from columns B, C and D (i.e. - - - = -, + - - = +, - + - = +
etc), it creates a column identical to column A. Therefore to estimate the B*C*D interaction effect
you would do the same calculation for A and you can’t determine which is responsible.
The table below shows which other effects are aliased. It is a good design because it can estimate
the main effects separately from any of the 2-way interactions and 2-way interactions are only
aliased with each other: A & BCD, B & ACD, C & ABD, D & ABC, AB & CD, AC & BD and AD &
BC.
Factors set
at these
levels
These are the interactions.
Note: the ones with identical columns are aliased with each
other.
Flight
No.
A
B C
D
A*B
1
-
-
-
-
+
+
+
+
+
+
-
-
-
-
4
+
+
-
-
+
-
-
+
-
-
-
+
+
-
6
+
-
+
-
-
-
+
-
+
-
-
+
-
+
7
-
+
+
-
-
+
-
-
-
+
-
-
+
+
10
+
-
-
+
-
+
-
-
-
+
+
+
-
-
11
-
+
-
+
-
-
+
-
+
-
+
-
+
-
13
-
-
+
+
+
-
-
+
-
-
+
-
-
+
16
+
+
+
+
+
+
+
+
+
+
+
+
+
+
B*C A*C C*D
B*D
A*D A*B*C B*C*D A*C*D A*B*D
The table of factors to investigate confirms the levels of each factor we are investigating. The lower
level is represented by a minus sign and the higher level is indicated by a plus sign.
Table of factors to investigate
Factor
Lower level (-)
Higher level (+)
Rotor length (r)
60 mm
120 mm
Rotor width (w)
40 mm
60 mm
Tail length (l)
30 mm
60 mm
Paper clip (p)
No
Yes
9
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Table of flight times
Flight
Number
Factors
Average
result
(seconds)
a) Rotor
Length (r)
b) Rotor
Width (w)
c) Tail
length (l)
d) Paperclip
on the tail?
1
60 mm -
40 mm -
30 mm -
No -
2
120 mm +
40 mm -
30 mm -
No -
3
60 mm -
60 mm +
30 mm -
No -
4
120 mm +
60 mm +
30 mm -
No -
5
60 mm -
40 mm -
60 mm +
No -
6
120 mm +
40 mm -
60 mm +
No -
43.75
7
60 mm -
60 mm +
60 mm +
No -
24.09
8
120 mm +
60 mm +
60 mm +
No -
9
60 mm -
40 mm -
30 mm -
Yes +
10
120 mm +
40 mm -
30 mm -
Yes +
26.32
11
60 mm -
60 mm +
30 mm -
Yes +
29.83
12
120 mm +
60 mm +
30 mm -
Yes +
13
60 mm -
40 mm -
60 mm +
Yes +
14
120 mm +
40 mm -
60 mm +
Yes +
15
60 mm -
60 mm +
60 mm +
Yes +
16
120 mm +
60 mm +
60 mm +
Yes +
25.56
37.86
29.06
39.75
To find the effect of the rotor length, we calculate the average of the results when rotor length is
at its high level (+) and subtract the average of the results when rotor length is at its low level (-).
In other words: [(row 4 + row 6 + row 10 + row 16)/4] – [(row 1 + row 7 + row 11 + row 13) /4]
= [(37.86+43.75+26.32+39.75)/4] – [25.56+24.09+29.83+29.06)/4] = 9.79 (to 2 decimal places)
Using a rotor length of 120 mm increases the flight time on average by 9.79 seconds when
compared to a rotor length of 60 mm.
Using similar calculations for the rotor width and tail length
Rotor width = [(37.86+24.09+29.83+39.75)/4] – [(25.56+43.75+26.36+29.06)/4] =1.71 (to 2 decimal
places)
A rotor width of 60 mm increases the flight time on average by 1.71 seconds when
compared to a rotor width of 40 mm.
10
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Tail length = [(43.75+24.09+29.06+39.75)/4] – [(25.56+37.86+26.32+29.83)/4] =4.23 (to 2 decimal
places).
A tail length of 60 mm increases the flight time on average by 4.23 seconds when compared
to a tail length of 30 mm.
7. Perform a similar calculation for the effect of the paperclip. Does adding the paperclip
increase or decrease the flight time?
Paper clip = [(26.32+29.83+29.06+39.75)/4] – [(25.56+37.86+43.75+24.09)/4] = -6.31 (to 2 decimal
place)
Therefore adding a paperclip decreases the flight time on average.
8. Try to use your knowledge of physics (mass, resistance, friction, stability) to explain the
effects of the rotor length, rotor width, tail length and paperclip results.
Factors that increase rotor area will increase flight time because it increases the resistance and
friction as it passes through the air. Factors that increase mass (such as the paperclip) can reduce
the flight time because they make the object heavier. However, sometimes the helicopters lack
stability and having a longer tail or adding a paperclip may keep the helicopter more stable which
allows it to capture more air in its rotor and spin better adding resistance through the air.
9. What combination of factors do you think would make up the helicopter with the
maximum flight time? Was this combination tested in the example?
Students may pick up that a larger rotor area (length and width) and long tail with no paperclip
would be the best combination (flight number 8) however this was not tested. This design enables
you to determine the best combination without even testing all the combinations.
Prepare to feedback to the rest of the class: a 5-minute summary of what you were tasked
with today: what statistical tools you used to solve the problem and what your conclusions
were.
Students should ensure that they have prepared a brief report on their findings to report back to
the rest of the class.
Credits
Produced by the RSS Careers in Statistics Workshop group with support from the Royal Statistical Society. Published July 2015.
Images:
Page 1 Gateshead Millennium Bridge: Public Domain from Wikimedia Commons user Mike1024
Page 1 Rollercoaster Dragon Khan, Port Aventura, Spain: Public domain from Wikimedia Commons user Boris23
Page 1 Atomium, Belgium: Flickr David Blaikie used under CC-BY licence
Page 2 Helicopter: Flickr/Ross Elliot, used under CC-BY licence
Page 2 Diagram by Tim Davis, RSS Careers in Statistics Workshop member
Page 6 Diagram by RSS Careers in Statistics Workshop group
Page 7 Diagram by RSS Careers in Statistics Workshop group
Statistical engineering helicopter templates drawn by Lyn Taylor on behalf of the RSS Careers in Statistics Workshop group
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