How far until it stops? Investigating stopping distances
using the PPDAC statistical enquiry cycle
Unit Outline
Introduction
This unit of work uses the PPDAC (Problem, Plan, Data, Analysis and Conclusion)
statistical enquiry cycle to investigate stopping distances in cars. The unit of work is
aimed at Level 5 of the curriculum (most likely a Year 10 class). The context is one that
should appeal to students due to its relevance to them as they and their peers prepare to
learn to drive. The unit allows them to investigate factors which affect stopping
distances, such as speed and road conditions. This will give more meaning to applying
appropriate following distances and the ‘2 second rule’.
The investigation analyses data collected during a study on total car stopping distance
with different road conditions and different initial speeds.The total stopping distance of a
car can be broken down into two major components: the reaction distance and the
braking distance. The reaction distance takes account of the time it takes for a driver to
perceive that they need to stop and the amount of time it takes for them to react to the
situation. The braking distance depends on the vehicle reaction time and the vehicle
braking capability. Although the study is fictitious, the findings in terms of averages for
the different stopping components under different conditions reflect reality.
The stopping distance investigation ‘How far before it stops?’ includes a template with
scaffolding to guide students through the components of the PPDAC cycle in the context
described above. It can be printed for students to write on directly, projected in a
classroom so that students can write out the answers in their books or can be distributed
electronically for students to type into directly. The in-class investigation assumes some
prior statistical knowledge, which is described below with resources and teaching
suggestions. Detailed lesson plans with suggested duration are also included and can
be adapted to meet the needs of the learners. It is expected that the lessons will consist
of discussion, analysis and some drafting of responses but the students will write up
their answers in more detail for homework. Alternatively more time could be allowed in
class for the project to be completed. An assessment rubric for the in-class investigation
has been developed and includes a self-assessment component to allow students to
reflect on their own learning. The in-class project could be used independently of the
lesson plans as a homework-based assessment tool at the end of a unit of work.
However, the depth of the conclusion sections is greatly enhanced by in-class
discussions so that students can draw on their peers’ and their teacher’s knowledge.
Prior
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knowledge (with
suggestions):
websites
for
resources
and
Calculating averages, quartiles and inter-quartile range
o NZ Maths statistical investigations: nzmaths.co.nz/statisticalinvestigations-units-work?parent_node=
teaching
o NZ Census at School classroom activities:
www.censusatschool.org.nz/classroom-activities
o Collect data about students in the class, e.g. heights, distance from home
to school, number of hours studying the Road Code so far, number of
hours supervised driving time they think they should have before sitting
their first driving test (compare this to the NZTA recommended time of a
minimum of 120 hours). Calculate statistics using this data.
•
Graphing data
o NZ Maths statistical investigations: nzmaths.co.nz/statisticalinvestigations-units-work?parent_node=
o NZ Census at School classroom activities:
www.censusatschool.org.nz/classroom-activities
o NZ Census at School data viewer:
www.censusatschool.org.nz/2010/data-viewer
o NZ Assessment Resource Banks (Mathematics):
arb.nzcer.org.nz/searchmaths.php
o Box and whiskers graph, dot plot, histogram, stem and leaf graphs of
class data.
•
Random sampling
o NZ Maths statistical investigations: nzmaths.co.nz/statisticalinvestigations-units-work?parent_node=
o NZ Census at School data viewer:
www.censusatschool.org.nz/2010/data-viewer
o Use class data or select a random selection of students from the class
(asking them to stand up) to demonstrate simple random sampling and/or
systematic sampling
•
Population and variables
o NZ Census at School classroom activities:
www.censusatschool.org.nz/classroom-activities
•
PPDAC cycle
o NZ Census at School classroom activities:
www.censusatschool.org.nz/classroom-activities
•
Posing investigative questions
o NZ Maths statistical investigations: nzmaths.co.nz/statisticalinvestigations-units-work?parent_node=
o NZ Census at School classroom activities:
www.censusatschool.org.nz/classroom-activities
o Use class data to pose summative, comparative and relationship
questions.
Curriculum links:
Learning area
Mathematics and Statistics – Statistics Level 5
Values
Excellence
Innovation, inquiry and curiosity
Community and participation
Respect
Achievement objectives
Level 5 Statistical Investigation:
Plan and conduct surveys and experiments using the statistical enquiry cycle:
• Determining appropriate variables and measures;
• Considering sources of variation;
• Gathering and cleaning data;
• Using multiple displays, and re-categorising data to find patterns, variations,
relationships, and trends in multivariate data sets;
• Comparing sample distributions visually, using measures of centre, spread, and
proportion;
• Presenting a report of findings.
Key competencies based on achievement objectives related to each
competency specific to Mathematics and Statistics (source: Team
Solutions New Zealand, Auckland University).
Thinking:
• Think logically
• Justify
• Co-construct knowledge
• Investigate
• Discern if answers are reasonable
• Interpret
• Deal with uncertainty and variation
• Make connections
• Hypothesise
• Seek patterns and generalisation
• Explore and use patterns and relationships in data
• Demonstrate and develop relational understanding
• Evaluate
• Analyse
Using language, symbols and text:
• Understand mathematics as a language
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Interpret statistical information
Process and communicate mathematical ideas
Know, use and interpret specialised vocabulary
Communicate findings
Use ICT as appropriate
Interpret visual representations such as graphs, diagrams
Use appropriate units
Demonstrate statistical literacy
Managing self:
• Work independently
• Self-assessment – What can/can’t I do
• Manage time effectively
Relating to others:
• Listen actively
• Share ideas
• Accept being wrong as part of learning
• Work cooperatively
• Communicate thinking
• Think-pair-share
• Remain open to learning from others
Participating and contributing:
• Share strategies and thinking
• Work in groups with everyone contributing
• Contribute to thinking groups
• Build on prior knowledge
• Contribute to a culture of inquiry and learning
Learning Intentions
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Identify suitable variables for the statistical investigation.
Identify the population.
Describe the problem of interest.
Calculate the summary statistics for relevant data.
Graph relevant data.
Analyse the data.
Compare/contrast the data.
Explain the likely result of repeating the sampling process.
Explain why different groups and organisations would be interested in the
findings of your investigation.
Peer-critique each others’ work.
Generate an investigative question.
Hypothesise the answer to the investigative question.
From the analysis carried out, reflect on and justify your findings.
Reflect on how reasonable the results of the investigation are.
•
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Generate a question that could be further investigated based on your
investigation.
Present your findings in a way that will inform others.
ICT resources
New Zealand Maths statistics investigations
nzmaths.co.nz/statistical-investigations-units-work?parent_node=
NZ Census at School classroom activities
www.censusatschool.org.nz/classroom-activities
NZ Census at School statistical investigation and the PPDAC cycle
www.censusatschool.org.nz/resources/statistical-investigation
www.censusatschool.org.nz/resources/how-kids-learn
NZ Census at School informal inference
www.censusatschool.org.nz/2009/informal-inference
NZ Census at School data viewer
www.censusatschool.org.nz/2010/data-viewer
NZ Assessment Resources Banks (Mathematics)
arb.nzcer.org.nz/searchmaths.php
Websites on stopping (and following) distances
www.nzta.govt.nz/resources/roadcode/about-driving/following-distance.html
www.rulesoftheroad.ie/rules-for-driving/speed-limits/speed-limits_stopping-distancescars.html
www.transportpolicy.org.uk/Future/20mph/20mph.htm (see subheading of ‘Safety’)
www.sdt.com.au/safedrive-directory-STOPPINGDISTANCE.htm
Youtube videos on stopping distances
www.youtube.com/watch?v=CzHklqaiTXI
www.youtube.com/watch?v=Z_n-HIBnfts
Website with information on Fathom software
softwareforlearning.tki.org.nz/Products/Fathom
Excel spreadsheet of stopping distances
Available on the NZTA website
‘How far until it stops?’ lesson plans
Lesson 1: Introduction to the ‘How far until it stops?’
investigation
Content:
• Introduction to the ‘How far until it stops?’ investigation
• Stopping distances
• PPDAC cycle: Stopping distances problem
Activities:
1. Introduce the investigation.
a) Tell students that the investigation will be about car stopping distances.
b) Use think-pair-share discussion to find out students’ prior knowledge about
stopping distances in cars using the sentence starter: ‘The stopping distance of a
car depends on...’
c) Have students read through the introduction to the investigation ‘How far until it
stops?’ in the student workbook.
d) Discuss the problem being investigated. Use think-pair-share to discuss why this
study might have been carried out.
e) Discuss the population for this investigation.
2. Problem
a) Students to write a description of the problem being investigated.
b) Students to pose a comparative question for this investigation and identify the
variables and population.
c) Students’ questions are then checked and if required students are given
feedback on their question and given the opportunity to rewrite it (this may take
more than one lesson).
Notes for teachers:
• There are a number of factors which affect the overall stopping distance. Some of these
are:
o Perception time
o Reaction time
o Vehicle reaction time
o Vehicle braking capability
o How good the brakes are
o Grip of the tyres on the road
o Conditions of the road surface
o Weather conditions
o Weight of the car and its contents
• Example of a description of the problem being investigated: ‘The problem being
investigated is stopping distances (in metres) for a variety of cars, road conditions and
drivers in New Zealand’.
• Students should make two attempts to pose their own comparative question. An example
of the format that students can use is: ‘Do total stopping distances in wet conditions
collected in the NZ road and vehicle safety study tend to be further than the total stopping
distances in dry conditions collected in the NZ road and vehicle safety study?’.
• Ideally questions should be written to reflect the hypothesis although this is not
necessary. The question above implies that the hypothesis is that stopping distances in
wet conditions are further than those in dry conditions (for the NZ road and vehicle safety
study).
• A suitable comparison investigative question is one that reflects the population, has a
clear variable to investigate, compares the values of a continuous variable across
different categories, and can be answered with the data.
• Questions need to be checked by the teacher and feedback given as required.
• If students are unable to write a question, even with guidance, they should then be given
a question to investigate. It is useful to make a note of this as this should be considered
when awarding the final grade for the investigation.
• Students who were able to write a question will then use one of their approved questions
for the statistical investigation.
• You may wish to have all students use the same question in order to simplify the
monitoring of the work and the marking. However, more valuable discussions can be had
at the end if students are working on different questions and this will be more interesting
to students.
• Population for this investigation is: Trials measured during the NZ road and vehicle safety
study.
• Students must identify the two variables for their investigative question, the categorical
variable (road conditions or initial speed) and the continuous variable (reaction distance,
braking distance or total stopping distance).
• Students could complete part or all of this investigation in pairs or threes. Such use of
cooperative work will enhance students understanding and communication of context
through shared knowledge as well as help them through the PPDAC statistical
investigation cycle.
Resources:
• Copy of the investigation for each of the students or an electronic copy to project in the
classroom.
ICT resources:
• NZ Census at School classroom activities
www.censusatschool.org.nz/classroom-activities
• Websites on stopping (and following) distances
www.nzta.govt.nz/resources/roadcode/about-driving/following-distance.html
www.rulesoftheroad.ie/rules-for-driving/speed-limits/speed-limits_stopping-distancescars.html
www.transportpolicy.org.uk/Future/20mph/20mph.htm (see subheading of ‘Safety’)
www.sdt.com.au/safedrive-directory-STOPPINGDISTANCE.htm
• You tube videos on stopping distances
www.youtube.com/watch?v=CzHklqaiTXI
www.youtube.com/watch?v=Z_n-HIBnfts
Lesson 2: ‘How far until it stops?’ investigation: Plan, Data and
Analysis
Content:
• PPDAC cycle: Stopping distances: Plan, Data and Analysis
Activities:
1. Plan
a) Use small group or whole class discussion for students to come up with what
they think the answer to their question is.
b) Discuss with students what data they will need to use from the sample of 120
trials given to them.
c) Students to write up their hypothesis and describe the data that they will use.
2. Data
a) Have students consider whether the data seems reasonable or if it needs
cleaning.
b) Discuss the larger stopping distances, why might these have occurred?
3. Analysis
a) Students choose two comparative graphs to display different features of the data.
Ideally a side-by-side box and whisker graph should be chosen, as well as one of
a dot plot, histogram or stem and leaf graph.
b) Students calculate summary statistics and construct graphs, using appropriate
technology if available (e.g. Fathom) or by highlighting the relevant trials on a
printout of the dataset provided and carrying out this process by hand.
Notes for teachers:
• Depending on the amount of time available for this investigation, an extra lesson could be
included for students to spend class time on the internet researching stopping distances.
Alternatively this could be done as homework prior to this lesson in order to aid students
in making their hypothesis and increasing their knowledge of the context (which will also
increase the depth of their discussion in the conclusion).
• Students need to appreciate that the dataset they have been provided with is a random
sample of all the trials of stopping distances carried out in the NZ study. They need to use
all of the data appropriate to their question and not sample further.
• As the sample is random the students can assume that it is representative of all the trials
carried out in the study.
• When considering whether the data needs cleaning students need to consider whether
the values seem reasonable. Avoid the use of the word ‘outlier’ to describe extreme
values in this dataset as none of the values presented can be considered outliers. All
values are reasonable and the data does not need cleaning.
• Students are likely to point out some of the larger values and question whether these are
valid. Discuss possible reasons for these values. For example a large reaction distance
may be due to the driver being distracted (e.g. by using their mobile phone). A large
braking distance may be due to worn tyres, which reduces the grip that the tyres has on
the road.
• Graphs must allow for direct comparison, i.e. share the same axes.
• The choice of graphs will depend on whether technology is being used to produce the
graphs. If suitable ICT resources are available then it is useful for students to become
familiar with using programs such as Fathom to produce the statistics and graphs,
especially if this is how the analysis is carried out in the assessment of AS 1.10 in your
school.
• A side-by-side box and whisker graph allows many features to be compared such as the
middle 50% of the data, medians and overlap of the data. A dot plot can be overlaid or
stacked with the box and whisker graph in order to provide insight into the distribution of
the data. Overlaying these two types aids students in developing a deeper understanding
of what the box and whisker graph represents.
Students should aim to have two or more graphs. However, one graph is sufficient.
• Avoid calculating and discussing the range or focusing on the maximum and minimum
values as these do not provide evidence when answering the investigative question.
Resources:
• Copy of the investigation for each of the students or an electronic copy to project in the
classroom.
ICT resources:
• Excel spreadsheet of stopping distances (available from the NZTA website)
• NZ Census at School classroom activities
www.censusatschool.org.nz/classroom-activities
• NZ Assessment Resource Banks (Mathematics)
arb.nzcer.org.nz/searchmaths.php
• Websites on stopping (and following) distances
www.nzta.govt.nz/resources/roadcode/about-driving/following-distance.html
www.rulesoftheroad.ie/rules-for-driving/speed-limits/speed-limits_stopping-distancescars.html
www.transportpolicy.org.uk/Future/20mph/20mph.htm (see subheading of ‘Safety’)
www.sdt.com.au/safedrive-directory-STOPPINGDISTANCE.htm
• You tube videos on stopping distances
www.youtube.com/watch?v=CzHklqaiTXI
www.youtube.com/watch?v=Z_n-HIBnfts
• Website with information about Fathom software
•
softwareforlearning.tki.org.nz/Products/Fathom
Lesson 3: ‘How far until it stops?’ investigation: Analysis
Content:
• PPDAC cycle: Stopping distances analysis
Activities:
1. Analysis
a) Discuss with students what they notice about their graphs and summary
statements when comparing the two groups. Features to consider include:
median/mean, middle 50% of data, shape, overlap, spread, usual or interesting
features.
b) Have students draft 3-5 statements about what they notice about their summary
statistics and graphs.
c) Peer-critique of statements, either as a class or in small groups. In each case
discussing what makes the statement good or how it could be improved.
d) Students to write final versions of 3-5 analysis statements after considering the
feedback given by their peers (may be completed for homework).
Notes for teachers:
• There are a number of features of the graphs that the students can comment on. The
investigation template guides them through these features.
• When discussing what students notice about their statistics and graphs, repeat back to
students their ideas, altering their words to model correct statistical language, ensuring it
is also in context and includes values.
• Students do not need to write an analysis statement about each of the listed features,
instead they should focus on what is relevant for their analysis and what will help them
answer their question.
• Analysis statements must be made in the context of stopping distances.
• Analysis statements should include values and units.
• Analysis statements should make it clear that it is the sample being referred to, not the
population.
• An example of an analysis statement is: ‘In the sample analysed, the median total
stopping distance for dry conditions is 21.1 m, which is 8 m less than the median
stopping distance for wet conditions (29.1 m). Each median value lies outside of the
middle 50% of the data for the other road condition.’ Further examples can be seen in the
assessment schedule.
• Avoid analysis statements about the range and extreme values as these do not add to the
discussion.
• Peer-critiquing and class discussion encourages students to consider what makes a
quality analysis statement.
• If students are doing different investigations, rather than a single question for the whole
class, it may be worthwhile to start by showing students a single set of summary statistics
and graphs and discuss analysis statements based on these. Then students can write
and peer-critique statements based on their own results.
• Some of the literature has students writing analysis statements starting with the words ‘I
notice…’ This can be incorporated into the instructions for writing analysis statements if
desired.
Resources:
• Copy of the investigation for each of the students or an electronic copy to project in the
classroom.
ICT resources:
• NZ Census at School classroom activities
www.censusatschool.org.nz/classroom-activities
Lessons 4-5: ‘How far until it stops?’ investigation: Conclusion
Content:
• PPDAC cycle: Stopping distances conclusion
Activities:
1. Discussion of the conclusion:
a) Discuss with students what the answer to their question about stopping distances
is and how they know that this is the answer, i.e. what evidence do they have
from their analysis of the sample.
b) Have students make an informal inference, answering their question about the
population and select 2-3 pieces of supporting evidence for their inference.
c) Discuss with students the concept of variability in sampling. Ask students if their
summary statistics and graphs would be the same and whether or not they think
they would be able to make the same claim, i.e. answer their question in the
same way.
d) Use think-pair-share or class discussion to discuss how reasonable they think
their results are. Students can draw on their previous knowledge and research to
answer this question. If they have not had a chance to do any research yet about
the context then this would be a good opportunity for them to do so by using the
internet or being encouraged to discuss the context with their parents and
whānau.
e) Use think-pair-share to discuss the following questions: ‘Why would your friends
who are learning to drive be interested in these results?’, ‘Why would other
drivers be interested in these results?’ and ‘Which organisations or other groups
of people might be interested in these results, and why?’.
f) Discuss with students what other questions this investigation has generated.
Encourage students to think about variables which have not been presented in
this investigation. Students should identify possible questions which would lead
to meaningful investigations (comparative or relationship questions), be able to
identify what data would need to be collected to carry out this investigation and
which organisations or groups of people might be interested in this investigation.
Students are not expected to carry out this further investigation.
2. Students to write up their conclusions based on the in-class discussions.
Notes for teachers:
• Allow at least 2 periods for this section of the investigation as students need to be given
time to work on the conclusion so that they have the chance to understand how to make
an informal inference and the context for the statistical investigation.
• Discussion allows the students to consider others’ viewpoints and experiences (both their
peers and the teacher’s) in order to enrich their understanding of the context and hence
the quality of the conclusion.
• The quality of the discussion in the Conclusion section largely separates students’
achievement level for this assessment.
• Students need to answer their question about the population (Trials measured during the
NZ road and vehicle safety study) using their summary statistics, graphs and analysis
statements. For example ‘‘The total stopping distances in wet conditions collected in the
NZ road and vehicle safety study do tend to be further than the total stopping distances in
dry conditions collected in the NZ road and vehicle study’.
• When selecting supporting evidence for their conclusion, students need to be able to
identify the relevant points from their analysis statements, graphs and summary
statements. The supporting evidence should not just repeat all the analysis statements
written previously. One quality statement is sufficient.
• Examples of supporting evidence for the conclusion is: ‘The reason I am able to make the
claim about total stopping distances is because the median total stopping distance for dry
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conditions is 21.1 m, which is less than the stopping distance for wet conditions (29.1).
This difference is significant because each median value lies outside of the middle 50%
of the data for the other road condition. Also, from the box and whisker graphs I can see
that the furthest 75% of total stopping distances in wet conditions are greater than the
shortest 50% of total stopping distances in dry conditions.’
Variability in sampling is a difficult concept for students to describe even if they have
some understanding of it. There are two aspects to consider. Firstly, that a different
sample (from the same initial study) is expected to produce differences in the sample
statistics and graphs. Secondly, the students need to consider whether they are likely to
be able to reach the same conclusion. If their first sample shows a large difference in the
distributions then it is likely that this difference will hold if the sample was repeated. If the
sample distributions are similar (such is the case for reaction distances at 40 km/h and at
50 km/h) then the conclusion would be that a difference can not be claimed and another
sample is unlikely to show a large difference and therefore they are unlikely to reach a
different conclusion.
When discussing how reasonable the results are there are a number of things to consider
depending on students’ questions:
o Braking (and total) stopping distances tend to increase with speed, this is why we
increase our following distances when the speed increases (‘2 second rule’ for
following distances).
o Braking (and total stopping) distances tend to increase when it is wet because the
tyres do not grip the road so well (‘4 second rule’ for following distances).
o Reaction distances would not be expected to change significantly with speed or
conditions as this is dependent on how fast humans perceive the need to brake and
then how quickly they can apply the brakes.
o When considering how braking (and total stopping) distances increases with initial
speed, the differences in the distributions might be initially surprising. They are
unlikely to realise that doubling the speed causes the braking distance to increase by
a factor of 4.
The interest that different groups of people and organisations will have in the results will
depend on the question posed. Some examples are given below
o My friends who are learning to drive might be interested in these results because
they might not realise how much of a difference 10 km/h has on how far it will take
them to stop if there is a hazard on the road or if a child runs out onto the road in
front of the car that they are driving.
o Other drivers might be interested in this result as it will show them the importance of
obeying the ‘2 second rule’ for following distances. The two second rule means that
the faster you are driving, the more space a driver needs to leave between their car
and the one in front in order to avoid a collision if the car in front brakes suddenly.
o Other people who might be interested in these results are parents of teenagers who
are teaching their children to drive so that they can tell them about the effect that
speed and road conditions has on how far the car will continue to travel after you spot
a hazard and need to brake.
o Organisations that make decisions about speed limits may be interested in these
results. It would help them to decide if the speed limit along shopping streets, near
schools or in residential areas with lots of children should be reduced from 50 km/h to
40 (or even 30) km/h in order to reduce injuries to pedestrians.
o Organisations who control variable speed signs on the highways may be interested in
these results as it would give them information on what to reduce the speed limits to
if the road conditions are wet.
When students identify further questions that this investigation has lead to they should
consider the groups that might be interested in the results of their initial investigation.
Students should identify variables outside of the five that were given to them.
An example of a possible further investigation: This investigation could lead to a further
investigation into the effect that distractions have on reaction time, as reaction time
affects the reaction distance (and therefore the total stopping distance). An investigation
question could be ‘Do reaction times to a stimulus tend to be longer when people are
texting compared to when they are not texting’. In order to answer this question I would
need to find a website or a program that measures reaction time and have people take
the test several times, sometimes texting and other times not texting.
Resources:
• Copy of the investigation for each of the students or an electronic copy to project in the
classroom.
ICT resources:
• NZ Census at School informal inference
www.censusatschool.org.nz/2009/informal-inference
• Websites on stopping (and following) distances
www.nzta.govt.nz/resources/roadcode/about-driving/following-distance.html
www.rulesoftheroad.ie/rules-for-driving/speed-limits/speed-limits_stopping-distancescars.html
www.transportpolicy.org.uk/Future/20mph/20mph.htm (see subheading of ‘Safety’)
www.sdt.com.au/safedrive-directory-STOPPINGDISTANCE.htm
• You tube videos on stopping distances
www.youtube.com/watch?v=CzHklqaiTXI
• www.youtube.com/watch?v=Z_n-HIBnfts
Name:____________________________________ Teacher:________________
How far until it stops?
Investigating a given multivariate data set of stopping distances using the
statistical enquiry cycle
INTRODUCTION
A recent study was carried out in New Zealand by a group of people interested in vehicle
and road safety. During the study the stopping distances of numerous drivers, using a
variety of makes and models of cars, on different road surfaces, were measured.
Hundreds of measurements were taken over the course of the study.
The total stopping distance of a car can be broken down into two major components, the
reaction distance and the braking distance. The reaction distance takes account of the
time it takes for a driver to perceive that they need to stop and the amount of time it
takes them to react to the situation. The braking distance depends on the vehicle
reaction time and the vehicle braking capability.
You have been provided with a random sample of 120 of the stopping distance trials
collected during the New Zealand vehicle and road safety study and have been asked to
carry out a statistical investigation using the PPDAC (Problem, Plan, Data, Analysis and
Conclusion) statistical enquiry cycle. The sample is random so it is considered to be
representative of all the measurements made during the New Zealand vehicle and road
safety study.
The table below shows a portion of the sample that you will be provided with to
complete the investigation. Values for five variables are included for each of the trials
carried out.
Conditions
Dry
Dry
Wet
Dry
Wet
Wet
Wet
Dry
Wet
Speed
40
50
40
50
40
50
50
50
40
Reaction distance
8.6
8.1
7.8
8.2
7.3
9.7
6.4
7.9
5.9
Braking distance
7.2
14.3
14.0
12.3
16.2
27.5
18.2
14.4
15.1
Total stopping distance
15.8
22.4
21.8
20.5
23.5
37.2
24.6
22.3
21.0
The variables in the sample and the details of the measurements made were:
Variable
Conditions
Speed
Reaction distance
Braking distance
Total stopping distance
Measurement made
Road conditions: Wet or dry
Initial speed before braking in km/h
The distance travelled before applying the brakes, in metres
The distance travelled between applying the brakes and coming to
a complete stop, in metres
Reaction distance + braking distance, in metres
PROBLEM
1. Briefly describe the problem being investigated.
2. Pose two investigative questions about the data collected from a New Zealand study
on stopping distances. Your investigative questions must be comparison questions.
A suitable comparison investigative question is one that reflects the population, has a
clear variable to investigate, compares the values of a continuous variable across
different categories, and can be answered with the data.
For each question state the variables you are investigating and the population.
Question One:
Categorical variable:
Continuous variable:
Population:
Question Two
Categorical variable:
Continuous variable:
Population:
You must show your questions to your teacher before continuing with this investigation.
Your teacher will check that your question is a suitable one and give you feedback if
required to improve your question.
PROBLEM
Select one of the investigative questions about stopping distances from the two you
have posed to use to complete the investigation.
If you only have one suitable investigative question then use this question. If you have
been unable to pose a suitable investigative question, discuss with your teacher how to
refine one of your questions.
Write the question below that you will be investigating:
Variables:
Population:
PLAN
1. What do you think the answer to your question about stopping distances is?
2. From the dataset provided, what data will you use?
DATA
1. Does your stopping distance data need cleaning? Justify your answer.
ANALYSIS
1. What graphs do you plan to produce to display your stopping distance data? You
must choose at least two comparative graphs which show different features. You can
choose from a box and whisker graph, dot plot, histogram and stem and leaf graph.
2. Calculate the summary statistics for the two categories and fill in the table below
(include units):
Title:
Category
Minimum
Maximum
Median
Upper quartile
Lower quartile
IQR
Mean
3. Graphs (draw below, print from the computer, or copy and paste from a suitable
program):
Ensure that your graphs have a title, axes are labelled (include units) and that the
graphs are a sensible size.
4. Describe features of the distributions comparatively. This means each statement must
be comparing both categories. You must use correct statistical terms, be specific (use
the values for each group) and refer to the samples not the population.
Aim to make five statements describing such things as: shape, overlap, middle 50%,
spread, shift, unusual or interesting features, summary statistics. You do not need to
discuss all of these aspects.
1) Centre (median or mean):
2)
Middle 50%:
3) Shape of the distributions:
4) Overlap:
5) Spread:
6) Unusual or interesting features:
7) Other observations about the distributions:
CONCLUSION
1. Write a conclusion answering your investigative question about stopping distances
from the original population. This should be a full sentence not just a yes/no answer.
2. Provide two or three pieces of evidence from your samples using your summary
statistics, graphs and summary statements as support for your conclusion.
1)
2)
3)
3. Describe what you would expect to see if the random sampling process was repeated
from the same population. Comment on what you would expect to see in your summary
statistics/graphs and whether you think you would be able to reach the same conclusion.
4. Comment on how reasonable you think your results are, based on your knowledge of
stopping distances.
5. Explain why your friends who are learning to drive might be interested in these results.
6. Explain why other drivers might be interested in these results.
7. Which organisations or other groups of people might be interested in these results,
and why?
8. What other questions has this investigation generated? (You do not need to carry out
this investigation)
(a) Write a comparative or relationship investigative question
(b) What data would need to be collected in order to answer this question?
(c) Who would be interested in this further investigation and why?
9. Other comments
APPENDIX 1: Random sample of trials from the NZ road and vehicle safety
study.
Conditions
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Speed
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
Reaction
distance
7.2
6.0
7.7
10.3
5.7
7.4
14.2
8.6
8.2
9.8
7.0
5.9
7.0
8.8
5.6
7.4
11.6
6.3
6.2
6.6
12.1
6.7
5.9
9.9
6.5
7.4
7.9
12.3
5.4
6.0
9.2
6.4
17.0
10.2
11.5
9.4
6.6
9.5
6.4
6.9
9.9
18.2
6.2
12.4
7.0
Braking
distance
9.3
8.3
9.3
11.9
8.0
9.1
9.7
7.2
9.3
9.6
7.2
8.3
12.5
6.6
12.1
9.9
9.8
10.0
11.4
8.4
13.7
12.0
9.0
9.5
7.6
9.9
7.4
9.2
9.1
8.6
16.5
13.4
18.3
15.5
14.0
14.8
17.7
15.0
16.3
14.9
13.1
14.3
15.8
15.6
13.9
Total stopping
distance
16.5
14.3
17.0
22.2
13.7
16.5
23.9
15.8
17.5
19.4
14.2
14.2
19.5
15.4
17.7
17.3
21.4
16.3
17.6
15.0
25.8
18.7
14.9
19.4
14.1
17.3
15.3
21.5
14.5
14.6
25.7
19.8
35.3
25.7
25.5
24.2
24.3
24.5
22.7
21.8
23.0
32.5
22.0
28.0
20.9
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Dry
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
50
50
50
50
50
9.8
14.8
9.3
8.2
7.6
11.1
6.9
8.1
8.6
7.5
10.6
7.0
13.2
18.9
7.9
7.8
5.9
6.1
8.0
7.8
6.8
12.4
6.3
6.8
7.5
7.8
8.1
6.1
8.5
6.2
9.3
7.5
8.6
6.6
14.6
6.6
10.0
6.7
5.5
5.2
10.7
9.2
7.3
7.1
6.0
7.6
7.5
12.3
9.2
6.4
16.6
15.0
15.4
12.3
14.9
14.6
14.7
14.3
13.4
13.2
13.5
12.8
11.9
14.8
14.4
14.0
15.1
13.7
16.7
17.2
22.3
17.5
14.6
22.3
15.5
12.3
23.6
15.1
15.3
15.8
11.8
18.8
19.0
16.7
13.5
17.8
16.2
16.8
14.2
18.6
18.9
17.0
16.2
17.5
16.6
27.3
24.8
23.2
29.9
29.6
26.4
29.8
24.7
20.5
22.5
25.7
21.6
22.4
22.0
20.7
24.1
19.8
25.1
33.7
22.3
21.8
21.0
19.8
24.7
25.0
29.1
29.9
20.9
29.1
23.0
20.1
31.7
21.2
23.8
22.0
21.1
26.3
27.6
23.3
28.1
24.4
26.2
23.5
19.7
23.8
29.6
26.2
23.5
24.6
22.6
34.9
32.3
35.5
39.1
36.0
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
Wet
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
8.6
7.2
6.1
9.7
15.6
11.8
8.4
9.1
9.7
14.8
18.3
8.9
10.1
7.1
14.1
6.2
9.9
8.0
17.2
8.2
11.2
9.6
6.4
9.7
10.5
28.2
17.5
16.9
27.5
26.1
32.2
23.2
28.3
26.0
25.7
30.6
24.2
27.8
30.1
16.8
26.0
31.9
18.3
23.0
30.3
21.8
25.4
18.2
29.4
24.5
36.8
24.7
23.0
37.2
41.7
44.0
31.6
37.4
35.7
40.5
48.9
33.1
37.9
37.2
30.9
32.2
41.8
26.3
40.2
38.5
33.0
35.0
24.6
39.1
35.0
Self-assessment: How far until it stops?
Tick one box for each row to assess how you went in the investigation
I was given an
investigative question
I constructed one
comparative graph
My graph(s) has a
heading
I have calculated the
mean or median for both
categories
I have written one
analysis statement about
the sample which
includes the variable,
values and units
I have answered my
question
I have not stated
whether or not I think the
results are reasonable
I have not described
what would happen if the
sampling process was
repeated
I have explained why
one group of people
would be interested in
my results (e.g. my
friends)
I have written a question
for further investigation
I wrote a suitable
investigative question
with guidance from my
teacher
I constructed two
comparative graphs
My graph(s) has a
heading and the
categories are labelled
I have calculated 3
statistics for both the
categories
I have written two
analysis statements
about the sample which
includes the variable,
values and units
I have answered my
question and given one
piece of supporting
evidence
I have stated whether or
not I think the results are
reasonable
I have described what
would likely happen to
the statistics and graphs
if the sampling process
was repeated
I have explained why two
groups of people would
be interested in my
results (e.g. my friends
and other drivers)
I have written a question
for further investigation
and described what data
would need to be
collected
I wrote a suitable
investigative question
without guidance from my
teacher
I constructed three
comparative graphs
My graph(s) has a
heading, the categories
are labelled and the axes
are labelled (including
units)
I have calculated 5+
statistics for each of the
categories
I have written three or
more analysis statements
about the sample which
includes the variable,
values and units
I have answered my
question and given two or
three pieces of
supporting evidence
I have described why the
results are or are not
reasonable
I have described what
would likely happen to
the statistics and graphs
if the sampling process
was repeated and
whether I would be able
to make the same claim
I have explained why
three groups of people
would be interested in my
results
I have written a question
for further investigation,
described what data
would need to be
collected and explained
who would be interested
in the further study
Assessment schedule: How far until it stops?
Evidence/Judgements for C
(Achievement)
Evidence/Judgements for B
(Achievement with Merit)
Evidence/Judgements for A
(Achievement with Excellence)
Comparison question provided to the student.
Poses an appropriate comparison question with
guidance.
Poses an appropriate comparison question
without guidance.
Draws one graph and gives summary
statistic(s) that allow features of the data to be
described in relation to the question.
Draws one graph and gives summary statistics
that allow features of the data to be described in
relation to the question.
Writes two statements that describe different
comparative features of the distributions in
context (i.e. units, variable).
Writes two statements with evidence that
describe different comparative features of the
distributions in context (i.e. units, variable,
specific values).
Draws two or more graphs that show different
features and gives summary statistics that allow
features of the data to be described in relation
to the question.
Answers the comparison question in the
context of the investigation or makes a correct
comparison using an informal inference about
the population.
Makes a correct informal inference about the
population from the sample data. Answers the
comparison question, with at least one
statement of supporting evidence.
Example of possible evidence using the
comparative question:
Shows an understanding of the stopping
distance context by correctly answering one of
questions 4-8 in the conclusion section.
‘Do total stopping distances in wet conditions
collected in the NZ road and vehicle safety
study tend to be further than the total stopping
distances in dry conditions collected in the NZ
road and vehicle safety study?’
Example of possible evidence using the
comparative question:
Population: Trials measured during the NZ
road and vehicle safety study.
Variables: Total stopping distances and road
conditions
‘Do total stopping distances in wet conditions
collected in the NZ road and vehicle safety
study tend to be further than the total stopping
distances in dry conditions collected in the NZ
road and vehicle safety study?’
Population: Trials measured during the NZ
road and vehicle safety study.
Writes at least three statements with evidence
that describe different key comparative features
of the distributions in context (i.e. units,
variable, specific values).
Makes a correct informal inference about the
population from the sample data. Answers the
comparison question, with two statements of
supporting evidence.
Shows an understanding of the stopping
distance context by correctly answering at least
two of questions 4-8.
OR Shows an understanding of the stopping
distance context by correctly answering at least
one of questions 4-8 and demonstrates an
understanding about sampling variability.
Example of possible evidence using the
comparative question:
‘Do total stopping distances in wet conditions
collected in the NZ road and vehicle safety
study tend to be further than the total stopping
distances in dry conditions collected in the NZ
• Draws at least one graph, for example a dot
plot or a box and whisker graph.
• Analysis:
o
•Collection
Gives summary
statistics.Box Plot
1
The total stopping distances for dry
conditions are clustered around 15-25
m but for wet conditions they are more
spread with most stopping distances
being from 20-42 m.
The mean total stopping distance for
dry conditions is 21.0 m, which is less
than the mean total stopping distance
for wet conditions of 30.0 m.
• Conclusion:
o The total stopping distances in wet
conditions tends to be further than the
total stopping distances in dry
conditions.
road and vehicle safety study?’
Population: Trials measured during the NZ
road and vehicle safety study.
Variables: Total stopping distances and road
conditions
• Draws at least two graphs, for example, a dot
plot and a box plot.
• Gives summary statistics.
Wet
Dry
Wet
15 20 25 30 35 40 45 50
Total_stopping_dis tance
o
• Draws at least one graph, for example a dot
plot or a box and whisker plot.
Dry
• Gives evidence of at least one summary
Dot group
Plot
statistics,
Collectionfor
1 example a modal
Variables: Total stopping distances and road
conditions
15 20 25 30 35 40 45 50
Total_s topping_dis tance
Category
Dry
Wet
Minimum
13.7
19.7
LQ
16.8
23.7
Median
21.2
29.1
UQ
24.3
35.9
Maximum
35.3
48.9
Mean
21.0
30.0
Note: the summary statistics may only be
evident in the description. Not all are needed,
just sufficient statistics to support the
description. They can be read off the graph, or
given by statistical software.
• Analysis:
Note: the summary statistics may only be
evident in the description. Not all are needed,
just sufficient statistics to support the
description. They can be read off the graph, or
given by statistical software.
• Analysis:
In the sample analysed:
o The median total stopping distance for
dry conditions is 21.2 m, which is
approximately 8 m less than the median
stopping distance for wet conditions
(29.1 m). Each median value lies outside
the middle 50% of the data for the other
road condition.
o The middle 50% of total stopping
distances for dry conditions (16.8-24.3
m) are further down the scale (lower)
than for wet conditions (23.7-35.9 m)
o 75% of the total stopping distances for
wet conditions are above the lower 50%
of total stopping distances for the dry
conditions.
Note: The extreme values in the total stopping
distances are not outliers and to say they are
In the sample analysed:
o The middle 50% of total stopping
distances for dry conditions are less
spread (from 16.8 to 24.3 m) than the
middle 50% of total stopping distances
for wet conditions (from 23.7 to 35.9 m).
o The median total stopping distance for
dry conditions is 21.2 m, which is
approximately 8 m less than the median
stopping distance for wet conditions
(29.1 m).
Conclusion:
o I would claim that total stopping
distances for wet conditions tend to be
further than the total stopping distances
for dry conditions for the NZ road and
vehicle safety study.
o I would make this claim as the median
total stopping distance for the wet
conditions is about 8 m greater than that
for dry conditions in the sample.
o These results seem reasonable because
from my general knowledge I know that
when it is wet on the road it takes longer
to stop in a car because the tyres do not
grip to the road as well. This is why in
wet conditions drivers should apply the ‘4
second rule’ rather than the ‘2 second
rule’ to following distances. Then they
will have more time to come to a stop if
the car in front of them brakes suddenly.
would be incorrect.
• Conclusion:
o I would claim that total stopping
distances for wet conditions tend to be
further than the total stopping distances
for dry conditions for the NZ road and
vehicle safety study.
o My claim is based on the evidence
present in the sample. The median total
stopping distance for wet conditions is
about 8 m more than for dry conditions
and the median for each lies outside of
the middle 50% of data for the other
road condition. Also, the furthest 75% of
total stopping distances for wet
conditions are greater than the shortest
50% of total stopping distances for dry
conditions.
o If I was to repeat this sampling process I
would expect there to be some
difference in my sample statistics and
my graphs. However, I expect to be able
to make the same claim as the distance
between the medians in my sample is
large and the medians lie outside the
middle 50% of data for the other road
condition.
o Other drivers might be interested in this
result as it will show them the
importance of using the ‘4 second rule’
rather than the ‘2 second rule’ for
following distances. The ‘4 second rule’
is used for wet conditions as it means
that there is further to the car in front,
giving a greater distance to come to a
stop and avoiding collisions if the car in
front brakes suddenly.
o This investigation could lead to a further
investigation into the effect that
distractions has on reaction time, as
reaction time affects the reaction
distance (and therefore the total
stopping distance). An investigation
question could be ‘Do reaction times to a
stimulus tend to be longer when people
are texting compared to when they are
not texting’. In order to answer this
question I would need to find a website
or a program that measures reaction
time and have people take the test
several times, sometimes texting and
other times not texting.
Final grades will be decided using a holistic examination of the evidence.