Qualification
Accredited
GCSE (9–1)
Delivery Guide
MATHEMATICS
J560
For first teaching in 2015
Statistics
Version 2
www.ocr.org.uk/maths
Introduction
GCSE (9–1) Mathematics
Delivery Guide
GCSE (9–1)
MATHEMATICS
Delivery guides are designed to represent a body of knowledge about teaching a particular
topic and contain:
• Content: A clear outline of the content covered by the delivery guide;
• Thinking Conceptually: Expert guidance on the key concepts involved, common difficulties
students may have, approaches to teaching that can help students understand these
concepts and how this topic links conceptually to other areas of the subject;
• Thinking Contextually: A range of suggested teaching activities using a variety of themes so
that different activities can be selected which best suit particular classes, learning styles or
teaching approaches.
If you have any feedback on this Delivery Guide or suggestions for other resources you would
like OCR to develop, please email [email protected]
Curriculum content
Page 3
Thinking conceptually
Page 5
Thinking contextually
Page 6
Learner resources
Page 8
2
© OCR 2016
Curriculum content
GCSE (9–1) Mathematics
GCSE (9–1)
content Ref.
Subject content
OCR 12
Statistics
12.01
Sampling
12.01a
Populations and samples
Delivery Guide
Initial learning for this
qualification will enable learners
to…
Foundation tier learners should
also be able to…
Higher tier learners should
additionally be able to…
DfE
Ref.
Define the population in a study,
and understand the difference
between population and sample.
Infer properties of populations or
distributions from a sample.
S1
Understand what is meant by simple
random sampling, and bias in
sampling.
12.02
Interpreting and representing data
12.02a
Categorical and numerical data
Interpret and construct charts
appropriate to the data type;
including frequency tables, bar
charts, pie charts and pictograms for
categorical data, vertical line charts
for ungrouped discrete numerical
data.
Design tables to classify data.
Interpret and construct line graphs
for time series data, and identify
trends (e.g. seasonal variations).
S2
Interpret multiple and composite bar
charts.
12.02b
Grouped data
Interpret and construct diagrams
for grouped data as appropriate, i.e.
cumulative frequency graphs and
histograms (with either equal or
unequal class intervals).
3
S3
S4
© OCR 2016
Curriculum content
GCSE (9–1) Mathematics
GCSE (9–1)
content Ref.
Subject content
12.03
Analysing data
12.03a
Summary statistics
Delivery Guide
Initial learning for this
qualification will enable learners
to…
Foundation tier learners should
also be able to…
Calculate the mean, mode, median
and range for ungrouped data.
Find the modal class, and calculate
estimates of the range, mean and
median for grouped data, and
understand why they are estimates.
Higher tier learners should
additionally be able to…
DfE
Ref.
Calculate estimates of mean,
median, mode, range, quartiles and
interquartile range from graphical
representation of grouped data.
S4,
S5
Draw and interpret box plots. Use the
median and interquartile range to
compare distributions.
Describe a population using statistics.
Make simple comparisons.
Compare data sets using ‘like for like’
summary values.
Understand the advantages and
disadvantages of summary values.
12.03b
Misrepresenting data
Recognise graphical
misrepresentation through incorrect
scales, labels, etc.
12.03c
Bivariate data
Plot and interpret scatter diagrams
for bivariate data.
Recognise correlation.
S4
Interpret correlation within the
context of the variables, and
appreciate the distinction between
correlation and causation.
S6
Draw a line of best fit by eye, and use
it to make predictions.
Interpolate and extrapolate from
data, and be aware of the limitations
of these techniques.
12.03d
Outliers
Identify an outlier in simple cases.
Appreciate there may be errors in
data from values (outliers) that do
not ‘fit’.
S4
Recognise outliers on a scatter graph.
4
© OCR 2016
Thinking conceptually
GCSE (9–1) Mathematics
Delivery Guide
Current students are learning statistics at a time when vast quantities of data are readily
available and presented to us. Studying statistics provides an opportunity for students to
develop both the analytical skills and the critical skills needed to consider and evaluate
inferences about data, which are frequently presented by both the media and the internet as
factual.
Numerical data on its own does not provide a sense of the inferences that can be made.
Representing data graphically is an effective tool to reveal potential trends and allows easier
comparisons between data sets. Grouped data sometimes causes students difficulties,
for example plotting frequency instead of cumulative frequency and when calculating
frequency density.
Statistics can be thought of as descriptive and inferential. With descriptive statistics, students
use graphical representation and find summary measures to convey information about the
available data. With inferential statistics, students use samples of data, which will be subject
to random variation, in order to learn about the population from which the sample of data is
thought to represent, and then to make deductions and draw conclusions.
A common misconception when analysing data is to pick any values to make comparisons
rather than comparing ‘like for like’ measures, such as two mean values. To address the
assessment objectives for mathematics, teaching should focus on enabling students to make
deductions within the context of the data. For example, when investigating correlations,
students should be able to explain any correlation in terms of the variables and at the same
time never assuming correlation implies causation.
The three broad areas of statistics content (sampling, interpreting and representing data, and
analysing data) provide a structure to teach statistics. Students’ learning and understanding
will benefit through always engaging with all three areas as part of an investigative cycle
when interrogating samples of data.
It is strongly suggested that where possible, teachers provide current examples to emphasise
the importance of statistical reasoning. Examples could include analysing graphs presented
in the media; making decisions on the efficacy of ‘beneficial’ food products; using time series
data to make long term predictions. Links to other areas of the school curriculum will be
enhanced through using examples from other subjects, in particular science, geography,
psychology and economics.
Whilst sophisticated sampling techniques are not included in the content, it is important
for students to understand the distinction between a population and a sample and also
to understand how sampling bias and variability between samples may affect summary
measures and subsequent inferences.
5
© OCR 2016
Thinking contextually
GCSE (9–1) Mathematics
Delivery Guide
Activities
[12.03a] Averages and Range Jigsaw
This activity works best in pairs or groups of 3. It addresses misconceptions and provides
practice of statistical concepts.
A major advantage of teaching statistics is the real life context within which the application
exists.
It is suggested that:
Cut out the 24 triangular jigsaw pieces for each group (Learner Resource 1).
•
real data is used
The group task is to match sides to create a hexagon from the triangular pieces.
•
the context is accessible in order to avoid confusion
Use matching pairs to further consolidate concepts, for example:
•
the context interests the students
•
∑ = 42 n = 3 range 2 matches mean 14 - ask students what are the possible numbers
•
the results and inferences are not obvious in order to maintain interest.
•
n = 5 median 2 matches mean 2 - ask students how many possible sets of numbers
they can find.
Additionally:
•
This leads to using the NRICH activity M, M and M, available at http://nrich.maths.
org/6267. This activity challenges students to find sets of numbers giving rise to the same
summary data.
sometimes it is okay to amend the data to be given, in order to avoid confusion before
students are ready for the challenge of dealing with raw data or contradictory results.
However, raw data is beneficial; it provides challenges and introduces the need for
students to make judgement calls. It is important for students to realise data is rarely
pristine; data could be incomplete and people do give silly answers or make mistakes
when data is collected, such as using the current year for the year of their birth for
instance
•
inconclusive results are still valid to use in the classroom – they happen in real life
•
using a variety of contexts is better than one long project – this enables students to
understand what inferences are specific to the context and what is general to most
contexts.
Further jigsaws can be created yourself using freely available Tarsia software,
www.mmlsoft.com/index.php/products/tarsia
[12.03a, 12.03c] Exploring a queuing problem
This is a whole class simulation activity.
A retailer has more than one service point for customers. Should the retailer require
customers to individually choose and queue at one of the service points or all queue
in a single line? A simulation is used to find possible waiting times. Data is gathered by
students in a human simulation.
Cut out the set of 28 cards for the whole class simulation (Learner Resource 2). It is
suggested that these are used with 3 service points. The actual simulation should take
approximately 10 minutes.
For a more detailed explanation see the accompanying ‘Teacher Resource 2: Exploring a
queuing problem’.
The data generated is used to find means, medians, modal values, ranges (and quartiles for
Higher tier). It is also used to plot scatter graphs (and box plots for Higher tier).
6
© OCR 2016
Thinking contextually
GCSE (9–1) Mathematics
Delivery Guide
[12.02a, 12.03a] What’s the Weather Like? (NRICH)
http://nrich.maths.org/10470/index
This activity involves using secondary data from Shawbury, Eastbourne and Nairn
weather stations. The data sets are ideally suited to making comparisons over time or
between different places. It is recommended that spreadsheets are used. It may also be
a good task to undertake collaboratively with science or geography.
[12.03c] Olympic Records (NRICH)
http://nrich.maths.org/7489
This activity gives students the opportunity to make sense of graphical data and
challenges them to apply their own knowledge about athletics to explain and interpret
key features of the graphs.
7
© OCR 2016
Delivery Guide
Averages and range jigsaw
mean 2
4
9
mean 300
56
n=
8
7
ian
d
me
4
2
ge
∑=
2
7
2
3
ran
5
4.5
10
5
10
5.5
9
8
7
8
5
8
3
range 7
4 4 4 4 4 4
5
∑=
2
7
an
Me
7
a
me
3
5
6
n=
de
2
n1
5
9
∑=
5
mo
5
72
6 5 6 5 6
6
∑=
n=
n=
median 5.5
2 6 5 11 1 3 11
Learner resource 1
GCSE (9–1) Mathematics
n = 8 ∑ = 69
mode 4
8
mean 5
© OCR 2016
Delivery Guide
Averages and range jigsaw
mean 0
-6
8
4
ian
d
me
0
ge
ra
ran
m
e4
ng
ran
ge
1
51
1
ian
ed
range 7
range 5
median 8
median 5 mode 11
5
7
2
5
5
∑
1
6
15
3
8
=8
a
me
12
8
1
d
me
5
6
5
n8
3
7
15
3
ian
2
5
7
11
1
6
10
8
15
5
8
12
9
14
e4
1
2
9
7
14
g
ran
2
median 5.5
-2
range 10
me
an
7
mean 1
61
60
66
61
Learner resource 1
GCSE (9–1) Mathematics
14
n=6
n = 5 ∑ = 1500
9
© OCR 2016
Delivery Guide
Averages and range jigsaw
∑ = 42
n = 3 range 2
7.5
6
9
5
8
8
7
6
7
7
n=
2
2
7
1
an
e
m
4
7
7
3
6.5
6
an
me
6.5
2
2
ian
d
me
7.5
4
8
4
7
4
7
6
1.2 2.4 0.5 0.3 0.1 1.5
mean 8
median 9
2 3 4 5 6 7 8 9
7
m
od
e5
ge
m
d
mo
12
6
ran
e5
5
13
9
ran
15
4
ge
de
mo
1
8
5
n=
6
9
n=
ia
ed
n6
mean 5 range 6
Learner resource 1
GCSE (9–1) Mathematics
mean 7
n = 11
median 7
10
© OCR 2016
Delivery Guide
m
od
e
4
1
12 8 11 7 8
6 7 8 8 6
n=
1
od
e
mo
de
n3
me
5
me
an
2
5
m
=6
5
15
6
ge
ran
dia
n2
an
15
2 1 2
range 6
dia
me
1
30
0
4
range 4 median 6 mode 5
mean 14
me
∑ = 56 n = 8
1.2 2.4 0.5 0.3 0.1 1.5
5
9
mean 5
n=
10
10
15
9
3
8
13
8
7
6
5
4
12
9
n8
∑=
an
12
∑=
15
88
a
me
me
∑ = 69
an
8
9
n=8
6.5
0
7
ran
ge
-2
me
-6
8
3
6
7
4
n7
7
7
an
7
me
4
6.5
1
14
n
8
7.5
dia
ge
2
mean 7
11
2
7.5
me
ran
3
n9
7
dia
9
me
2
7
ge
7
2
11
5
6
7
0
0
15
4
m
4
ran
5
4
14
11
61
66
60
61
2
3
1
8
0
4
5
5
an
ge
ran
4
∑=
11
n4
dia
me
4
de
mean 1
n
=5
mo
2
ean
4
∑=
=2
me
5 7 2 4 8 5 4
1
n5
n
5
6
5
6
6
7
5
2
5
5
6
9 8 10 7 3 12 6 14
n
=6
6
9 5
11
5
5
6
m
4
ge
ran
dia
∑ = 42 n = 3 range 2
n=
7
me
5
ge
6 7 5
n=
ran
51
n1
ia
ed
5
de
mo
∑=5 n=3
e7
ng
ra
.5
62
3
n5
an
range 5 median 8
dia
me
mean 7 median 7
me
2 3 4.5 5 5.5 7 8 9
Averages and range jigsaw solution
mean 5.5 range 7
72
9
Teacher resource 1
GCSE (9–1) Mathematics
10
© OCR 2016
Learner resource 2
GCSE (9–1) Mathematics
Delivery Guide
Wait time cards
Arrival time:
10:13
Transaction time:
9 minutes
Arrival time:
10:17
Transaction time:
6 minutes
Arrival time:
10:21
Transaction time:
3 minutes
Arrival time:
10:14
Transaction time:
7 minutes
Arrival time:
10:18
Transaction time:
7 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Arrival time:
10:07
Transaction time:
9 minutes
Arrival time:
10:16
Transaction time:
5 minutes
Arrival time:
10:20
Transaction time:
3 minutes
Arrival time:
10:14
Transaction time:
3 minutes
Arrival time:
10:04
Transaction time:
5 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Arrival time:
10:10
Transaction time:
8 minutes
Arrival time:
10:09
Transaction time:
9 minutes
Arrival time:
10:19
Transaction time:
4 minutes
Arrival time:
10:12
Transaction time:
6 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
12
© OCR 2016
Learner resource 2
GCSE (9–1) Mathematics
Delivery Guide
Wait time cards
Arrival time:
10:08
Transaction time:
3 minutes
Arrival time:
10:13
Transaction time:
4 minutes
Arrival time:
10:22
Transaction time:
2 minutes
Arrival time:
10:03
Transaction time:
8 minutes
Arrival time:
10:06
Transaction time:
7 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Arrival time:
10:07
Transaction time:
10 minutes
Arrival time:
10:24
Transaction time:
4 minutes
Arrival time:
10:23
Transaction time:
5 minutes
Arrival time:
10:16
Transaction time:
1 minutes
Arrival time:
10:20
Transaction time:
2 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Arrival time:
10:25
Transaction time:
7 minutes
Arrival time:
10:02
Transaction time:
9 minutes
Arrival time:
10:21
Transaction time:
6 minutes
Arrival time:
10:15
Transaction time:
10 minutes
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Single queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
Multiple queue wait time:
mins
13
© OCR 2016
Teacher resource 2
GCSE (9–1) Mathematics
Delivery Guide
Exploring a queuing problem
Explain the scenario:
A retailer has more than one service point for customers open. Should the retailer require customers to individually choose and queue at one of the service points or just all queue in a single
line? A simulation is used to find possible waiting times.
Discuss with the class:
• Places where people queue to use a service point (bank / shop / post office / toll booth / etc)
• Potential issues with the two types of queuing system
• What do they think may be best? Best for whom? Why?
Each student is issued with a card showing arrival time and transaction time. There is space on the card to write in wait times using each of the queuing systems.
For example:
Arrival time:
10:13
Transaction time:
9 minutes
• Arrival time 10:13 means they join the queue at 10:13.
• If they reach the service point at 10:20 the wait time is 7 minutes.
• Transaction time 9 minutes means they leave the service point at 10:29.
Single queue wait time:
mins
Multiple queue wait time:
mins
14
© OCR 2016
Teacher resource 2
GCSE (9–1) Mathematics
Delivery Guide
Exploring a queuing problem
Run the simulation first with a single queue and then with multiple queues.
With multiple queues, allow students to switch lines.
Use chairs as service points.
The teacher calls out the times at minute intervals, adjust the cadence to give students time to complete the wait time. (You may need a trial run using a few students).
The data generated is used to find means, medians, modal values, ranges (and quartiles for Higher tier). It is also used to plot scatter graphs (and box plots for Higher tier).
Discuss with the class:
• Is it better to have a lower mean wait time or lower modal wait time?
• Would a single queue or a multiple queue system deal better with situations in which one or two people have long wait times (outliers) and everyone else had a very short wait time?
• What difference would it have made if you weren’t allowed to change queues?
• Which queuing system is fairer? Why?
• Which queuing system would you choose? Justify your choice using the results.
The arrival times and transaction times are randomly generated. A further set of results will enable meaningful discussion of sampling and bias.
Produce a second set of cards and run the whole simulation again to find a second set of summary values and plot graphs.
Discuss with the class:
Are the results similar or different for the second simulation?
Which queuing system would you choose now? Why?
15
© OCR 2016
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