HANDLING DATA
COURSEWORK
World Data
Main Menu
What is
Coursework???
Specify and Plan
Collect, Process &
Represent
Interpret and
Discuss
What You Should
Do?
Planning the
Investigation
Sample
Mean, Median,
Mode and Range
Pie Charts
Bar Charts
Histograms and
Freq Polygons
Scatter Plots
Stem and Leaf
Plots
Cumulative
Frequency
Box and Whisker
Plots
Your Task
 Is given in detail on the task sheet.
 Basically your task is to:
“investigate what influences the amount a student drinks.”
 The database has been selected for you from Rondam
Secondary school.
What Will Happen
A MIX OF THE FOLLOWING:
•Direct Teaching – statistics skills, ICT,
investigation cycle
•Group Work –
planning, discussing,
•Individual Time – writing up, working
Hypothesis
Specify
and
Plan
How could you
make it better?
Interpret and
discuss
Investigation
cycle
Collect, process
and represent
Specify and
plan
What to do in this section?
 Examine the Writing Frame and what decisions you
must make to fill it in.
 Decide on the hypothesis you are going to test. Make
sure it is well explained.
 Write a clear and detailed description of the task and
your plan to test the hypothesis.
 Do a draft first. Your final write up will come later.
Hypothesis
Collect,Specify
Process
and
Plan
Represent
How could youand
make
it better?
Interpret and
discuss
Investigation
cycle
Collect, process
and represent
Specify and plan
What to do in this section?
 Collect the data – fully explain your sampling technique
and sample size.
 Tabulate the data. Only include the information relevant
to your hypothesis.
 Using statistical and graphical methods to process and
examine the data.
Hypothesis
Interpret
Specify
and
and
Plan
How could you
make itDiscuss
better?
Interpret
and discuss
Investigation
cycle
Collect, process
and represent
Specify and plan
What to do in this section?
 This is the big crunch section.
 Draw conclusions from all of your calculations and relate
these to your initial hypothesis.
 Make sure you:
 Compare results to show differences/similarities.
 Use facts and statistics taken directly from your calculations.
 Evaluate your approach and explain any changes you would make
if you were doing it again.
 Consider bias in your results.
And Now …….
Challenge
What will a good piece of maths investigative work look like ???
You should consider:
• What will it contain?
• How will it be presented?
• How will it be marked?
• What will it look like?
15 mins in groups of 5 or 6
Formulating a hypothesis
The first step in planning a statistical enquiry is to decide
what problem you want to explore.
This can be done by asking questions that you want your
data to answer and by stating a hypothesis.
A hypothesis is a statement that you believe to be true but
that you have not yet tested.
The plural of hypothesis is hypotheses.
For example,
Year Eleven pupils with paid
jobs don’t do as well in their
exams.
Forming a hypothesis
“Year Eleven pupils with paid jobs
don’t do as well in their exams.”
How could you find out if this statement is true?
Think about:
What data (information) would you need to collect?
How will you collect it?
Which Year Elevens does this statement cover?
How could you ensure the data you collect represents all
of these Year Elevens?
What would you do with the data?
What would you expect to find?
Key vocabulary
hypothesis – a statement that can be tested
population – the group (often of people) referred to in the
hypothesis
sample – a selection from the population
biased sample – an unfair selection
representative sample – a fair selection
cross section – a selection that reflects all the subgroups
within the population
objective data – information that is not affected by people’s
opinions
Key vocabulary
subjective data – information that is affected by people’s
opinions
primary data – information you collect yourself, by asking
people, measuring, carrying out experiments, and so on
secondary data – information that has been collected already,
that you get from books, the internet, and so on
ethical issues – problems to do with confidentiality and
personal questions
reliable results – results that will be repeated if the
experiment or survey is carried out again with a new sample
Extending a hypothesis
Once you have collected data and drawn conclusions about
your hypothesis, you could ask further questions and pursue
other lines of enquiry.
You will need to plan what these might be beforehand if you
are carrying out a survey. For example,
“People feel stressed when they have exams.”
“You get less work done when it is noisy.”
“Sleep deprivation affects concentration.”
“Coffee can help you revise better.”
“The more revision you do, the better your exam results.”
How could you extend these hypotheses?
What extra information might it be worth collecting?
Sampling – Soap Wars
Westenders
12
Carnation Street
10
Millions
8
6
4
2
JAN
FEB
MAR
APR
MAY
AUG
VIEWING FIGURES
How are TV viewing figures compiled?
Television viewing figures
When compiling television viewing figures, it is impractical to
find out what everyone in the country is watching at a
particular time.
Instead, the viewing habits of a sample of households is
carefully monitored and the data collected is used to compile
the figures.
To avoid bias, it is important that the sample is representative
of all television viewing households across the country.
This is done by dividing households into categories and taking
samples in proportion to the size of each category.
This is an example of a stratified sample.
Different sampling methods
Random sampling
People are chosen at random e.g. names picked from a hat or
using a random number generator on a calculator.
Every member of the population has an equal
chance of being chosen.
Systematic sampling
Members of the population are chosen at regular intervals,
such as every 100th person from a telephone directory.
Quota sampling
You keep asking until you have enough people from each
category. An example would be a survey in the street where you
stop when you have enough people from each age category.
Evaluating different sampling methods
Random sampling
 Every member of the population has an equal chance of
being chosen, which makes it fair.
 It can be very time consuming and usually impractical.
Systematic sampling
 You are unlikely to get a biased sample.
 It is not strictly random: some members of the population
cannot be chosen once you have decided where to start on
the list.
Evaluating different sampling methods
Quota sampling
 This is easier to manage.
 It could be biased. For example, if you are only asking
people on the street or in a shop, the sample might not
represent people at work all day.
Stratified sampling
 It is the best way to reflect the population accurately.
 It is time consuming and you have to limit the number of
relevant variables to make it practical.
The three averages and range
There are three different types of average:
MODE
MEAN
MEDIAN
most common
sum of values
number of values
middle value
The range is not an average, but tells you how the data is
spread out:
RANGE
largest value – smallest value
Comparing sets of data
Here is a summary of Chris and Rob’s performance in the 200
metres over a season. They each ran 10 races.
Mean
Range
Chris
Rob
24.8 seconds
1.4 seconds
25.0 seconds
0.9 seconds
Which of these conclusions are correct?
Robert is more reliable.
Robert is better because his mean is higher.
Chris is better because his range is higher.
Chris must have run a better time for his quickest race.
On average, Chris is faster but he is less consistent.
Pie charts
A pie chart is a circle divided up into sectors which are
representative of the data.
In a pie chart, each category is shown as a fraction of the
circle.
Methods of travel to work
For example, in a
survey half the people
asked drove to work, a
quarter walked and a
quarter went by bus.
Car
Walk
Bus
Pie charts
To convert raw data into angles for n data items:
360 ÷ n represents the number of degrees per data item.
For example, 40 people take part in a survey. What angle
represents
one person?
360° ÷ 40 = 9°
two people?
9° × 2 = 18°
eight people? 9° × 8 = 72°
How many people are represented by an angle of 36°?
There are 9° per person. 36° ÷ 9° = 4 people.
Drawing pie charts
There are 30 people in the survey and 360º in a full pie chart.
Each person is therefore represented by 360º ÷ 30 = 12º
We can now calculate the angle for each category:
Newspaper
No of people
Working
Angle
The Guardian
8
8 × 12º
96º
Daily Mirror
7
7 × 12º
84º
The Times
3
3 × 12º
36º
The Sun
6
6 × 12º
72º
Daily Express
6
6 × 12º
72º
Total
30
360º
Drawing pie charts
Once the angles have been calculated you can draw the pie
chart.
Start by drawing a circle using
a compass.
The Daily
The
Express
Draw a radius.
Guardian
72º 96º
Measure an angle of 96º from
72º
the radius using a protractor
84º
The Sun
36º The Daily
and label the sector.
Mirror
The
Measure an angle of 84º from
Times
the the last line you drew and
label the sector.
Repeat for each sector until the pie chart is complete.
Drawing bar charts
When drawing bar chart remember:
Give the bar chart a title.
Use equal intervals on the axes.
Label both the axes.
Leave a gap between each bar.
Drawing bar charts
Use the data in the frequency table to complete a bar chart
showing the the number of children absent from school from
each year group on a particular day.
Year
Number of
absences
7
74
8
53
9
32
10
11
11
10
Bar charts for two sets of data
Two or more sets of data can be shown on a bar chart.
For example, this bar chart shows favourite subjects for a
group of boys and girls.
Girls' and boys' favourite subjects
8
Number of pupils
7
6
5
Girls
4
Boys
3
2
1
0
Maths
Science
English
Favourite subject
History
PE
Frequency diagrams
Frequency diagrams can be used to display grouped
continuous data.
For example, this frequency diagram shows the distribution of
heights for a group students:
Heights of students
Frequency
35
30
25
20
15
10
5
0
150
155
160
165
170
175
180
185
Height (cm)
This type of frequency diagram is often called a histogram.
Drawing frequency diagrams
Use the data in the frequency table to complete the frequency
diagram showing the time pupils spent watching TV on a
particular evening:
Time spent
(hours)
Number
of people
0≤h<1
4
1≤h<2
2≤h<3
3≤h<4
4≤h<5
6
8
5
3
h≤5
1
Histograms and Frequency Polygons
We can show the trend of these graphs more clearly using
a FREQUENCY POLYGON.
Using a previous example, you first need to draw a histogram
Then joint the midpoints of each column.
Heights of Year 8 pupils
35
Frequency
30
25
20
15
10
5
0
140
145
150
155
160
Height (cm)
165
170
175
Scatter Graphs
Scatter graphs
What does this scatter graph show?
Life expectancy
85
80
75
70
65
60
55
50
0
20
40
60
80
100
Number of cigarettes smoked in a week
120
It shows that life expectancy decreases as the number of
cigarettes smoked increases.
This is called a negative correlation.
Interpreting scatter graphs
Scatter graphs can show a relationship between two variables.
This relationship is called correlation.
Correlation is a general trend. Some data items will not fit this
trend, as there are often exceptions to a rule. They are called
outliers.
Scatter graphs can show:
positive correlation: as one variable increases, so does
the other variable
negative correlation: as one variable increases, the other
variable decreases
zero correlation: no linear relationship between the variables.
Correlation can be weak or strong.
The line of best fit
The line of best fit is drawn by eye so that there are roughly
an equal number of points below and above the line.
Look at these examples,
25
25
25
25
20
20
20
20
15
15
15
15
10
10
10
10
5
0
0
0
5
10
15
20
25
Strong positive
correlation
5
5
5
0
0
0
5
10
15
20
Weak positive
correlation
25
0
5
10
15
20
25
Strong negative
correlation
0
5
10
15
20
25
Weak negative
correlation
Notice that the stronger the correlation, the closer the points
are to the line.
If the gradient is positive, the correlation is positive and if the
gradient is negative, then the correlation is also negative.
Line of best fit
When drawing the line of best fit remember the following points,
The line does not have to pass through the origin.
For an accurate line of best fit, find the mean for each
variable. This forms a coordinate, which can be plotted. The
line of best fit should pass through this point.
The line of best fit can be used to predict one variable from
another.
It should not be used for predictions outside the range of
data used.
The equation of the line of best fit can be found using the
gradient and intercept.
Constructing stem-and-leaf diagrams
The data below represents the numbers of cigarettes smoked
in a week by regular smokers in Year 11.
7
15
5
38
13
10
41
23
30
22
45
20
20
7
7
11
5
17
24
30
17
19
Put this data into a stem-and-leaf diagram.
The stem should represent ____
tens and the leaf should
units
represent _____.
Work out the mode, mean, median and range.
Calculations with stem-and-leaf diagrams
Stem
(tens)
Leaf (units)
0 5 5 7 7 7
1 0 1 3 5 7 7 9
Mode
The mode is __
7 .
Mean
There are ___
22 people in the
survey and they smoke a total
427 cigarettes a week.
of ____
427 ÷ 22 =___
19
2 0 0 2 3 4
3 0 0 8
4 1 5
Median
The median is halfway between
17 and ___.
19 This is ___.
18
___
Range
___
45 – ___
5 = ___
40
Solving problems with stem-and-leaf diagrams
Stem
(tens)
Leaf (units)
0 5 5 7 7 7
1 0 1 3 5 7 7 9
2 0 0 2 3 4
3 0 0 8
4 1 5
What fraction of the group
smoke more than 20 cigarettes a
week? What is this as a
percentage?
The mean number smoked is 19.
How many smoke less than the
mean? What is this as a
percentage?
What percentage smoke less
than 10 cigarettes?
A packet of 20 cigarettes costs
about £4. Work out the average
amount spent on cigarettes
using the median.
Cumulative Freq - Choosing class intervals
You are going to record how long each member of your class
can keep their eyes open without blinking.
How could this information be recorded?
What practical issues might arise?
Time is an example of continuous data.
You will have to decide how accurately to measure the times,
to the nearest tenth of a second?
to the nearest second?
to the nearest five seconds?
Holding Your Breath
You will also have to decide what size class intervals to use.
When continuous data is grouped into class intervals it is
important that no values are missed out and that there are no
overlaps.
For example, you may decide to use class intervals with a
width of 5 seconds.
If everyone holds their breath for more than 30 seconds the first
class interval would be more than 30 seconds, up to and
including 35 seconds.
This is usually written as 30 < t ≤ 35, where t is the time in
seconds.
< t ≤ 40
The next class interval would be 35
_________.
Cumulative frequency
Cumulative frequency is a running total. It is calculated by
adding up the frequencies up to that point.
Here are the results of 100 people holding their breath:
Time in
seconds
30 < t ≤ 35
35 < t ≤ 40
40 < t ≤ 45
45 < t ≤ 50
50 < t ≤ 55
55 < t ≤ 60
Frequency
9
12
24
28
16
11
Cumulative
frequency
9
9 + 12 = 21
21 + 24 = 45
45 + 28 = 73
73 + 16 = 89
89 + 11 = 100
Time in seconds
0 < t ≤ 35
0 < t ≤ 40
0 < t ≤ 45
0 < t ≤ 50
0 < t ≤ 55
0 < t ≤ 60
Plotting a cumulative frequency graph
100
The upper boundary for each
class interval is plotted against
its cumulative frequency.
90
Cumulative frequency
80
70
A smooth curve is then drawn
through the points.
60
50
We can use the graph to
estimate the median by finding
the time for the 50th person.
40
30
This gives us a median time of
47 seconds.
20
10
0
30
35
40
45
50
55
Time in seconds
60
The interquartile range
Remember, the range is a measure of spread. It is the
difference between the highest value and the lowest value.
When the range is affected by outliers it is often more
appropriate to use the interquartile range.
The interquartile range is the range of the middle 50% of the
data.
The lower quartile is the data item ¼ of the way along the list.
The upper quartile is the data item ¾ of the way along the list.
interquartile range = upper quartile – lower quartile
Finding the interquartile range
100
The cumulative frequency graph
can be used to locate the upper
and lower quartiles and so find
the interquartile range.
90
Cumulative frequency
80
70
The lower quartile is the time
of the 25th person. 42 seconds
60
50
The upper quartile is the time
of the 75th person. 51 seconds
40
30
20
10
0
30
35
40
45
50
55
Time in seconds
60
The interquartile range is the
difference between these two
values.
51 – 42 = 9 seconds
A box-and-whisker diagram
A box-and-whisker diagram, or boxplot, can be used to
illustrate the spread of the data in a given distribution using the
median, the lower quartile and the upper quartile.
These values can be found from a cumulative frequency graph.
100
For example, for this cumulative
frequency graph showing the results of
100 people holding their breath,
Cumulative frequency
90
80
70
Minimum value = 30
60
50
Lower quartile = 42
40
30
Median = 47
20
Upper quartile = 51
10
0
30 35 40 45 50 55 60
Time in seconds
Maximum value = 60
A box-and-whisker diagram
The corresponding box-and-whisker diagram is as follows:
Minimum value
Median
Lower quartile
30
42
Maximum value
Upper quartile
47
51
60
Lap times
James takes part in karting
competitions and his Dad records
his lap times on a spreadsheet.
One of the karting tracks is at
Shenington. In 2004, 378 of James’
lap times were recorded.
The track is 1108 metres long. James’ fastest time in a race
was 51.8 seconds.
In which position in the list would the median lap time be?
There are 378 lap times and so the median lap time will be the
378 + 1 th
value ≈ 190th value
2
Lap times
In which position in the list would the lower quartile be?
There are 378 lap times and so the lower quartile will be the
378 + 1 th
value ≈ 95th value
4
In which position in the list would the upper quartile be?
There are 378 lap times and so the upper quartile will be the
378 + 1 th
3×
value ≈ 284th value
4
Lap times at Shenington karting circuit
James’ lap times are displayed in the following cumulative
frequency graph.
400
Cumulative frequency
350
300
250
200
150
100
50
0
52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92
Lap times in seconds
Box and whisker plot for James’ race times
Minimum value
Maximum value
Lower quartile
Median
Upper quartile
52 54
53
58
What conclusions can you draw about James’ performance?
91