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Mathematics
Representing and interpreting data Š Boardworks Ltd 2004
Contents
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Representing and interpreting data A 1 Bar charts 1 A 2 Pie charts 1 A 3 Frequency diagrams 1 A 4 Line graphs 1 A 5 Scatter graphs 1 A 6 Comparing data 1 Š Boardworks Ltd 2004
Categorical data
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Categorical data is data that is non-numerical. For example,
favourite football team, eye colour, birth place.
Sometimes categorical data can contain numbers. For example,
favourite number, last digit in your telephone number, most used bus route.
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Discrete and continuous data
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Numerical data can be discrete or continuous. Discrete data can only take certain values. For example,
shoe sizes, the number of children in a class, the number of sweets in a packet.
Continuous data comes from measuring and can take any value within a given range. For example,
the weight of a banana, the time it takes for pupils to get to school, the height of 13 year-olds. Š Boardworks Ltd 2004
Discrete or continuous data
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Bar charts for categorical data
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Bar charts can be used to display categorical or nonnumerical data. For example, this bar graph shows how a group of children travel to school. How children travel to school Number of children
12 10 8 6 4 2 0
walk
train
car
bicycle
bus
other
Method of travel
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Bar charts for discrete data
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Bar charts can be used to display discrete numerical data. For example, this bar graph shows the number of CDs bought by a group of children in a given month. Number of CDs bought in a month
Number of children
25 20 15 10 5 0 0
1
2
3
4
5
Number of CDs bought
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Bar charts for grouped discrete data
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Bar charts can be used to display grouped discrete data. For example, this bar graph shows the number of books read by a sample of people over the space of a year. Books read in one year
Number of books
20+ 16-19 12-15 8-11 4-7 0-3 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
Number of people
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Bar charts for two sets of data
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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 Number of pupils
8 7 6 5
Girls
4
Boys
3 2 1 0
Maths
Science
English
History
PE
Favourite subject
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Bar line graphs
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Bar line graphs are the same as bar charts except that lines are drawn instead of bars. For example, this bar line graph shows a set of test results.
Number of pupils
Mental maths test results
Mark out of ten
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Drawing bar charts
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When drawing bar chart remember: Give the bar chart a title. Use equal intervals on the axes. Draw bars of equal width. Leave a gap between each bar. Label both the axes. Include a key for the chart if necessary.
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Contents
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D3 Representing and interpreting data A D3.1 Bar charts 1 A D3.2 Pie charts 1 A D3.3 Frequency diagrams 1 A D3.4 Line graphs 1 A D3.5 Scatter graphs 1 A D3.6 Comparing data 1 Š Boardworks Ltd 2004
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.
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Car Walk Bus
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Pie charts This pie chart shows the distribution of drinks sold in a cafeteria on a particular day.
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Drinks sold in a cafeteria
coffee soft drinks tea
Altogether 300 drinks were sold. Estimate the number of each type of drink sold. Coffee:
75
Soft drinks: 50 Tea:
175
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Pie charts These two pie charts compare the proportions of boys and girls in two classes.
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Mr Humphry's class
Number of boys Number of girls
Mrs Payne's class
Number of boys Number of girls
Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?
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Drawing pie charts To draw a pie chart you need compasses and a protractor.
The first step is to work out the angle needed to represent each category in the pie chart.
We need to work out how many degrees are needed to represent each person or thing in the sample.
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Drawing pie charts
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For example, 30 people were asked which newspapers they read regularly. The results were : Newspaper
Number of people
The Guardian
8
Daily Mirror
7
The Times
3
The Sun
6
Daily Express
6
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Drawing pie charts Method 1 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:
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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 © Boardworks Ltd 2004
Drawing pie charts Once the angles have been calculated you can draw the pie chart. Start by drawing a circle using compasses. Daily The Express Guardian Draw a radius. 72º 96º Measure an angle of 96º from 72º 84º the radius using a protractor The Sun 36º 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.
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Drawing pie charts Use the data in the frequency table to complete the pie chart showing the favourite colours of a sample of people.
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Favourite colour
No of people
Pink
10
Orange
3
Blue
14
Purple
5
Green
4
Total
36
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Drawing pie charts Use the data in the frequency table to complete the pie chart showing the holiday destinations of a sample of people.
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Holiday destination
No of people
UK
74
Europe
53
America
32
Asia
11
Other
10
Total
180
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Reading pie charts The following pie chart shows the favourite crisp flavours of 72 children.
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Prawn cocktail 55º
Smokey bacon 35º
Salt and 135º vinegar 85º 50º Cheese and onion
Ready salted
How many children preferred ready salted crisps? How many degrees repesents one child? 360 = 5º. 72 The number of children who preferred ready salted is: 135 ÷ 5 =
27 © Boardworks Ltd 2004
Contents
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D3 Representing and interpreting data A D3.1 Bar charts 1 A D3.2 Pie charts 1 A D3.3 Frequency diagrams 1 A D3.4 Line graphs 1 A D3.5 Scatter graphs 1 A D3.6 Comparing data 1 Š Boardworks Ltd 2004
Frequency diagrams Frequency diagrams are used to display grouped continuous data. For example, this frequency diagram shows the distribution of heights in a group of Year 8 pupils:
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Heights of Year 8 pupils 35
Frequency
30 25 20
The divisions between the bars are labelled.
15 10 5 0 140
145
150
155
160
165
170
175
Height (cm)
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Contents
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D3 Representing and interpreting data A D3.1 Bar charts 1 A D3.2 Pie charts 1 A D3.3 Frequency diagrams 1 A D3.4 Line graphs 1 A D3.5 Scatter graphs 1 A D3.6 Comparing data 1 Š Boardworks Ltd 2004
Line graphs
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Line graphs are most often used to show trends over time. For example, this line graph shows the temperature in London, in ºC, over a 12-hour period.
Temperature (ºC)
Temperature in London 20 18 16 14 12 10 8 6 4 2 0 6 am 7 am 8 am 9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm
Time
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Line graphs
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This line graph compares the percentage of boys and girls gaining A* to C passes at GCSE in a particular school. Percentage of boys and girls gaining A* to C passes at GCSE 70 60 50 40
Girls Boys
30 20 10 0 1998
1999
2000
2001
2002
2003
2004
What trends are shown by this graph? Š Boardworks Ltd 2004
Contents
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D3 Representing and interpreting data A D3.1 Bar charts 1 A D3.2 Pie charts 1 A D3.3 Frequency diagrams 1 A D3.4 Line graphs 1 A D3.5 Scatter graphs 1 A D3.6 Comparing data 1 Š Boardworks Ltd 2004
Scatter graphs and correlation
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We can use scatter graphs to find out if there is any relationship or correlation between two sets of data. For example, Do tall people weigh more than small people? If there is more rain, will it be colder? If you revise longer, will you get better marks? Do second-hand car get cheaper with age? Is more electricity used in cold weather? Are people with big heads better at maths?
Š Boardworks Ltd 2004
Scatter graphs and correlation
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When one variable increases as the other variable increases, we have a positive correlation.
Length of spring (cm)
For example, this scatter graph shows that there is a strong positive correlation between the length of a spring and the mass of an object attached to it. The points lie close to an upward sloping line. This is the line of best fit. Mass attached to spring (g) Š Boardworks Ltd 2004
Scatter graphs and correlation
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Sometimes the points in the graph are more scattered. We can still see a trend upwards.
Science score
This scatter graph shows that there is a weak positive correlation between scores in a maths test and scores in a science test. The points are scattered above and below a line of best fit. Maths score Š Boardworks Ltd 2004
Scatter graphs and correlation
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When one variable decreases as the other variable increases, we have a negative correlation.
Temperature(°C)
For example, this scatter graph shows that there is a strong negative correlation between rainfall and hours of sunshine. The points lie close to a downward sloping line of best fit. Rainfall (mm) Š Boardworks Ltd 2004
Scatter graphs and correlation
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Sometimes the points in the graph are more scattered.
Outdoor temperature (ÂşC)
We can still see a trend downwards.
Electricity used (kWh)
For example, this scatter graph shows that there is a weak negative correlation between the temperature and the amount of electricity a family used.
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Scatter graphs and correlation
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Sometimes a scatter graph shows that there is no correlation between two variables. Number of hours worked
For example, this scatter graph shows that there is a no correlation between a person’s age and the number of hours they work a week. The points are randomly distributed. Age (years) Š Boardworks Ltd 2004
Plotting scatter graphs
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This table shows the temperature on 10 days and the number of ice creams a shop sold. Plot the scatter graph. Temperature (°C)
14
16
20
19
23
21
25
22
18
18
Ice creams sold
10
14
20
22
19
22
30
15
16
19
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Plotting scatter graphs
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We can use scatter graphs to find out if there is any relationship or correlation between two set of data. Hours watching TV
2
4
3.5
2
Hours doing homework 2.5 0.5 0.5
2
1.5 2.5 3
2
3
5
1
0.5
1
0
2
3
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Contents
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D3 Representing and interpreting data A D3.1 Bar charts 1 A D3.2 Pie charts 1 A D3.3 Frequency diagrams 1 A D3.4 Line graphs 1 A D3.5 Scatter graphs 1 A D3.6 Comparing data 1 Š Boardworks Ltd 2004
Comparing distributions
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The distribution of a set of data describes how the data is spread out. Two distributions can be compared using one of the three averages and the range. For example, the number of cars sold by two salesmen each day for a week is shown below. Matt
5
7
6
5
7
8
6
Jamie
3
6
4
8
12
9
8
Who is the better salesman?
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Comparing distributions
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Matt
5
7
6
5
7
8
6
Jamie
3
6
4
8
12
9
8
To decide which salesman is best let’s compare the mean number cars sold by each one. Matt: 44 5+7+6+5+7+8+6 = Mean = = 6.3 (to 1 d.p.) 7 7 Jamie: 3 + 6 + 4 + 8 + 12 + 9 + 8 50 Mean = = = 7.1 (to 1 d.p.) 7 7 This tells us that, on average, Jamie sold more cars each day.
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Comparing distributions
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Matt
5
7
6
5
7
8
6
Jamie
3
6
4
8
12
9
8
Now let’s compare the range for each salesman. Matt: Range = 8 – 5 = 3 Jamie: Range = 12 – 3 = 9 The range for the number of cars sold each day is smaller for Matt. This means that he is a more consistent or reliable salesman. We could argue that Jamie is better because he sells more on average, or that Matt is better because he is more consistent. © Boardworks Ltd 2004