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KS3 Mathematics. D3 Representing and interpreting data. D3 Representing and interpreting data. Contents. D3.2 Pie charts. D3.3 Frequency diagrams. D3.1 Bar charts. D3.4 Line graphs. D3.5 Scatter graphs. D3.6 Comparing data.

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### KS3 Mathematics

### Contents

### Bar charts for categorical data

### Bar charts for discrete data

### Bar charts for grouped discrete data

### Bar charts for two sets of data

### Bar line graphs

### Drawing bar charts

### Drawing bar charts

### Contents

### Pie charts are drawn instead of bars.

### Pie charts are drawn instead of bars.

### Pie charts are drawn instead of bars.

### Drawing pie charts are drawn instead of bars.

### Drawing pie charts are drawn instead of bars.

### Drawing pie charts

### Drawing pie charts

### Drawing pie charts

### Drawing pie charts are drawn instead of bars.

### Drawing pie charts

### Drawing pie charts

### Reading pie charts

### Contents

### Frequency diagrams

### Drawing frequency diagrams

### Contents

### Line graphs

### Line graphs

### Drawing line graphs

### Contents

### Scatter graphs

### Scatter graphs and correlation

### Scatter graphs and correlation

### Scatter graphs and correlation

### Scatter graphs and correlation

### Scatter graphs and correlation

### Scatter graphs and correlation

### Plotting scatter graphs

### Plotting scatter graphs

### Contents

### Comparing distributions

### Comparing distributions

### Comparing distributions

### Comparing the shape of distributions

### Comparing the shape of distributions

D3 Representing and interpreting data

D3 Representing and interpreting data

D3.2 Pie charts

D3.3 Frequency diagrams

D3.1 Bar charts

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data

Bar charts can be used to display categorical or non-numerical data.

For example, this bar graph shows how a group of children travel to school.

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.

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.

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.

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.

Give the bar chart a title. are drawn instead of bars.

Use equal intervals on the axes.

Label both the axes.

Leave a gap between each bar.

When drawing bar chart remember:

Year are drawn instead of bars.

Number of absences

7

74

8

53

9

32

10

11

11

10

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.

D3 Representing and interpreting data are drawn instead of bars.

D3.1 Bar charts

D3.3 Frequency diagrams

D3.2 Pie charts

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data

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.

For example, in a survey half the people asked drove to work, a quarter walked and a quarter went by bus.

This pie chart shows the distribution of drinks sold in a cafeteria on a particular day.

Altogether 300 drinks were sold.

Estimate the number of each type of drink sold.

Coffee:

75

Soft drinks:

50

Tea:

175

These two pie charts compare the proportions of boys and girls in two classes.

Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?

To draw a pie chart you need a compass and a protractor.

The first step is to work out the angle needed to represent each category in the pie chart.

There are two ways to do this.

The first is to work out how many degrees are needed to represent each person or thing in the sample.

The second method is to work out what fraction of the total we want to represent and multiply this by 360 degrees.

For example, 30 people were asked which newspapers they read regularly.

The results were :

Total are drawn instead of bars.

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:

96º

8 × 12º

84º

7 × 12º

36º

3 × 12º

72º

6 × 12º

72º

6 × 12º

30

360º

8 are drawn instead of bars.

7

× 360º =

× 360º =

30

30

Method 2

Write each category as a fraction of the whole and find this fraction of 360º.

8 out of the 30 people in the survey read The Guardian so to work out the size of the sector we calculate

96º

7 out of the 30 people in the survey read the Daily Mirror so to work out the size of the sector we calculate

84º

Newspaper are drawn instead of bars.

No of people

Working

Angle

6

3

7

8

6

The Guardian

8

× 360º

30

30

30

30

30

Daily Mirror

7

× 360º

The Times

3

× 360º

The Sun

6

× 360º

Daily Express

6

× 360º

Total

Method 2

These calculations can be written into the table.

96º

84º

36º

72º

72º

30

360º

Once the angles have been calculated you can draw the pie chart.

Start by drawing a circle using a compass.

The Daily Express

The Guardian

Draw a radius.

72º

Measure an angle of 96º from the radius using a protractor and label the sector.

96º

72º

84º

The Sun

36º

The Daily Mirror

Measure an angle of 84º from the the last line you drew and label the sector.

The Times

Repeat for each sector until the pie chart is complete.

Total are drawn instead of bars.

36

Use the data in the frequency table to complete the pie chart showing the favourite colours of a sample of people.

Favourite colour

No of people

Red

10

Yellow

3

Blue

14

Green

5

Purple

4

Holiday destination are drawn instead of bars.

No of people

UK

74

Europe

53

America

32

Asia

11

Other

10

Total

180

Use the data in the frequency table to complete the pie chart showing the holiday destinations of a sample of people.

Smokey are drawn instead of bars.

bacon

Prawn cocktail

35º

55º

Salt and vinegar

135º

85º

Ready salted

135

=

50º

360

Cheese and onion

The following pie chart shows the favourite crisp flavours of 72 children.

How many children preferred ready salted crisps?

The proportion of children who preferred ready salted is:

0.375

The number of children who preferred ready salted is:

0.375 × 72 =

27

D3 Representing and interpreting data are drawn instead of bars.

D3.1 Bar charts

D3.2 Pie charts

D3.3 Frequency diagrams

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data

Heights of Year 8 pupils are drawn instead of bars.

35

30

25

20

Frequency

15

10

5

0

145

150

155

160

165

170

175

140

Height (cm)

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:

The divisions between the bars are labelled.

Time spent (hours) are drawn instead of bars.

Number of people

0 ≤ h < 1

4

1 ≤ h < 2

6

2 ≤ h < 3

8

3 ≤ h < 4

5

4 ≤ h < 5

3

h ≤ 5

1

Use the data in the frequency table to complete the frequency diagram showing the time pupils spent watching TV on a particular evening:

D3 Representing and interpreting data are drawn instead of bars.

D3.1 Bar charts

D3.2 Pie charts

D3.3 Frequency diagrams

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data

Line graphs are drawn instead of bars. 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.

This line graph compares the percentage of boys and girls gaining A* to C passes at GCSE in a particular school.

What trends are shown by this graph?

Age (years) gaining A* to C passes at GCSE in a particular school.

Weight (kg)

1

9.5

2

12.0

3

14.2

4

16.3

5

18.4

This data shows the weight of a child taken every birthday.

Plot the points on the graph and join them with straight lines.

D3 Representing and interpreting data gaining A* to C passes at GCSE in a particular school.

D3.1 Bar charts

D3.2 Pie charts

D3.3 Frequency diagrams

D3.5 Scatter graphs

D3.4 Line graphs

D3.6 Comparing data

We can use scatter graphs to find out if there is any relationship or correlation between two set of data.

Handspan (cm)

18

16

20

15

16

21

19

17

20

18

Foot length (cm)

24

21

28

20

22

30

25

22

27

23

Do tall people weigh more than small people? relationship or

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?

We can use scatter graphs to find out if there is any relationship or correlation between two sets of data.

For example,

Length of spring (cm) relationship or

Mass attached to spring (g)

When one variable increases as the other variable increases, we have a positive correlation.

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.

Science score relationship or

Maths score

Sometimes the points in the graph are more scattered. We can still see a trend upwards.

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.

Temperature( relationship or °C)

Rainfall (mm)

When one variable decreases as the other variable increases, we have a negative correlation.

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.

Outdoor temperature ( relationship or ºC)

Electricity used (kWh)

Sometimes the points in the graph are more scattered.

We can still see a trend downwards.

For example, this scatter graph shows that there is a weak negative correlation between the temperature and the amount of electricity a family used.

Number of hours worked relationship or

Age (years)

Sometimes a scatter graph shows that there is no correlation between two variables.

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.

Temperature ( relationship or °C)

14

16

20

19

23

21

25

22

18

18

Ice creams sold

10

14

20

22

19

22

30

15

16

19

This table shows the temperature on 10 days and the number of ice creams a shop sold. Plot the scatter graph.

Hours watching TV relationship or

2

4

3.5

2

1.5

2.5

3

5

1

0.5

Hours doing homework

2.5

0.5

0.5

2

3

2

1

0

2

3

We can use scatter graphs to find out if there is any relationship or correlation between two set of data.

D3 Representing and interpreting data relationship or

D3.1 Bar charts

D3.2 Pie charts

D3.3 Frequency diagrams

D3.6 Comparing data

D3.4 Line graphs

D3.5 Scatter graphs

Matt relationship or

5

7

6

5

7

8

6

Jamie

3

6

4

8

12

9

8

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.

Who is the better salesman?

3 + 6 + 4 + 8 + 12 + 9 + 8 relationship or

5 + 7 + 6 + 5 + 7 + 8 + 6

7

7

=

44

50

Matt

5

7

6

5

7

8

6

7

7

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:

Mean =

= 6.3 (to 1 d.p.)

Jamie:

Mean =

= 7.1 (to 1 d.p.)

This tells us that, on average, Jamie sold more cars each day.

Matt relationship or

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.

This distribution is relationship or skewed to the left.

This distribution is symmetrical (or normal).

This distribution is skewed to the right.

This distribution is random.

We can comparing distributions by looking at the shape of their graphs.

Group A relationship or

Group B

Group C

Group D

Frequency

Frequency

Frequency

Frequency

1-10

11-20

21-30

31-40

41-50

1-10

11-20

21-30

31-40

41-50

1-10

11-20

21-30

31-40

41-50

1-10

11-20

21-30

31-40

41-50

Four groups of pupils sat the same maths test. These graphs show the results.

One of the groups is a top set, one is a middle set, one is a bottom set and one is a mixed ability group.

Use the shapes of the distribution to decide which group is which giving reasons for your choice.

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