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KS3 Mathematics

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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

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

Bar charts for categorical 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.

Bar charts for discrete 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.

Bar charts for 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.

Bar charts for two sets of data

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.

Bar line graphs

For example, this bar line graph shows a set of test results.

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

When drawing bar chart remember:

Year

Number of absences

7

74

8

53

9

32

10

11

11

10

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.

D3 Representing and interpreting data

Contents

D3.1 Bar charts

D3.3 Frequency diagrams

D3.2 Pie charts

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data

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.

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

Pie charts

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

Pie charts

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?

Drawing pie charts

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.

Drawing pie charts

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

The results were :

Total

Method 1

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:

96º

8 × 12º

84º

7 × 12º

36º

3 × 12º

72º

6 × 12º

72º

6 × 12º

30

360º

8

7

× 360º =

× 360º =

30

30

Method 2

Drawing pie charts

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

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

Drawing pie charts

These calculations can be written into the table.

96º

84º

36º

72º

72º

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 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

36

Drawing pie charts

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

No of people

UK

74

Europe

53

America

32

Asia

11

Other

10

Total

180

Drawing pie charts

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

Smokey

bacon

Prawn cocktail

35º

55º

Salt and vinegar

135º

85º

Ready salted

135

=

50º

360

Cheese and onion

Reading pie charts

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

Contents

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

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.

Frequency diagrams

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)

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:

Drawing frequency diagrams

D3 Representing and interpreting data

Contents

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 most often used to show trends over time.

Line graphs

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.

Line graphs

What trends are shown by this graph?

Age (years)

Weight (kg)

1

9.5

2

12.0

3

14.2

4

16.3

5

18.4

Drawing line graphs

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

Contents

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.

Scatter graphs

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?

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.

Scatter graphs and correlation

For example,

Length of spring (cm)

Mass attached to spring (g)

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

Scatter graphs and 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

Maths score

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

Scatter graphs and correlation

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(°C)

Rainfall (mm)

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

Scatter graphs and 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 (ºC)

Electricity used (kWh)

Scatter graphs and correlation

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

Age (years)

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

Scatter graphs and correlation

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 (°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.

Plotting scatter graphs

Hours watching TV

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.

Plotting scatter graphs

D3 Representing and interpreting data

Contents

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

5

7

6

5

7

8

6

Jamie

3

6

4

8

12

9

8

Comparing distributions

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

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

=

Comparing distributions

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

5

7

6

5

7

8

6

Jamie

3

6

4

8

12

9

8

Comparing distributions

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 skewed to the left.

This distribution is symmetrical (or normal).

This distribution is skewed to the right.

This distribution is random.

Comparing the shape of distributions

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

Group A

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.

Comparing the shape of distributions

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.