representing and interpreting data 1

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Mathematics

Representing and interpreting data


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Contents

Representing and interpreting data

1 Bar charts

2 Pie charts

3 Frequency diagrams

4 Line graphs

5 Scatter graphs

6 Comparing data


Categorical data

Categorical data is data that is non-numerical.

For example,

Sometimes categorical data can contain numbers.

For example,

favourite football team, eye colour, birth place.

favourite number, last digit in your telephone number, most used bus route.


Discrete and continuous data

Discrete data can only take certain values.

Continuous data comes from measuring and can take any value within a given range.

Numerical data can be discrete or continuous.

For example,

For example,

shoe sizes, the number of children in a class, the number of sweets in a packet.

the weight of a banana, the time it takes for pupils to get to school, the height of 13 year-olds.


Discrete or continuous data


Bar charts for categorical 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.

How children travel to school

0

2

4

6

8

10

12

walk train car bicycle bus other

Method of travel

Num

ber o

f chi

ldre

n


Bar charts for discrete data

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

0

5

10

15

20

25

0 1 2 3 4 5

Number of CDs bought

Num

ber o

f chi

ldre

n


Bar charts for grouped discrete data

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

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0-3

4-7

8-11

12-15

16-19

20+

Num

ber o

f boo

ks

Number of people


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

0

1

2

3

4

5

6

7

8

Maths Science English History PE

Favourite subject

Num

ber o

f pup

ils

GirlsBoys


Bar line graphs

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.

Mental maths test results

Mark out of ten

Num

ber o

f pup

ils


Drawing bar charts

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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D3 Representing and interpreting data

D3.2 Pie charts

D3.1 Bar charts

D3.3 Frequency diagrams

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.

Methods of travel to work

CarWalkBus


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

Drinks sold in a cafeteria

coffeesoft drinkstea


Pie charts

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

Mr Humphry's class

Number ofboysNumber ofgirls

Mrs Payne's class

Number ofboysNumber ofgirls

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 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.


Drawing pie charts

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


Drawing pie charts

Method 1There are 30 people in the survey and 360º in a full pie chart.Each person is therefore represented by 360º ÷ 30 = 12ºWe can now calculate the angle for each category:

Newspaper No of people Working AngleThe Guardian 8Daily Mirror 7The Times 3The Sun 6Daily Express 6

8 × 12º 96º7 × 12º 84º3 × 12º 36º6 × 12º 72º6 × 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 compasses.Draw a radius.Measure an angle of 96º from the radius using a protractor and label the sector.

96º

The Guardian

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

84º

Daily Mirror

Repeat for each sector until the pie chart is complete.

36º

The Times

72º

72º

The Sun

Daily Express


Drawing pie charts

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

No of people

10

3

14

5

4

Favourite colour

Pink

Orange

Blue

Purple

Green

Total 36


Drawing pie charts

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

Holiday destination

No of people

UK 74

Europe 53

America 32

Asia 11

Other 10

Total 180


Reading pie charts

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

35º

Smokeybacon

135º Ready salted50º

Cheese and

onion

85º

55ºSalt and vinegar

Prawn cocktail

How many children preferred ready salted crisps?

How many degrees repesents one child?

360 72 = 5º.

The number of children who preferred ready salted is:

135 ÷ 5 = 27


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D3.2 Pie charts

D3.1 Bar charts

D3.4 Line graphs

D3.5 Scatter graphs

D3.6 Comparing data


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:

The divisions between the bars are labelled.Fr

eque

ncy

Height (cm)

0

5

10

15

20

25

30

35

140 145 150 155 160 165 170 175

Heights of Year 8 pupils


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D3.4 Line graphs


D3.2 Pie charts

D3.1 Bar charts

D3.5 Scatter graphs

D3.6 Comparing data


Line graphs

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 in London

02468

101214161820

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

Tem

pera

ture

(ºC

)


Line graphs

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?

Percentage of boys and girls gaining A* to C passes at GCSE

0

10

20

30

40

50

60

70

1998 1999 2000 2001 2002 2003 2004

GirlsBoys


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D3.5 Scatter graphs

D3.4 Line graphs


D3.2 Pie charts

D3.1 Bar charts

D3.6 Comparing data


Scatter graphs and correlation

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

For example,

If you revise longer, will you get better marks?

Do second-hand car get cheaper with age?

Are people with big heads better at maths?

Do tall people weigh more than small people?

Is more electricity used in cold weather?

If there is more rain, will it be colder?



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.

Mass attached to spring (g)

Leng

th o

f spr

ing

(cm

)

The points lie close to an upward sloping line.

This is the line of best fit.



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.

Maths score

Sci

ence

sco

re

The points are scattered above and below a line of best fit.



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.

Rainfall (mm)

Tem

pera

ture

(°C

)

The points lie close to a downward sloping line of best fit.



Sometimes the points in the graph are more scattered.

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

Electricity used (kWh)

Out

door

tem

pera

ture

(ºC

)

We can still see a trend downwards.



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.

Age (years)

Num

ber o

f hou

rs w

orke

d


Plotting scatter graphs

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

Temperature (°C)

Ice creams sold

14

10

16

14

20

20

19

22

23

19

21

22

25

30

22

15

18

16

18

19


Plotting scatter graphs

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

Hours watching TV

Hours doing homework

2

2.5

4

0.5

3.5

0.5

2

2

1.5

3

2.5

2

3

1

5

0

1

2

0.5

3


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D3.6 Comparing data

D3.5 Scatter graphs

D3.4 Line graphs


D3.2 Pie charts

D3.1 Bar charts


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.

Matt

Jamie

5

3

7

6

6

4

5

8

7

12

8

9

6

8

Who is the better salesman?



To decide which salesman is best let’s compare the mean number cars sold by each one.

Matt

Jamie

5

3

7

6

6

4

5

8

7

12

8

9

6

8

Matt:

Mean = 5 + 7 + 6 + 5 + 7 + 8 + 67 = 44

7 = 6.3 (to 1 d.p.)

Jamie:

Mean = 3 + 6 + 4 + 8 + 12 + 9 + 87 = 50

7 = 7.1 (to 1 d.p.)

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



Now let’s compare the range for each salesman.

Matt

Jamie

5

3

7

6

6

4

5

8

7

12

8

9

6

8

Matt: Range = 8 – 5 = Jamie: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.

3Range = 12 – 3 = 9

We could argue that Jamie is better because he sells more on average, or that Matt is better because he is more consistent.

representing and interpreting data 1

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