© boardworks ltd 2005 1 of 44 d3 presenting data ks4 mathematics

© Boardworks Ltd 2005 1 of 44

D3 Presenting data

KS4 Mathematics


D3.1 Bar graphs

D3 Presenting data

Contents

D3.2 Line graphs

D3.3 Pie charts

D3.4 Stem-and-leaf diagrams

D3.5 Scatter graphs


Make a list of mistakes.

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1412 5

1 3 4 6 7 9

Bars should be separate.

The bars must be the same width.

The gaps must be the same width.

The scales must go up by equal intervals.

The numbers on the horizontal axis must appear in the middle of the bar.

The axes must be labelled.

There should be a title.

Bar graphs

What is wrong with this bar graph?


Number of pupils per primary school teacher

0 10 20 30 40 50 60

Sweden

Saudi Arabia

USA

United Kingdom

Zambia

Tanzania

Uganda

Education around the world


This graph show the responses of Year 7 to 11s in England to being offered drugs.

Pe

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nta

ge

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80

100

1999 2003

YesRefusedNever offered

Stacked bar graphs and drinking habits

What conclusions can you draw?


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100

East Asia &Pacific

Europe &Central Asia

Latin America &Caribbean

Middle East &North Africa

South Asia Sub-SaharanAfrica

Percentage of male literacy Percentage of female literacy

Comparative bar graphs and world literacy


D3.2 Line graphs

Contents

D3.1 Bar graphs

D3.3 Pie charts


D3.5 Scatter graphs

D3 Presenting data


Here are the percentages of Year 7 to Year 10 that smoke regularly in Great Britain.

Smoking among young people

Year 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Boys 11 13 7 7 9 9 10 11 9 8 9

Girls 11 13 12 9 11 10 13 15 12 10 12

Total 11 13 10 8 10 10 12 13 11 9 10

What would be the most appropriate graph to illustrate this data?


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16

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Pe

rcen

tag

e o

f re

gu

lar

smo

kers

Boys

Girls

Regular smoking in Years 7 to 10

Year


Regular smoking in Years 7 to 10

Year

0

2

4

6

8

10

12

14

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Per

cen

tag

e o

f re

gu

lar

smo

kers


Smoking among young people

Here are the percentages of Year 11s that smoke regularly in Great Britain.

Year 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Boys 24 28 18 17 25 21 26 28 19 21 21

Girls 25 28 27 22 25 25 30 33 29 25 26

Total 25 28 22 20 25 23 28 30 24 23 23

How does this data compare to that of Years 7 to 10?

Again we can show the data using a line graph.


Regular smoking in Year 11

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5

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15

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25

30

35

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Pe

rcen

tag

e o

f re

gu

lar

smo

kers

Boys

Girls

Year


Regular smoking in Year 11

0

5

10

15

20

25

30

35

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Per

cen

tag

e o

f re

gu

lar

smo

kers

Year


Comparing Years 7 to 10 to Year 11

0

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30

35

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Per

cen

tag

e o

f re

gu

lar

smo

kers

Years 7 to 10

Year 11

Year


D3.3 Pie charts

Contents

D3.2 Line graphs

D3.1 Bar graphs


D3.5 Scatter graphs

D3 Presenting data


Percentages

1 140 000

330 000

360 000

540 000

600 000

Numbers for 1988

39%

10%

12%

15%

25%

Percentage for 2003

Numbers for 2003

Never

Last week

1 – 4 weeks ago

1 – 6 months ago

More than 6 months ago

Percentage for 1988

When did you last have a

drink?

Over 10 000 Year 7 to Year 11 pupils took part in a survey in 2003 carried out by the Department of Health. There are about 3 000 000 people in England in this age range. Fill in the table.

1 170 000

300 000

360000

450 000

750 000

38%

20%

18%

12%

11%


Drinking habits among young people in 2003

Compare the results for 2003 with 1988.

25%

15%

12%10%

38%

20%

18%

12%11%

39%

Last week

1-4 weeks

1-6 months

More than 6 months

Never


Pie charts

There are 9° per person.

To convert raw data into angles for n data items:

360 ÷ n represents the number of degrees per data item.

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°?

36° ÷ 9° = 4 people.


Converting data into angles


Displaying data as a pie chart

There should be no gaps in your pie chart.

The angles should add up to 360o.

Angles should be rounded off to the nearest degree if necessary.

If you have had to round off, the angles may add up to slightly more or less than 360o.

Each section should be labeled or a key should be used. You may want to include actual numbers or percentages. Angles are not normally included.

When drawing a pie chart, it is helpful to note the following points.


Solving problems with pie charts

This data represents the smoking habits among Year 7s in England. Calculate the angles.

Can you explain the totals?

Angle

85Never

1Regular

2Occasional

2Used to

11Tried

PercentageSmoking habits

364° 101Total

306°

40°

7°

7°

4°


1% 2% 2%

11%

84%

RegularOccasionalUsed to TriedNever

Pie chart of Year 7 smoking habits


36% of Year 11s have never smoked. What angle represents this category?

72o represents the Year 11s who have tried smoking. What percentage is this?

There are approximately 600 000 Year 11s in England. 22% smoke regularly. How many people is this?

One third of this group have a cigarette within 30 minutes of waking. How many is this?

Pie chart of Year 11 smoking habits

RegularOccasionalUsed to TriedNever



Contents

D3.3 Pie charts

D3.2 Line graphs

D3.1 Bar graphs

D3.5 Scatter graphs

D3 Presenting data


Stem (pounds) Leaf (pence)

4 00 40 50 70 70 80

5 00 30 40 50 50

6 20 50 50 60 70

7 00 30 50 50 50 90

8 00 40 50 70

9 50 50 60

10 00 00 20 40 50 60

Interpreting stem-and-leaf diagrams

A stem-and-leaf diagram can be used to display data items in order without grouping them.

A stem-and-leaf diagram can be used to display data items in order without grouping them.

For example, this table shows how much pocket money some regular smokers in Year 11 spend on cigarettes in a fortnight.


The data below represents the numbers of cigarettes smoked in a week by regular smokers in Year 11.

7 38 41 22 20 7 5 24 1715 13 23 45 7 11 17 30 19 5 10 30 20

Constructing stem-and-leaf diagrams

Put this data into a stem-and-leaf diagram.

The stem should represent ____ and the leaf should represent _____.

Work out the mode, mean, median and range.

tensunits


5 5 7 7 70

1 54

0 0 83

0 0 2 3 42

0 1 3 5 7 7 91

Leaf (units)Stem (tens)

Calculations with stem-and-leaf diagrams

427 ÷ 22 =___19

This is ___.

427

22

7

17 19 18

45 5 40

Mode

The mode is __ .

Mean

There are ___ people in the survey and they smoke a total of ____ cigarettes a week.

Median

The median is halfway between ___ and ___.

Range

___ – ___ = ___


Solving problems with stem-and-leaf diagrams

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.

5 5 7 7 70

1 54

0 0 83

0 0 2 3 42

0 1 3 5 7 7 91

Leaf (units)Stem (tens)


Use job advertisements in newspapers and the internet to investigate how much graduates leaving university get paid compared with school leavers of 16 or 18.

Investigation using stem-and-leaf diagrams

Record your results in stem-and-leaf diagrams.

Calculate the mean and median incomes for each group.

What conclusions can you draw about the financial advantages of getting a degree?

An extension task could involve comparing the incomes of new graduates with graduates after ten years. Which careers offer greater opportunities for promotion or financial rewards?


D3.5 Scatter graphs

Contents


D3.3 Pie charts

D3.2 Line graphs

D3.1 Bar graphs

D3 Presenting data


Scatter graphs

As height increases, weight increases.

What does this scatter graph show about the relationship between the height and weight of twenty Year 10 boys?

This is called a positive correlation.This is called a positive correlation.

40

45

50

55

60

140 150 160 170 180 190Height (cm)

Wei

gh

t (k

g)


Scatter graphs

What does this scatter graph show?

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0 20 40 60 80 100 120Number of cigarettes smoked in a week

Lif

e e

xp

ect

ancy

It shows that life expectancy decreases as the number of cigarettes smoked increases.

This is called a negative correlation.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.


Identifying correlation from scatter graphs

strong positive correlation

strong negative correlation

zero correlation

weak positive correlation

weak negative correlation.

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A

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C

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F

Decide whether each of the following graphs shows,


Relationships between two variables


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Strong positive correlation

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Strong negative correlation

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Weak negative correlation

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,

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0 5 10 15 20 25

Weak positive 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

The line does not have to pass through the origin.

When drawing the line of best fit remember the following points,

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.


The line of best fit

50

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0 20 40 60 80 100Number of cigarettes smoked in a week

Lif

e ex

pec

tan

cyThis graph shows the relationship between life expectancy and the number of cigarettes smoked in a week.


Solving problems with lines of best fit

Work out an estimate for the equation of the line of best fit using the gradient and intercept.

Use the equation to estimate the life expectancy for someone who smokes 10 cigarettes a day.

Why would an estimate of the number of cigarettes smoked for a life expectancy of 40 years not be reliable?

Can you explain why there are so many outliers for this data?


Cause and effect

A study finds a positive correlation between the number of cars in a town and the number of babies born.

“Buying a new car can help you get pregnant!”

The local newspaper reports,

Does the study support this conclusion?

What might this be in the example above?

Correlation does not necessarily imply that there is a causal relationship between the two variables. There may be some other cause.


Cause and effect

A study finds a negative correlation between the number of sledges sold and the temperature.

Explain.

Does the study support this conclusion?

“If you want it to snow, go out and buy a sledge!”

The local newspaper reports,


Cause and effect

Discuss these headlines.

“Taller students do better in new Maths test!”

“Counselling can make you depressed.”

“The more coffee you drink, the more stressed you are.”

“Chocolate causes lower grades at university.”

“New exercise regime causes dramatic rise in injuries at local hospital.”


Review

List the types of graph you have covered in this topic.

What kind of data would you use for each kind of graph?

What possible mistakes could you make with a bar graph?

If you are investigating a relationship between two variables, what kind of graph would you use?

How do you calculate the angles in a pie chart if you know how many data items there are altogether?


Review

What other problems can you solve with a pie chart? Give examples and outline the method for each.

How would you calculate the three averages from a stem and leaf diagram?

What are the different types of correlation? Give examples.

What is a line of best fit and how would you draw one?

© boardworks ltd 2005 1 of 44 d3 presenting data ks4 mathematics

Documents

year slide

bar graphs d3

line graphs d3

scatter graphs d3

data slide

data contents d3

line graphs contents

world slide