You should be able to

■ carry out basic arithmetic

1 Find the midpoint of each pair.

a 5 and 15 b 20 and 30 c 0 and 40

d 10 and 15 e 25 and 30

2 Work out each of these.

a (20 � 0.2) � (10 � 1.4) � (5 � 0.8)

b (30 � 0.6) � (20 � 1.2) � (5 � 0.6)

3 Work out each of these.

a 13 � (50 � 40) b 12 � (10 � 5)

c 9 � (40 � 15) d 9 � (0.5 � 0.2)

Check in

Ric

h t

ask What is the most likely time during a football game for a team to score a

goal?

It is frequently stated by football commentators that teams are most likely to

concede a goal within fi ve minutes of scoring a goal themselves. Is this true?

Investigate and write a report on your results

Orienta

tion

■ Construct and

interpret frequency

polygons, histogram

and scatter diagrams

A-level

Maths, Biology,

Economics, Geography

Business

D1 Calculate summary

statistics

D2 Compare data sets

N1 Basic arithemetic

What I need to know What I will learn What this leads to

Interpreting frequency graphsOver 30% of the numbers in everyday use begin with the digit 1 whilst less that 5%

begin with a 9. ‘Benford’s law’, as it is called, makes it possible to detect when a list

of numbers has been falsifi ed. This is particularly useful in fraud investigations for

detecting ‘made-up’ entries on claim forms and expense accounts.

What’s the point?One of the tasks statisticians work on is fi nding ways to display and characterise

different data sets using statistical ‘fi ngerprints’. This is so that they can fi rst identify

and then measure any possible differences to decide if they are signifi cant or not.

Interp

reting freq

uen

cy graphs

D4

276276 277277

AO

2Fu

nct

ional

Math

s 1 The frequency tables show the ages of the first 100 people to visit a shopping centre on a Monday and a Saturday.

Draw frequency polygons for these data. Compare the ages of the shoppers on the two days.2 The frequency polygons show the time taken by 120 teachers and

120 office workers to travel home from work.

Compare the two sets of journey times.3 The frequency polygons show the number of miles Jayne travelled

each day in her car during December and January.

Compare the daily mileages for the two months. Suggest reasons for any differences or similarities.

Age, a, years Frequency

0 � a � 10 11

10 � a � 20 3

20 � a � 30 14

30 � a � 40 12

40 � a � 50 15

50 � a � 60 19

60 � a � 70 26

Monday

Age, a, years Frequency

0 � a � 10 7

10 � a � 20 18

20 � a � 30 31

30 � a � 40 20

40 � a � 50 12

50 � a � 60 9

60 � a � 70 3

Saturday

Teachers

100 20 30 40 500

10

20

30

40

50

60

minutes

Fre

qu

ency

2010 30 40 50 600

10

20

30

40

50

60

minutes

Fre

qu

ency

Office workers

Fre

qu

ency

December

200 40 60 80 1000

2

4

6

8

10

12

Miles200 40 60 80 100

0

2

4

6

8

10

12

Fre

qu

ency

Miles

January

Grade BExa

mple The tables show the age distribution of people in a gym and a bowls club.

Draw frequency polygons for this data.Compare the ages of the people in the two clubs.

Age, a, years Frequency

0 � a � 10 32

10 � a � 20 56

20 � a � 30 23

30 � a � 40 14

40 � a � 50 5

Gym

Age, a, years Frequency

20 � a � 30 6

30 � a � 40 14

40 � a � 50 34

50 � a � 60 45

60 � a � 70 24

Bowls

100 20 30 40 7050 600

10

20

30

40

50

60

Fre

qu

ency

Age (years)

Gym

100 20 30 40 7050 600

10

20

30

40

50

60

Age (years)

Fre

qu

ency

Bowls

The modal age is greater in the bowls club.Highest frequency for gym is in age range 10–20 years.Highest frequency for bowls is in age range 50–60 years.

The range of ages is the same at both clubs.Estimated range for gym: 50 � 0 � 50Estimated range for bowls: 70 � 20 � 50

The estimated mean age for bowls is much higher than that for gym.

Gym: (5 � 32) � (15 � 56) � (25 � 23) � (35 � 14) � (45 � 5)

_____________________________________________________ 130

� 2290 � 130 � 17.6 years (1 dp)

Bowls: (25 � 6) � (35 � 14) � (45 � 34) � (55 � 45) � (65 � 24)

_______________________________________________________ 123

� 6205 � 123 � 50.4 years (1 dp)

For each

class you

plot (midpoint,

frequency).

For the end

points you plot

(0, lower bound

of first class)

and (0, upper

bound of last

class).

Join the points

with straight

lines.

To estimate

the range, use

the highest and

lowest possible

data values.

Use midpoint

frequency to

estimate the

means.

The data needs

to be grouped

in equal sized

class intervals. It

can be discrete

or continuous.

This spread will show you how to:

● Draw frequency polygons to represent grouped data

● Compare data sets using summary statistics

Exercise D4.1

● You can represent grouped data in a frequency polygon.● You can use frequency polygons to compare data sets.

p.90

Frequency polygonsKeywords

Frequency

polygon

Mean

Midpoint

Modal

Range

D4.1

278278 279279

AO

3

Grade A

For each set of dataa Copy and complete the table.b Draw a histogram to represent the data.

1 Reaction times of a sample of students.

Time, t seconds 1 � t � 3 3 � t � 4 4 � t � 5 5 � t � 6 6 � t � 9

Class width

Frequency 12 17 19 11 18

Frequency density

2 Amounts spent by the first 100 customers in a shop one Saturday.

Amount spent, £a 0 � a � 5 5 � a � 10 10 � a � 20 20 � a � 40 40 � a � 60 60 � a � 100

Class width

Frequency 6 10 23 29 24 8

Frequency density

3 Distance travelled to work by 100 office workers.

Distance, d miles 0 � d � 2 2 � d � 5 5 � d � 10 10 � d � 20 20 � d � 30

Class width

Frequency 8 15 27 44 6

Frequency density

4 Times of goals scored in Premiership football matches one Saturday.

Time, t minutes 0 � t � 10 10 � t � 40 40 � t � 45 45 � t � 55 55 � t � 85 85 � t � 90

Class width

Frequency 12 48 18 11 30 22

Frequency density

5 Distances swum by children in a sponsored swim.

Distance, d km 0.1 � d � 0.2 0.2 � d � 0.5 0.5 � d � 1 1 � d � 2 2 � d � 5

Class width

Frequency 3 12 22 25 18

Frequency density

Exercise D4.2Exa

mple

p Ursula collected data on the time taken to complete a simple jigsaw.Draw a histogram to represent these data.

Time, t seconds 40 � t � 60 60 � t � 70 70 � t � 80 80 � t � 90 90 � t � 120

Frequency 6 6 10 7 6

Time, t seconds 40 � t � 60 60 � t � 70 70 � t � 80 80 � t � 90 90 � t � 120

Class width 20 10 10 10 30

Frequency 6 6 10 7 6

Frequency density 0.3 0.6 1 0.7 0.2

● Frequency density � frequency

_________ class width

Add rows to the

table to calculate

class width

and frequency

density.

This spread will show you how to:

● Understand frequency density

● Represent grouped data on a histogram

You can represent grouped continuous data in a histogram.

● In a histogram, the area of each bar represents the frequency.

The data can be in equal or unequal sized class intervals.The vertical axis represents frequency density.

Area � frequency �

class width � bar height

So bar height � frequency

__________ class width

5040 60 70 80 90 100 110 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (seconds)

Fre

qu

ency

den

sity

Pro

ble

m 6 Times dog owners spend on daily walks.

Time, t minutes 10 � t � 20 20 � t � 40 40 � t � 60 60 � t � 90 90 � t � ?

Class width 30

Frequency 8 9

Frequency density 0.8 1.4 1.3

HistogramsKeywords

Class width

Frequency

Frequency

density

Histogram

D4.2

280280 281281

Grade A/A*

1 The histogram shows thetimes a sample of studentsspent watching TV oneevening.

a How many students spent

longer than 2 1 _ 2 hours

watching TV?b Copy and complete the

frequency table for thesedata.

c How many students werein the sample?

Time, t hours 0 � t � 0.5 0.5 � t � 1 1 � t � 2 2 � t � 2.5 2.5 � t � 3

Frequency

2 The histogram shows thedistances a sample of teacherstravel to work each day.a How many teachers travel

between 10 and 30 kilometres?b Copy and complete the

frequency table for these data.c How many teachers were in

the sample?

Distance, d km 0 � d � 5 5 � d � 10 10 � d � 20 20 � d � 30 30 � d � 50

Frequency

3 The histograms show the heights of some boys aged 11 and 16. For each histogram, draw a frequency table and calculate the

number of boys in each sample a aged 11 b aged 16.

Exercise D4.3

Frequency density �

frequency

__________ class width

Exa

mple

p The histogram shows the times a sample of students spent on the internet one evening.

a Estimate how many students spent longer than 50 minutes on the internet?b Complete the frequency table for these data.

Time, t minutes 0 � t � 20 20 � t � 30 30 � t � 35 35 � t � 45 45 � t � 60

Frequency

c How many students were included in the sample?

a The area in the histogram that represents 50 minutes is only part of the last bar. Area 50 � t � 60 � height � width of 50 � 60 class interval � 0.4 � 10 � 4 Four students spent longer than 50 minutes on the internet.b

Time, t minutes 0 � t � 20 20 � t � 30 30 � t � 35 35 � t � 45 45 � t � 60

Frequency 0.1 � 20 � 2

0.8 � 10 � 8

2.8 � 5 � 14

1.5 � 10 � 15

0.4 � 15 � 6

c 2 � 8 � 14 � 15 � 6 � 45 45 students were included in the sample.

2010 30 500 40 600

3

2.5

2

1.5

1

0.5

Time (minutes)

Fre

qu

ency

den

sity

● Frequency � frequency density � class width

You can calculate frequencies from a histogram.For each bar, the area represents the frequency.

The area of each

bar gives the

frequency.

This spread will show you how to:

● Understand frequency density

● Calculate frequencies from a histogram

Keywords

Class width

Frequency

Frequency

density

Histogram

210 30

60

50

40

30

20

10

hours

Fre

qu

ency

den

sity

130 140120110 150 160 1700

3

2.5

2

1.5

1

0.5

11-year-old heights (cm)

Fre

qu

ency

den

sity

150 160140130 170 180 1900

3

2.5

2

1.5

1

0.5

16-year-old heights (cm)

Fre

qu

ency

den

sity

2010 30 500 400

3

2.5

2

1.5

1

0.5

km

Fre

qu

ency

den

sity

Interpreting histogramsD4.3

282282 283283

Grade A*Exercise D4.4

1 The incomplete table and histogram give some information about the weights, in grams, of a sample of apples.

a Use the information in the histogram to work out the missing frequencies in the table.

b Copy and complete the histogram.

2 The incomplete table and histogram give some information about distances travelled by sales representatives on one day.

a Copy and complete the table.b Copy and complete the histogram.

120 130 140 150 160 170 180 190 200

Weight, (grams)

Weight, g grams Frequency

120 � g � 140 8

140 � g � 150 6

150 � g � 155

155 � g � 160

160 � g � 165

165 � g � 175 16

175 � g � 185 12

185 � g � 200 6

0 40 80 120 160

Distance, (miles)

200

Miles travelled, m Frequency

0 � m � 80 32

80 � m � 100 24

100 � m � 120

120 � m � 140

140 � m � 160 44

160 � m � 200 28

Exa

mple

p The incomplete table and histogram give some information about the lengths of phone calls Wendy made at work one day.

a Use the information in the histogram to complete the table.b Complete the histogram.

a The class 0 � t � 1 has frequency density � frequency

__________ class width

� 17

___ 1 � 17

On the histogram, 0 � t � 1 bar has height 3.4 cm, so vertical scale � 17 � 3.4 � 5 per cm

From the histogram the class, 1 � t � 4 has frequency � frequency density � class width � 7 � 3 � 21 calls. The class 4 � t � 6 has frequency � 4 � 2 � 8 calls. The completed table is:

b

0 4 8 12 202 6 10 14 16 8Time

Time, t minutes Frequency

0 � t � 1 17

1 � t � 4

4 � t � 6

6 � t � 10 12

10 � t � 20 10

You can use information from tables and histograms to fi ll in gapsin data.

This spread will show you how to:

● Use the information given in tables and histograms to deal with

problems such as missing data

Keywords

Class width

Frequency

Frequency

density

Histogram

Use the

information in

the table to work

out the scale on

the vertical axis.

Calculate

the frequency

densities:

6 � t � 10:

fd � 12/4 � 3

10 � t � 20:

fd � 10/10 � 1

Remember to

scale and label

the vertical axis.

0 4 8 12 202 6 10 14 16 8Time

0

20

10

15

5

Fre

qu

ency

den

sity

Time, t min Frequency Class width

Frequency density

0 � t � 1 17 1 17

1 � t � 4 21 3 7

4 � t � 6 8 2 4

6 � t � 10 12 4 3

10 � t � 20 10 10 1

AO

3P

roble

m 3 Copy the tables and work out the height, in terms of f, for the second bar in each of these histograms.a

b

Time, t minutes Frequency Bar height Frequency density

0 � t � 10 8 3.2 cm 8 � 10 � 0.8

10 � t � 40 f

Time, t minutes Frequency Bar height Frequency density

0 � t � 25 10 2 cm

25 � t � 30 f

Further histogramsD4.4

284284 285285

Grade A*Exercise D4.5

1 Compare the heights of two samples of boys of different ages summarised in these histograms.

2 Compare the weights of samples of apples and pears summarised in these histograms.

3 Compare the reaction times of girls and boys summarised in these histograms.

120 130110 140 160150 1700

3

2.5

2

1.5

1

0.5

Height (cm)

Fre

qu

ency

den

sity

Aged 11

140 150130 160 180170 1900

3

2.5

2

1.5

1

0.5

Height (cm)

Fre

qu

ency

den

sity

Aged 16

120 140 160 180130 150 170 190 200 Weight (grams)

0

2.5

1

1.5

2

0.5

Fre

qu

ency

den

sity

Apples

70 90 110 13080 100 120 140Weight (grams)

0

2.5

1

1.5

2

0.5

Fre

qu

ency

den

sity

Pears

2 3 4 5 6 7 8 9 100

12

10

8

6

4

2Fre

qu

ency

den

sity

Time (seconds)

Girls

3 42 5 76 80

14

12

10

8

6

4

2

Time (seconds)

Fre

qu

ency

den

sity

Boys

The range is

an estimate, as

you do not have

the actual data

values.

This is the

class with the

highest frequency

density. It may

not be the class

with the highest

frequency.

Exa

mple

p The histograms show the times taken by a sample of boys and a sample of girls to complete the same puzzle.Compare the times taken by the boys and the girls.

The range of times is the same for both boys and girls: 120 � 40 � 80.

The modal class for boys (70–80 seconds) is a slower time than the modal class for girls (60–70 seconds), so the boys were generally slower.

The girls’ times are more positively skewed than the boys’ times indicating that girls’ times were shorter, that is, the girls were quicker.

5040 60 70 80 90 1001101200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (seconds)

Fre

qu

ency

den

sity

Boys

5040 60 70 80 90 1001101200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (seconds)

Fre

qu

ency

den

sity

Girls

You can use histograms to compare data sets.● The highest bar on a histogram represents the modal class.● You can estimate the range.

The shape of a histogram shows whether the data is skewed.

● To compare histograms for two or more data sets, consider the modal class, range and skewness.

This spread will show you how to:

● Use histograms to compare two or more data sets, considering the

modal class, range and skewness

Keywords

Frequency

density

Histogram

Modal class

Skewness

Positive skewHeights of jockeys

No skew – symmetricalHeights of random sample of men

Negative skewHeights of basketball players

p.152

Using histograms to compare data setsD4.5

286286 287287

Grade C/B

AO

2Fu

nct

ional

Math

s 3 The table shows the number of hot water bottles sold per month in a chemist’s shop and the average temperature for each month.

Month Jan Feb Mar April May June July Aug Sept Oct Nov Dec

Average monthly temperature c

2 4 7 10 14 19 21 20 18 15 11 5

Sales of hot water bottles

32 28 10 4 6 0 2 3 7 15 22 29

a Draw a scatter graph for the data.b Describe the correlation shown and the relationship between the two sets of

results.c Draw in a line of best fit.d Use the line to predict the average temperature in a month when the shop sold

20 hot water bottles.e The weather forecast predicts an average January temperature of �5°C. Could

you use your graph to find how many hot water bottles would be sold?

1 Louise recorded the average number of minutes per day spent playing computer games and the reaction times of nine students.

Minutes per day spent playing computer games

40 60 75 40 35 20 80 50 45

Reaction time, seconds 5.2 4.3 3.9 5.5 6.0 7.2 3.6 4.8 5.0

a Draw a scatter diagram of these data.b Describe the correlation shown.c Describe the relationship between the average number of minutes

per day spent playing computer games and reaction time.

2 Bob caught nine fish during one angling session. He recorded the weights and lengths of the fish he caught in this table.

Weight, g 500 560 750 625 610 680 600 650 580

Length, cm 30 32 50 44 39 48 40 45 36

a Draw a scatter diagram of these data.b Describe the correlation shown.c Describe the relationship between the weights and lengths of

the fish.

Exercise D4.6

There could

be a non-linear

relationship, for

example y � x2,

or y � 2x.

Read up from

the horizontal

axis to the line

of best fit, then

across to the

vertical axis.

A scatter graph shows the correlation between two sets of data.

A line of best fit – Passes through the point (mean of set A, mean of set B). – Has a similar number of points above and below the line.● The stronger the correlation, the closer the points lie to the

line of best fi t.● Positive or negative correlation implies variable A increases or

decreases, respectively, as variable B increases.● Zero correlation implies no linear relationship.

You can use a line of best fi t to predict data values within the range of data collected.

This spread will show you how to:

● Draw and use scatter graphs

● Draw lines of best fit, understanding positive, negative and zero

correlation

Set

B

Set A

Exa

mple

p The table gives the marks earned in two exams by 10 students.

Maths % 70 76 61 70 89 65 59 58 73 82

Statistics % 78 82 74 75 93 70 66 62 77 89

a Draw a scatter graph for the data.b Describe the correlation and the relationship shown.c Draw a line of best fit.d Predict the Statistics mark for a student who scored 62% in Maths.e Could you use your graph to predict the Maths mark for a student

who scored 32% in Statistics? Give your reasons.

a, c See graph.b Positive correlation. Students

who scored higher in Mathsalso tended to score higher inStatistics.

c Maths mean: 703 � 10 � 70.3 Statistics mean: 766 � 10 � 76.6d Predicted Statistics mark � 70%e No. This Statistics mark is outside

the range of the data.

Exam results

Maths %

Sta

tist

ics

%

6050 70 80 90 10050

60

70

80

90

100

Positive correlation Negative correlationZero correlation

Scatter graphsKeywords

Correlation

Line of best fit

Relationship

AO

3P

roble

m 4 For the data in question 3 find the equation of the line of best fit.

How should you interpret this equation?p.74

D4.6

288288 289289

Grade CExercise D4.7Exa

mple

p Jenny’s quarterly gas bills over a period of two years are shown in the table.

Jan–March April–June July–Sept Oct–Dec

2003 £65 £38 £24 £60

2004 £68 £42 £30 £68

Plot the data on a graph and comment on any pattern in the data.

Draw axes on graph paper with time on the horizontal axis.Plot the coordinates as crosses on the grid.Join them up with straight lines.

J–M A–JA–J J–S O–D J–M A–J J–S O–D20

30

40

50

60

70

2003 2004

Pou

nds

spen

t (£

)

Gas bills are highest in the Winter months and lowest in the Summer months. This annual pattern appears to repeat itself.

There is a slight trend for the bills to rise from year-to-year.

● Plotting the data on a line graph makes it easier to see any patterns.

Some types of data are collected over an extended period of time.For example, Electricity and gas bills are produced every quarter (three monts). Mobile phone bills are generated each month. Unemployment reates arre published each month.

This spread will show you how to:

● Draw and interpret line graphs

Keywords

Line graph

Seasonal

variation

Trend

Data often shows a short term, seasonal variation For example, icecream sales are higher in Summer than in Winter .and a longer term trend. For example, inflation measures the annual increase or decrease in prices.

Plot time on

the horizontal

axis, J–M means

Jan–March.

For each of questions 1–6a Plot the data on a graphb Comment on any patterns in the data.

1 The table shows Ken’s monthly mobile phone bills.

Jan Feb Mar April May June July Aug Sept Oct Nov Dec

£16 £12 £15 £18 £16 £18 £12 £10 £12 £15 £16 £20

2 The table shows Mary’s quarterly electricity bills over a two-year period.

Jan–March April–June July–Sept Oct–Dec

2004 £45 £20 £15 £48

2005 £54 £24 £18 £50

3 The table shows monthly ice-cream sales at Angelo’s shop during one year.

Jan Feb Mar April May June July Aug Sept Oct Nov Dec

£16 £12 £15 £18 £38 £48 £52 £58 £18 £15 £16 £40

4 A town council carried out a survey over a number of years to find the percentage of local teenagers who used the town’s library. The table shows the results.

year 1998 1999 2000 2001 2002 2003 2004 2005

% 14 18 24 28 25 20 18 22

5 Christabel kept a record of how much money she had earned from babysitting during three years.

Jan–April May–August Sept–Dec

2001 £12 £18 £30

2002 £21 £33 £60

2003 £39 £42 £72

6 Steve kept a record of his quarterly expenses over a period of two years.

Jan–March April–June July–Sept Oct–Dec

2003 £35 £56 £27 £12

2004 £39 £68 £29 £18

Line graphsD4.7

290

Summary

Check out

291

(Edexcel Limited 2004)

Worked exam questionThe histogram gives information about the times, in minutes, 135 students spent on the Internet last night.

Use the histogram to complete the table. (2)

You should now be able to:

● Use frequency polygons to compare two sets of data

● Draw and interpret histograms using frequency density

● Use histograms to compare two or more sets of data

● Draw and use scatter graphs

● Recognise correlation and draw and use lines of best fi t

● Look at data to fi nd patterns and exceptions

● Interpret line graphs

Exam questionD4

Time (t minutes) Frequency

0 � t � 10 20

10 � t � 15 18

15 � t � 30 45

30 � t � 50 52

TOTAL 135

10 � 2 � 20 5 � 3.6 � 18 15 � 3 � 45 20 � 2.6 � 52 � 135

Show these calculations.

00

1

2

3

4

10 20 30 40 50Time (t minutes)

Fre

qu

ency

den

sity

(st

uden

ts p

er m

inu

te)

Time (t minutes) Frequency

0 � t � 10

10 � t � 15

15 � t � 30

30 � t � 50

TOTAL 135

AO

2 1

0

10

0 10 20 30 40 50 60

20

30

40

50

60

Cups of coffee

Fre

qu

ency

A café collected information about the number of cups of coffee it sold each day. The frequency polygon shows the information about these sales. Each class interval is the width of 10 cups of coffee. a Write down the modal class interval. (1)

The café also collected information about the number of cups of tea it sold each day.

The frequency table gives the information about these sales.

Number of cups of tea sold each day Frequency

0 – 10 20

11 – 20 46

21 – 30 50

31 – 40 38

41 – 50 31

51 – 60 18

b Using the same scale as in the grid above, draw a frequencypolygon to show this information. (2)

Kath says “The range of sales of coffee is the same as the range of sales of tea.”

c Explain why this may not be true. (1)