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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)