Numeracy
Outcomes 1/2
[HIGHER]
Introduction
Section 1: Basic Excel Graphs
Tutor Assignment
Section 2: Graphs from Social Trends
Section 3: Graphs related to Geography
Tutor Assignments
Answers to SAQs
3
5
16
17
27
40
45
INTRODUCTION
The purpose of this pack is to make you familiar with a wide variety of graphs, charts,
tables and other diagrams which either illustrate or gather together data for the purpose of
analysis.
The following is an extract from the National Unit Specification statement of standards.
Don’t be alarmed by its formal vocabulary and style – this pack contains all you need.
Outcome 1
Analyse and interpret graphical information.
Performance criteria
(a)
Extract information accurately from complex diagrams.
(b)
Give full and correct interpretations of significant features.
Outcome 2
Select and use appropriate graphical forms to communicate information.
Performance criteria
(a)
Select an appropriate form.
(b)
Use the selected form of communication to present information clearly.
At this level it is not very constructive to distinguish between the interpretation of
graphs (Outcome 1) and drawing graphs (Outcome 2). Hence the two Outcomes have
been combined into one teaching pack.
The first section (graphs using Excel) has no SAQs. But don’t worry, the next two
sections have plenty! You can practise drawing graphs by hand on graph paper, or
using Excel, or using any other package you may have on your PC at home or at work.
All are acceptable. Your tutor may give you more guidance about this. If you don’t
have access to Excel, read this section anyway.
Note that the answers to the SAQs often contain a lot more than just the bare answer.
There are frequently useful hints and points to note in there, so make sure you always
check the answer to every question (after your own attempt, of course) even if you
know your own answer is correct.
If you find you have particular difficulties with a certain type of graph, contact your
tutor. I have deliberately not included any basic graph-drawing techniques here. If I
did, this unit would run to a couple of hundred pages.
OUTCOME 1/2: NUMERACY/HIGHER
3
INTRODUCTION
Your tutor will have access to this material from a similar Intermediate 2 package and
will be able to provide you with the necessary help.
Resources
You will need a scientific calculator, a protractor and a set of compasses. Access to
Excel or other graph-drawing software packages is not essential but may add to your
interest.
Assessment
Your tutor/college/training centre will keep you right about how and when to sit your
formal assessment to achieve this Outcome.
Good luck with the Outcomes.
4
OUTCOME 1/2: NUMERACY/HIGHER
BASIC EXCEL GRAPHS
SECTION 1
Since you are allowed to use computer software to draw graphs and charts, let’s have a
look first at what the most widely used program has on offer – Microsoft Excel.
Example 1a: Simple bar chart
Company Sales
00
99
20
98
19
97
19
96
19
95
19
94
19
93
19
92
19
91
19
90
8
7
6
5
4
3
2
1
0
19
Sales Value (£m)
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Total Sales
(£m)
5.3
4.9
4.8
5.1
5.3
5.7
6.2
6.9
7.0
7.2
7.1
19
Year
Year
Note the following features:
•
•
•
•
both axes are labelled and the chart has a title
the scale is clearly shown – numbers represent millions of £
the numerical axis correctly starts at 0 – a requirement for ALL bar charts
the horizontal axis shows discrete (i.e. separate) years and each bar refers to the total
sales for that year.
Because the horizontal axis refers to time, some people would prefer to display the
information as a simple line graph.
Example 1b: Simple line graph
Sales Value (£m)
Company Sales
8
6
4
2
0
1985
1990
1995
2000
2005
Year
This is based on the same information as Example 1a.
OUTCOME 1/2: NUMERACY/HIGHER
5
BASIC EXCEL GRAPHS
Note the following features:
• both axes are labelled and the graph has a title (I won’t repeat this again, but these are
generally requirements for every type of diagram)
• the dots, which are joined by lines, line up with the years.
The software automatically starts the vertical axis at 0, but this is not a requirement of a line
graph. The axis can start at any convenient place, so long as it does not distort the data it is
trying to portray.
So we can opt to change the origin from 0 to, say, 4 and get the following result, which
displays the data much more clearly. Differences in values become much more obvious.
Company Sales
Sales Value (£m)
7.5
7
6.5
6
5.5
5
4.5
4
1988
1990
1992
1994
1996
1998
2000
Year
It is argued that a line graph actually shows the data ‘better’ (whatever that means) than a
bar chart but you have to be careful about what you can do with the graph.
Normally a line graph should be used when both the variables (in this case the years and
the sales values) are continuously changing, and then you can interpolate (i.e. read between
the dots). Well you can’t do that here. The dot half way between 1992 and 1994 is clearly
1993, and certainly we had sales in 1993. But you can’t split the line between 1992 and
1993. It doesn’t split into quarters of the year, or into months, because the sales values are
end-of-year totals. The information is not broken down any more than that.
But I digress. Let’s go back to bar charts.
Suppose we want to compare more than one set of similar data. An ideal vehicle for this is
the compound bar chart.
6
OUTCOME 1/2: NUMERACY/HIGHER
BASIC EXCEL GRAPHS
Example 1c: Compound (or multiple) bar chart
Year
Total Sales
Company A
(£m)
5.3
4.9
4.8
5.1
5.3
5.7
6.2
6.9
7
7.2
7.1
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Total Sales
Company B
(£m)
2.5
2.9
3.1
3.1
3.6
3.8
4.2
4.5
4.4
4.1
3.9
Total Sales
Company C
(£m)
4.3
5.1
5.2
5.6
5.9
6.2
6.5
6.9
7.1
7.4
7.7
00
20
98
97
99
19
19
19
96
19
94
93
95
19
19
19
91
92
19
19
90
9
8
7
6
5
4
3
2
1
0
19
Sales value (£m)
Three-company comparison
Year
Total Sales Company A (£m)
Total Sales Company C (£m)
Total Sales Company B (£m)
Features of this, in addition to the normal features of any simple bar chart, are:
• the bars representing the sales of the three companies within any one year are close
together, usually with no spaces between them at all, but
• there is a space between each year and the next, otherwise we would be totally
confused as to what referred to what
• there must now be a key or legend attached so that we know which bar relates to which
company
A catchy-looking variation is a three-dimensional version as follows:
OUTCOME 1/2: NUMERACY/HIGHER
7
BASIC EXCEL GRAPHS
8
Total Sales Company A (£m)
Total Sales Company C (£m)
2000
1998
1996
1994
1992
1990
7
6
5
4
3
2
1
0
Total Sales Company B (£m)
However, you see that company B has disappeared somewhat, and to make everything
more visible it is sometimes necessary to change the order of the spreadsheet columns to
achieve the desired result:
8
6
4
2
Total Sales Company B (£m)
2000
1998
1996
1994
1992
1990
0
Total Sales Company A (£m)
Total Sales Company C (£m)
Multiple line graphs can also be drawn.
Example 1d: Multiple line graph
9
8
7
6
5
4
3
2
1
0
1988
1990
1992
1994
1996
1998
Total Sales Division A (£m)
Total Sales Division B (£m)
Total Sales Division C (£m)
8
OUTCOME 1/2: NUMERACY/HIGHER
2000
2002
BASIC EXCEL GRAPHS
Note that you must now have different styles of line – firm, dots, dashes, alternate dashes
and dots, or whatever. Or the points must be styled differently – dots, diamonds, triangles,
etc. The larger the number of data sources the more complicated the diagram, until
eventually one diagram no longer does a good job.
We can, of course, jazz it up with a 3D version (really only effective in colour):
8
Total Sales Division C (£m)
Total Sales Division A (£m)
6
4
Total Sales Division B (£m)
2
2000
1998
1996
1994
1992
1990
0
Or, we can draw an area graph.
Example 1e: Area graph
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
18
16
14
12
10
8
6
4
2
0
Total Sales Division A (£m)
Total Sales Division C (£m)
Total Sales Division B (£m)
This still shows the same information as the multiple line graph, but the shading shows the
sales values rather than the line.
The very dark shading shows the sales of Division A, equivalent in this case to the lowest of
the three lines on this graph.
The middle shading represents the sales of Division B, so the middle line on this graph
shows the cumulative sales of A and B together.
OUTCOME 1/2: NUMERACY/HIGHER
9
BASIC EXCEL GRAPHS
Similarly, the lightest shading represents the sales of Division C, and the top line shows the
cumulative sales of the three divisions all together.
Thus the line graph is showing each division's contribution to the total.
To my mind ithis is not really very satisfactory. To show each component's contribution to
a whole (assuming that you know what the whole is!), it’s better to use a component bar
chart.
Example 1f: Component bar chart
Activity
Walking
Climbing
Potholing
Sailing
Snorkelling
TOTALS
1990
125
24
17
45
0
211
1995
100
58
22
40
30
250
2000
75
100
20
40
90
325
350
300
123456
123456
123456
250
200
150
1234567
1234567
1234567
1234567
1234567
1234567
123456
123456
123456
123456
123456
123456
123456
123456
123456
123
123
123 Snorkelling
1234567
1234567
1234567
Sailing
123
123
123
Potholing
Climbing
Walking
100
50
0
1990
1995
2000
There are a few things to watch out for here:
• you must know what the ‘whole’ (or total) is so that you know how high to draw the
vertical axis
• each horizontal line across the bar is the cumulative total of all the components below it
• the key might be the same way round as the components are in the original table, or
they may be reversed – here the snorkelling part is at the top of the bar (and also in the
key) but at the bottom of the table.
It could be, of course, that you want to know what proportion of the whole is attributable
to each component. A good vehicle for this is the percentage component bar chart.
10
OUTCOME 1/2: NUMERACY/HIGHER
BASIC EXCEL GRAPHS
Example 1g: Percentage component bar chart
100%
90%
80%
70%
Snorkeling
Sailing
Potholing
Climbing
Walking
60%
50%
40%
30%
20%
10%
0%
1990
1995
2000
You really must know the whole here, so that you can calculate percentages.
(Reminder: in 1990, 125 people out of a total of 211 went walking, so the percentage
walking is calculated as:
125
× 100% = 59.2%
211
but if you are drawing this by hand you would round it off to the nearest whole number
(59%) as the graph probably won't be accurate enough to show decimal places.)
Note that all the columns are the same height (100%). You use this kind of graph when you
are not specially interested in the actual amount, but more in how various components
contribute to the total.
I have shown you a three-dimensional version, but two-dimensional ones are just as good.
Proportions of the whole are also usefully displayed by a pie chart.
Example 1h: Pie chart
0%
21%
Walking
8%
60%
Climbing
Potholing
Sailing
11%
OUTCOME 1/2: NUMERACY/HIGHER
11
BASIC EXCEL GRAPHS
The one here shows the information of the previous example for 1990. The angle at the
centre of each sector is the same proportion of 360° as the quantity is of the total.
(Reminder: walking is represented as 125 out of 211 altogether, so the angle is calculated by
125
× 360° = 213.3°
211
which would be rounded off to 213°.
Points to note:
• a legend is necessary as before
• a pie chart is not so good at illustrating situations where one of the components is 0, e.g.
snorkelling
• showing percentages on the chart is optional
• accurate figures can’t be easily derived from a pie chart – few of us carry protractors
about in our pockets or handbags!
• a pie chart is not so useful if there are too many components, or if all the components are
so similar to each other that the eye can’t see the different sector sizes.
Example 1i: Scatter diagram or scattergraph
A scatter diagram is used when you have two variables, various values of which are paired
off, and you are trying to see if there is a relationship between them.
Company Product Lines Value (£m)
A
5
40
B
18
127
C
7
64
D
24
200
E
10
35
F
10
70
G
20
22
Analysis of product line values
250
y = 5.5527x + 5.1494
Value(£m)
200
150
100
50
0
0
12
5
10
15
20
No. of Product Lines
OUTCOME 1/2: NUMERACY/HIGHER
25
30
BASIC EXCEL GRAPHS
Points to note:
• the axes may or may not start at 0, just wherever is convenient
• each point is plotted just as you would plot co-ordinates
• the diagram should have a title
The line drawn on the diagram is an optional extra. It is called the line of regression of y
on x, where y is the variable plotted vertically (value) and x is the variable plotted
horizontally (number of product lines). It allows you to estimate, within limits, a value of y
for a given x. So I could estimate that, given a company with 17 product lines, the total
value would be round about £100 million. (Go along to 17, up to the line, then across to
100.) The equation is the mathematical formula which allows you to calculate the value as
£99.5 million (5.5527 × 17 + 5.1494 = 99.5453).
For more information about this, look at Outcome 3 pack, Section 4 Correlation and
Regression.
Excel even allows you to draw a so-called bubble diagram.
Example 1j: Bubble diagram
Company
A
B
C
D
E
F
G
Lines
5
18
7
24
10
10
20
Value (£m)
40
127
64
200
35
70
22
558
Share (%)
7.2
22.8
11.5
35.8
6.3
12.5
3.9
This is a way of showing three variables on the same diagram.
250
35.8
Value (£m)
200
150
22.8
100
12.5
11.5
50
-50
6.3
7.2
0
0
5
10
3.9
15
20
25
30
No. of Product Lines
OUTCOME 1/2: NUMERACY/HIGHER
13
BASIC EXCEL GRAPHS
Each dot of the previous scatter diagram is expanded to form a circle. The areas of these
circles are in proportion to the values of the third variable.
Excel also gives the opportunity of mixing together two different graphs on different scales.
Example 1k
10
3450
9.5
3400
9
3350
8.5
3300
8
3250
7.5
3200
7
Number of General
Practitioners
Infant Mortality Rate (per 1000)
Infant Mortality Rate vs No. of GP's
3150
1985
1986
1987
1988
1989
1990
Year
Infant mortality rate
no of GP's
Here we have a line graph combined with a bar chart. The line graph shows the number of
general practitioners (GPs) in Scotland and refers to the right-hand vertical axis. The
numbers are actual numbers of people. The bar chart shows the infant mortality rate (per
thousand births) and refers to the left-hand vertical axis. Note that neither axis starts at 0
here; we’re breaking the rules for bar charts (must start at 0) but it would be difficult to
show these two sets of figures meaningfully otherwise.
14
OUTCOME 1/2: NUMERACY/HIGHER
BASIC EXCEL GRAPHS
You could show the same data on two line graphs like this:
10
3450
9.5
3400
9
3350
8.5
3300
8
3250
7.5
3200
7
1985
1986
1987
1988
1989
No. of General Practitioners
Infant Mortality Rate (per 1,000)
Infant Mortallity Rate vs No. of GP's
3150
1990
Year
Infant mortality rate
no of GP's
Note that there is no particular significance attached here to the place where the two lines
meet. If one of the scales had been a bit different the lines would have crossed somewhere
else. The data comes from the Scottish Abstract of Statistics 1991, published by the
Government Statistical Service.
*
As you see, there is a large choice of diagram available on Excel. More complex, specialised
software can produce even more specialised diagrams, but you don’t normally meet them in
everyday life.
OUTCOME 1/2: NUMERACY/HIGHER
15
TUTOR ASSIGNMENT
T1
For each question decide which of the types of graph from Section 1 is the most
appropriate and draw it. If you are using squared paper, then obviously two-dimensional
illustrations will be all you can manage; if you and your tutor have decided that you can do
the graphs on a computer, perhaps you can be a bit more adventurous!
Where you think there is more than one way to illustrate the data, or where the same
method seems to lend itself to two questions, draw one type of graph for one question, and
another for another question, to give yourself practice.
1.
The following information comes from Social Trends Volume 30, published by the
Office for National Statistics. It shows the percentage of the male population which
belongs to certain age groups in three selected years.
1901
1961
1998
2.
Under 16
33.6
24.8
21.3
55-64
5.8
11.2
9.9
0
15
5
20
10
24.4
15
28.4
20
32
25
35.2
30
38.2
Mon
23
17
Tues
25
22
Wed
28
23
Thurs
29
24
Fri
30
24
65-74
3.1
6.3
7.9
35
41
Sat
30
20
1940
5.3
7.2
1950
5.7
6.4
1960
6.4
5.3
1970
7.3
4.4
1980
8.2
3.3
A garage sells the following cars over a period of time:
Make
Number
16
45-54
8.8
13.9
13.3
75+
1.2
3.1
5.3
40
43.3
Sun
28
19
The table shows how many million hectares of land in Outer Monrovia are under
cultivation and how many are forested.
Year
Farmed
Forested
5.
35-44
12.2
13.9
14.7
The maximum and minimum daily temperature was noted over a period of a week
Day
Maximum
Minimum
4.
25-34
15.7
13.1
16.2
In an experiment, water at an initial temperature of 15°C is steadily heated and its
temperature noted every 5 minutes. The results are as shown.
Time from Start
Temperature
3.
16-24
19.7
13.9
11.4
Ford
15
Fiat
12
Rover
25
OUTCOME 1/2: NUMERACY/HIGHER
Nissan
18
Others
16
1990
8.7
2.2
2000
9.1
1.5
GRAPHS FROM SOCIAL TRENDS
SECTION 2
We will now look more closely at two sources of graphs.
First of all, we’ll look at some graphs from the publication produced by the Office for
National Statistics called Social Trends. The graphs will be familiar, the contexts may not
be. This doesn’t stop you using your common sense to interpret them, though.
The emphasis in the questions will be on interpretation.
One of the main ground rules is to check the small print. Somewhere, usually at or near the
top, will be a statement of the units being used – are they percentages, thousands of people,
millions of pounds? This should be one of the first things you look for.
But remember that you cannot read these graphs to any high degree of accuracy. Do the
best you can, perform any calculations, but round off your answer to something sensible.
Don't expect people to be impressed by lots of decimal places. If you feel that the nearest
million is all the graph can show, so be it!
2.1 Bar Charts
? 2.1A
The type of back-to-back comparative bar chart shown here is very popular, and is often
used instead of the compound or multiple bar chart.
Population of working age:1 by gender and social class, Spring 1999
1
Males aged 16 to 64, females aged 16 to 59.
Includes members of the armed forces, those who did not state their current or
last occupation, and those who had not worked in the last eight years.
Source: Labour Force Survey, Office for National Statistics
2
OUTCOME 1/2: NUMERACY/HIGHER
17
GRAPHS FROM SOCIAL TRENDS
The very top bar on the left shows that, of all males of working age, about 8% of them were
of professional class in 1999.
1.
For which social classes was the proportion of the working population roughly the
same for males and females?
In which classes was the difference most marked? Why do you think this is?
2.
? 2.1B
Look at the graph and the small print (the note at the foot of the graph). The numbers
relate to indices. Comparison is being made with the UK so the UK is given the number
100, and all others are referenced to that.
Relative price levels:1 EU comparison, 1997
1
Price level indices for private consumption – the ratio of purchasing power parities to the official
exchange rates.
Source: Eurostat
Sweden’s index is about 121. This means that price levels are, on average, 21% higher in
Sweden than in the UK.
1.
2.
18
How do price levels compare in (a) Finland? (b) Italy?
What difference would it have made to the chart if the countries had been placed in
alphabetical order?
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS FROM SOCIAL TRENDS
? 2.1C
Sight tests: by age and provider, 1997–98
Source: Department of Health
The figures here show rates per 10,000 people. So if you look at the 45–54 age group, out
of every 10,000, about 1,000 get a sight test from the NHS and 2,000 people get theirs
privately. That’s 10% and 20% respectively. Presumably the other 70% of that age group
didn’t get a sight test in 1997–98.
1.
2.
3.
4.
What percentage of people aged 75 and over got sight tests altogether in1997–98?
The bar for the under 16s appears longer than the combined two bars for 25–34.
Does this mean that lots more under 16s had to have sight tests? And why are they
all NHS?
The total number of people in the 55–64 age group was 19.8 million. How many
sight tests were made by the different providers for that age group?
Comment on the general pattern, and try to offer an explanation.
OUTCOME 1/2: NUMERACY/HIGHER
19
GRAPHS FROM SOCIAL TRENDS
? 2.1D
Here is a slightly confusing bar chart.
Percentage of women childless at age 25, 35 and 45: 1 by year of birth
1
Data for women aged 35 born in 1973, and those aged 45 born since 1963, are projections. All
other percentages are based on actual data up to the end of 1998.
Source: Office for National Statistics
At first glance that it might seem that, as far as 1923 was concerned, 45% of 25-year-old
women were still childless, 20% of 35-year-old women were childless and so on. But it
doesn’t mean that.
1.
2.
3.
20
What do the figures for 1923 actually mean?
What percentage of women born in 1943 appear never to have had children at all?
Discuss the general trend of the chart.
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS FROM SOCIAL TRENDS
? 2.1E
The table below shows the percentage of adults with a full driving licence, by gender and
age, in 1976 and also in 1998.
Age
17-20
21-29
30-39
40-49
50-59
60-69
70 & over
Males
1976
35
77
85
76
75
58
33
1998
47
78
88
89
84
82
64
Females
1976
1998
18
37
43
65
48
75
36
72
25
60
15
48
5
19
Illustrate this information in a similar way to the bar charts in this section.
2.2 Line Graphs
? 2.2A
1
1998-based
Data for 1901 to 1921 exclude the Irish Republic, which was constitutionally a part
of the United Kingdom during this period.
Source: Office for National Statistics; Government Actuary ’s Department; General
Register Office for Scotland; Northern Ireland Statistics and Research Agency.
2
1.
2.
3.
4.
The birth line shows three clear peaks. What do you think caused these?
If we define population increase as ‘births minus deaths’
(a) in which year was there no increase?
(b) what is the general trend over the past century?
Why is that a poor way to define population increase?
How many people were born in 1901?
OUTCOME 1/2: NUMERACY/HIGHER
21
GRAPHS FROM SOCIAL TRENDS
? 2.2B
1
Full-time pupils only.
Source: Department for Education and Employment
1.
2.
3.
What is the most obvious thing that strikes you about the two graphs? What is the
reason for this?
Why is there a sudden jump in the numbers of secondary school pupils about 1972?
The two graphs are closer together in 1998/99 than they were in 1946/47. Why do
you think this is?
? 2.2C
Incidence of malignant melanoma of the skin: by gender
1
Directly age-standardised using the European standard population.
Source: Office for National Statistics
Health statistics are usually given in the form of a rate, and a rate ‘per 100,000’ is probably
the most commonly used one.
1.
2.
22
In 1991 there were 28.2 million men in this country and 29.6 million women. How
many actual incidences of melanoma were there altogether?
Both graphs show similarities up to about 1984, then both graphs start doing
something wildly different. Can you suggest reasons for this?
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS FROM SOCIAL TRENDS
? 2.2D
Source: Department of the Environment, Transport and the Regions
1.
2.
3.
Does this show three line graphs drawn on the same diagram, or an area chart?
How do you know?
You may notice that, from about 1985 or so, the middle line is horizontal and the
other two are parallel. Why do you think they are connected like this? (This is
actually quite a cunning question!)
What is the general trend in car ownership?
? 2.2E
Volume of retail sales
Source: Office for National Statistics
OUTCOME 1/2: NUMERACY/HIGHER
23
GRAPHS FROM SOCIAL TRENDS
This is a beautiful example of a cyclical time series.
(Time series means data taken at regular intervals over a period of time; cyclical means
recurring in cycles. It is a feature of these graphs that certain conditions repeat themselves
every so often, leading to a graph which also tends to be cyclical.)
1.
2.
What is the general trend in retail sales?
When, during a cycle, does each violent peak occur? Why? What follows a peak?
Why?
? 2.2F
You may have to refer to Outcome 3 pack, Section 2 Median and Quartiles before you can
answer this question.
Distribution of real1 household disposable income2
1
Before housing costs adjusted to April 1999 prices using the retail prices index less
local taxes.
2
Equivalised disposable household income has been used for ranking the individuals.
3
Data from 1993 onwards are for financial years; data for 1994-95 onwards exclude
Northern Ireland.
Source: Institute for Fiscal Studies
The 10th percentile is the highest income of the poorest 10% of the population.
1.
2.
3.
24
What does the 10th percentile line show?
Make some comments about the 90th percentile line.
Make some comments for the median line.
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS FROM SOCIAL TRENDS
? 2.2G
Labour disputes: working days lost
Source: Office for National Statistics
1.
2.
What does the enormous peak show?
What effect has this had on the rest of the line graph, with reference to the vertical
scale?
? 2.2H
Workforce: by selected industry
1
There was no Census in 1941.
Source: Census, Office for National Statistics
1.
2.
3.
What is the general trend shown here?
What would your projections be for the year 2001?
Can you give two different reasons for the dotted lines in the middle of the diagram?
OUTCOME 1/2: NUMERACY/HIGHER
25
GRAPHS FROM SOCIAL TRENDS
? 2.2J
Illustrate the following information with a suitable diagram. It shows the death rates per
1,000 live births for children under one year old. The source is the Office for National
Statistics and the data was published in the year 2000.
Year
Males
Females
26
1961
26.3
18.2
1971
20.2
15.5
1981
12.7
9.5
OUTCOME 1/2: NUMERACY/HIGHER
1991
8.3
6.3
1998
6.4
5.2
2011
3.7
3.1
2021
3.1
2.4
GRAPHS RELATED TO GEOGRAPHY
SECTION 3
Various disciplines or areas of study have their own specialist, sometimes even obscure,
types of diagram.
As an example, I have chosen geography, mainly because many graphs of geographical
information need only the application of common sense to extract information (though
interpretation of this information may require knowledge of the subject).
These diagrams have all been taken from Higher Still materials.
3.1 Various Types of Bar Chart
? 3.1A
Age pyramid for a developed
country (Japan)
Male
Female
Percentage
Age pyramid for a developing
country (Bangladesh)
Male
Female
Percentage
Back-to-back bar charts like this are frequently used to compare the age distribution of a
population, either of the same country but at different periods of its history, or of two
countries. They are called population pyramids. In this case they are grouped in intervals
of 5 years, but other intervals are possible.
1.
2.
What is the most obvious difference between the two pyramids?
What do you think each one will look like in 30 or 40 years’ time?
OUTCOME 1/2: NUMERACY/HIGHER
27
GRAPHS RELATED TO GEOGRAPHY
? 3.1B
Here is another back-to-back bar chart.
Annual rainfall, 1940–90 (percentage above/below average)
An interesting question poses itself: what is meant by ‘average’? In this case it will be the
true mean, i.e. the total amount of rainfall over a certain number of years divided by the
number of years, but which years? Is it the same years all the time? Or do they update it
every so often?
Describe in general terms what happened
1.
between 1940 and 1958,
2.
between 1958 and 1969,
3.
between 1970 and 1990.
28
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
? 3.1C
This is quite a popular type of three-dimensional bar chart, requiring specialised software to
draw it, as it is impossible to draw easily by hand.
How do the population growths of the two countries compare?
? 3.1D
Farm land use
This is a variation on a component bar chart, an alternative to a pie chart. It can be used
only if you know the whole. In this case we are talking about agricultural land use and there
are no components other than the four mentioned.
To what use is one quarter of the land put?
OUTCOME 1/2: NUMERACY/HIGHER
29
GRAPHS RELATED TO GEOGRAPHY
3.2 Mixed Graphs
? 3.2A
Climate graph for Ibiza
Climate graph for Kano, Nigeria
Here we have a bar chart and a line graph superimposed on each other. In each one
• the bar chart refers to the rainfall and uses the right-hand vertical scale
• the line graph refers to the temperature and uses the left-hand vertical scale.
You could have two line graphs instead, but using two different types makes things clearer.
1.
2.
3.
4.
30
What obvious feature do you notice right away about the Ibiza temperature and
rainfall?
Does the same thing happen at Kano?
What obvious contrast is there between Ibiza’s rainfall and Kano’s?
Roughly what is Kano’s total rainfall for a year?
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
? 3.2B
Hotel demand in the Algarve, Portugal
Too many different lines can confuse the issue, especially if the lines are all in black and
white, even if they are drawn in different styles.
1.
2.
3.
Which two locations show steady demand throughout the year?
Which two locations vary substantially throughout the year?
Why does the percentage axis only go up to 25 or so and not all the way to 100?
? 3.2C
Life expectancy and agricultural population
OUTCOME 1/2: NUMERACY/HIGHER
31
GRAPHS RELATED TO GEOGRAPHY
I’m sure you didn’t expect to see a scatter diagram here! Don’t forget that existence of a
relationship does not mean that one variable is caused by the other.
Here we have an apparently almost perfect, straight line of best fit, but the countries
illustrated may well have been specially chosen to fit an agenda, though we have to assume
that they are representative of all countries in general.
What is the connection here between the two variables?
3.3 Chorographic Diagrams
Chorography is the name given to the technique of mapping regions, and next we have
several different examples of this.
? 3.3A
Wheat yields
Here we have the map of some land alongside a river. Parts of it are shaded according to
wheat yields (the key for this is on the right). J, K and L are not shaded, but the actual data
is also included, on the left.
1.
2.
32
Shade in fields J, K and L according to the yields.
Are the fields with the highest yields generally nearer the river or further away?
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
? 3.3B
Population density in Spain
This map shows population density. Notice that, in the key on1234567
the right, the numbers are
1234567
1234567
written alongside the joins of the shadings. Thus the shading 1234567 shows areas with a
population density of between 75 and 112.5 (but whether this is per hectare or per square
kilometre is not specified – no one is perfect!).
Where does the population of Spain tend to live? Why do you think this is?
A big drawback of this type of diagram is that the boundaries shown on the map are local
government boundaries, and so the figures are averages over that whole region, whereas in
fact part of a certain region may well be very densely populated and another part of the
same area might be almost deserted.
OUTCOME 1/2: NUMERACY/HIGHER
33
GRAPHS RELATED TO GEOGRAPHY
? 3.3C
Employment structure in Portugal
Employment
structure
1970
Primary
Secondary
Tertiary
Total
employment
750,000
500,000
200,000
100,000
50,000
Here is a similar type of diagram, with a pie chart representing an economic or
demographic feature of each local government area. A key at the side relates the circle
sizes to, in this case, the total employment of that area. (You may have seen similar maps
with little bar charts instead of pie charts.)
The two largest circles represent Lisbon almost half way up, and Oporto in the north.
What is the total employment of Oporto and Lisbon, and how does their employment
structure differ?
34
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
? 3.3D
Average gross weekly
household income, 1994
Average house price (£), 1994
Frequently we have to compare information from two separate graphs.
In general terms, what do you notice about the areas with a low/high household income,
and those areas with a low/high average house price?
3.4 Isoline Maps
You will be familiar with isoline maps. You see one on television every time you watch a
weather forecast. Isoline maps join with lines points which have the same numerical
feature. If places of equal temperature are joined, we have isotherms; if places with equal
air pressure are joined, we have isobars.
OUTCOME 1/2: NUMERACY/HIGHER
35
GRAPHS RELATED TO GEOGRAPHY
? 3.4A
Snowfall in Scotland
Depth of snow in the ten worst
winters this century (cm)
Braemar is shown with the letter B, Dingwall with the letter D and Glasgow with the letter
G. What was the average depth of snow during the ten worst winters this century at these
three places?
? 3.4B
Aircraft at Heathrow
Units: relative values taken from a noise meter
This a noise level chart for the area surrounding Heathrow airport.
What can you say about the noise level at (a) Staines (b) Hounslow?
36
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
3.5 Flowline Maps
Flowline maps are maps which show movements between places by lines, the thickness of
which is proportional to the quantity or size of flow.
? 3.5
Visitors to the Edinburgh International Festival
1 mm represents 200 people on the EIF mailing list
This shows how many people there are (or were at some point in the past) on the mailing
list for the Edinburgh International Festival.
1.
2.
3.
Which country had most?
Roughly how many were there from Spain?
There were 586 from Eire. How wide should the line from there be?
OUTCOME 1/2: NUMERACY/HIGHER
37
GRAPHS RELATED TO GEOGRAPHY
3.6 Cartograms
A cartogram is a form of topological map, i.e. one in which the shape, distance and
direction are disregarded, but the relative position of places is retained. It is used when the
aim of the map is to highlight some value, such as population or production of a commodity.
The area of the country is drawn proportional to the value.
? 3.6A
Gross national product (GNP) of countries
This shows you the gross national product of the countries of the world. The names of
some countries are written in. An area key is necessary if any values are to be extracted at
all, though such a diagram is more often used just for comparative purposes. Parts of the
world are easily recognisable. But can you spot Africa? Australia? Italy?
According to this cartogram, what is the approximate GNP of the UK?
38
OUTCOME 1/2: NUMERACY/HIGHER
GRAPHS RELATED TO GEOGRAPHY
? 3.6B
Oil production and consumption
Here we can easily make comparisons between the two maps without worrying about the
actual figures, but just by looking at the areas.
The top chart shows oil production, the bottom chart shows oil consumption in the same
units of measurement.
1.
2.
Name some countries which consume more oil than they produce.
Name some countries which produce more oil than they consume.
OUTCOME 1/2: NUMERACY/HIGHER
39
TUTOR ASSIGNMENTS
T2
The combined graph below compares the number of rail journeys in Scotland (bar chart,
left-hand axis) and the number of air journeys (line graph, right-hand axis) in a selected
period of time. (By air journeys we mean the number of passengers passing through all
Scottish airports.)
Rail/air passenger numbers – Scotland
12
Train Journeys (million)
57
10
56
8
55
54
6
53
52
4
51
2
50
49
Air Journeys (million)
58
0
1985
1986
1987
1988
1989
1990
Year
Rail
Air
(a)
In 1985 a total of £94.2 million was raised from the rail fares. What was the average
rail fare in 1985? In 1986 the receipts were £99.5 million. By how much had the
average fare risen during that year?
(b)
What is the general trend in each of the two modes of transport?
40
OUTCOME 1/2: NUMERACY/HIGHER
TUTOR ASSIGNMENTS
T3
We often have one type of graph illustrating the data in general and another looking more
closely at a specific part of the data.
The line graph below shows the total expenditure of the Scottish Arts Council over the
period 1984–89. The pie chart on the next page shows the breakdown of expenditure for
1987.
Scottish Arts Council Expenditure
16
£million
15
14
13
12
1984
1985
1986
1987
1988
1989
Year
OUTCOME 1/2: NUMERACY/HIGHER
41
TUTOR ASSIGNMENTS
Expenditure 1987
Music,opera & dance
Drama
Touring
Art
Literature
Film
Festivals
Combined arts
Surveys & seminars
Housing the arts
In this pie chart, the sectors follow the legend clockwise from the top.
(a)
(b)
(c)
42
Roughly how much money was spent in 1985?
What percentage of the expenditure in 1987 was on drama?
Roughly how much money was spent on literature in 1987?
OUTCOME 1/2: NUMERACY/HIGHER
TUTOR ASSIGNMENTS
T4
This bar chart shows the age structure of claimants for benefit one year.
Redraw it as best you can as a percentage component chart.
Numbers of claimants by age group
30,000
Number
25,000
20,000
15,000
10,000
5,000
0
17-under 20-under 30-under 40-under 50-under
20
30
40
50
65
Age Group
T5
The following chart is one that you are probably not familiar with. It can be used, broadly
speaking, in the same way as a component chart, but in situations where there are only
three components. Notice that each axis is a percentage axis 0–100%.
Suppose a company has a project A where the three costs are: wages £45,000, materials
£100,000 and overheads £20,000. We can turn these into percentages of total cost and get
wages 27%, materials 61% and overheads 12%. We can then plot these percentages on a
special triangular set of axes as shown on the next page.
Project A is shown by a dot, joined with broken lines (at right angles) to the appropriate
percentage on each of the three axes.
Use your skill and judgement to answer the following:
(a)
What are the three percentages for Project B?
(b)
If project C cost a total of £190,000 what was the cost of each component?
(c)
Describe in general terms a project which finds itself near the bottom left-hand
corner of the triangle.
OUTCOME 1/2: NUMERACY/HIGHER
43
TUTOR ASSIGNMENTS
100
90
80
.
70
60
Materials
0
.
50
10
60
70
80
90
100
B
20
50
20
40
40
30
40
50
20
10
0
OUTCOME 1/2: NUMERACY/HIGHER
Wages
0
10
Overheads
44
C
.
60
A
70
80
90
100
ANSWERS TO SAQs
ANSWERS
?2.1A: Answers
1.
Partly skilled and unskilled manual.
2.
Skilled non-manual has a far higher proportion of women than men, due perhaps to
large numbers of female office staff.
Skilled manual has a far higher proportion of men. This will be because large
sections of industry still employ men, e.g. engineering.
?2.1B: Answers
1.
Index for Finland is about 114, i.e. prices are, on average, 14% higher in Finland than
in the UK. Index for Italy is about 85, i.e. prices in Italy are, on average, 15% lower
than in the UK.
2.
It might have been easier to find the country, but the bars would all have been
intermingled, making the graph more difficult to read.
?2.1C: Answers
1.
NHS: 2,600 per 10,000 = 26%
Private: 2,100 per 10,000 = 21%
2.
Children under 16 get sight tests free on the NHS, which is why there appear to be
no under-16s being tested in the private sector (or there are so few they don’t show
up). The main thing to note is that the graph gives rates per 10,000. A longer bar
means that more people out of every 10,000 who are in that age group had eye tests,
but this might not translate into more bodies, e.g.
rate 170 per 10,000 with population 600,000 translates into 10,200 actual people
but
rate 234 per 10,000 with population 385,000 translates into 9,000 actual people.
3.
NHS: 1,500 per 10,000 times 19.8 million = 2,970,000 tests
Private: 2,000 per 10,000 times 19.8 million = 3,960,000 tests
OUTCOME 1/2: NUMERACY/HIGHER
45
ANSWERS TO SAQs
4.
NHS: initial high rates for young people, then much lower rates for middle aged (eyes
don’t change much, spectacles don’t need changed, don’t go for tests), then higher
rates for older people as eyes change more and need more frequent testing.
Private: gradual increase, then constant rate for ages 45 upwards; more affluent
people get private tests and are more liable to go back regularly.
?2.1D: Answers
1.
1923 refers to the year of birth. By 1948, 45% of the women (who are now 25
years old) are still childless. By 1958, only 20% of the women (who are by now 35
years old) are still childless. This reduces to 15% by 1968 when the women are 45
years old.
2.
About 10% of women born in 1943 are childless at age 45, i.e. by 1988. The data
was compiled for the end of 1998. Very, very few women have their first child after
the age of 45.
3.
The general trend shows that, after an initial decrease in the ages by which a first
child is born, the ages gradually increase, i.e. women are putting off having children
till they are older.
Women born in 1973 are 25 years old by 1988, and over 60% of them have not yet
had their first child.
The figures also show that increasing numbers are electing not to have any children
at all.
46
OUTCOME 1/2: NUMERACY/HIGHER
ANSWERS TO SAQs
?2.1E: Answers
Here is a suggested possible graph:
Full Driving Licence Holders
?2.2A: Answers
1.
First peak – immediately after end of World War I
Second peak – after end of World War II
Third peak – the so-called ‘baby boom’, i.e. children being born in the 1960s to
parents who were born during the second peak.
2.
(a)
Round about 1978 or 1979, births equalled deaths. Although it is difficult to
see because of lack of colour, it is the births line which continues as the high
line after 1980.
OUTCOME 1/2: NUMERACY/HIGHER
47
ANSWERS TO SAQs
(b)
Over the past century deaths have been surprisingly static, but births have, in
general, decreased.
3.
Doesn’t take into account emigration and immigration.
4.
Approximately 1.1 million (i.e. 1,100,000) were born in 1901.
?2.2B: Answers
1.
Secondary line matches the primary line, but about 7 years later. Pupils are at
primary school for 7 years, then move to secondary.
2.
School leaving age raised (from 15 to 16) hence a sudden jump.
3.
As more pupils stay on for fifth and sixth year, and fewer leave after S4, the total
numbers in secondary school get more like the total numbers in primary, but there
will always be a gap because secondary lasts for one year less than primary.
?2.2C: Answers
1.
Women: 8 per 100,000 out of a total of 28.2 million makes 2,256 cases.
Men: 6 per 100,000 out of a total of 29.6 million makes 1,776 cases.
A grand total of 4,032, i.e. approximately 4,000 cases.
2.
It could be that increases in foreign holidays in sunny climates lead to peaks. A
cancer scare, leading to people covering up more, may cause the rates to decrease,
then they increase again after the publicity campaign has stopped and people
become less cancer conscious.
?2.2D: Answers
1.
Three line graphs. You know because the percentages don’t add up to 100.
2.
‘One or more cars’ will include ‘two or more cars’. Subtract any ‘two or more’ value
from the ‘one or more’ value and you get ‘one car only’.
Since the percentage owning one car only is pretty well constant after 1970, the
other two are bound to be virtually the same, with one higher than the other.
3.
48
Going up. And up.
OUTCOME 1/2: NUMERACY/HIGHER
ANSWERS TO SAQs
?2.2E: Answers
1.
Sales going up.
2.
The peak occurs towards the end of each year (the horizontal axis is clearly a
continuous scale, and you can split each year into 12 (monthly) parts). Obviously
there is a large increase over the Christmas period, immediately followed by a drop
when people have no more money to spend.
?2.2F: Answers
1.
The poorest 10% don’t appear to be getting much better off.
2.
The 90th percentile line shows the disposable income of the bottom 90% of the
population. This is equivalent to the income of the top 10%. They are clearly
getting much richer, with the main increase occurring in the late 1980s.
3.
The median line has increased a bit but not much.
The overall picture is that, on average, people are better off (shown by the median
line, i.e. half earn below the line, half earn above) but the increase is not equitably
distributed, with the richest getting richer still.
?2.2G: Answers
1.
National strike of 1926.
2.
To accommodate the high value of 1926, the scale is very compressed, showing very
little definition between, say, 1931 and 1970.
?2.2H: Answers
1.
Agriculture and textiles decreasing throughout the period 1901–91.
Manufacturing increasing till the 1960s, then declining.
Office increasing steadily.
2.
Textiles decreasing further (to 1%?).
Agriculture decreasing too, or perhaps levelling off.
Manufacturing decreasing (to 15%?).
Office increasing (to 45%?).
OUTCOME 1/2: NUMERACY/HIGHER
49
ANSWERS TO SAQs
3.
No census in 1941 (because of the war) but it is assumed that the trends before and
after would be the same. Presumably this census is being taken every 10 years.
There is no information about whether years other than those mentioned have
records which contribute to the graph (i.e. the horizontal scale might not be
continuous).
?2.2J: Answers
Here is a possible solution.
Two points to note:
(a)
The last two sections are projections (data published in the year 2000) so they
should be shown as dotted lines.
(b)
The last available figures are presumably for 1998. To keep the horizontal scale
constant, where 1998 appears in the solution we should, strictly speaking, have
2001, and the dots should be lined up 6 spaces before this (where 1998 should be).
50
OUTCOME 1/2: NUMERACY/HIGHER
ANSWERS TO SAQs
The lines joining the 1991 figures to the 1998 ones will then appear slightly steeper.
In practice, however, we would probably plot 1998 where you actually see it, even
though we know it should be to the left of this.
?3.1A: Answers
1.
Bangladesh has a very young population, Japan is more middle aged.
2.
The Bangladesh pyramid will look like the current Japanese one.
The Japanese one may well look something like the current Bangladesh one upside
down.
?3.1B: Answers
1.
Wetter than average.
2.
About average.
3.
Drier than average.
?3.1C: Answers
Pakistan population increasing fairly steadily till the 1950s, than an acceleration in growth,
almost trebling in the next 40 years.
UK details not so easily seen, but growth is clearly slow. Note that figures for 1901 (and
also 1911?) are missing.
?3.1D: Answers
Cereals account for about one quarter of the land use. This would have been seen more
easily had it been at one end of the bar. Placing it in the middle makes it more difficult to
relate it to the whole.
?3.2A: Answers
1.
High temperature coincides with low rainfall.
Low temperature coincides with high rainfall.
OUTCOME 1/2: NUMERACY/HIGHER
51
ANSWERS TO SAQs
2.
Only to a certain extent – the peak rainfall coincides with a dip in the temperature,
but the lowest rainfall occurs at the same time as the lowest temperatures.
3.
When Ibiza is dry, Kano is wet.
4.
Approximately 10 + 60 + 120 + 200 + 300 + 130 + 10 + 5 = 835mm of rain per
year.
?3.2B: Answers
1.
Faro and Vilamoura (lines roughly horizontal).
2.
Albufeira and Praia (very busy in the summer, very quiet otherwise).
3.
Highest percentage is only 22 or so. A common fault when drawing graphs of
percentages is to assume automatically that they will go up to 100! The scale should
only go as far as it needs to go.
?3.2C: Answers
This scattergraph appears to show that the percentage of the workforce employed in
agriculture is very strongly related to the life expectancy. The more agricultural workers,
the shorter their lifespan!
One does not depend on the other, of course. Third-world countries, where most of the
people work on the land, have a low life expectancy, but this is caused by other factors (e.g.
poor health care) and not because agriculture there is necessarily any more dangerous than
anywhere else.
?3.3A: Answers
1.
12345678
12345678
12345678
J shaded with horizontal lines 12345678
12345678 because 6.5 is between 6.0 and 6.9.
K shaded with dots ••••••••••••••• •••••• because 4.5 is between 4.0 and 5.9.
L shaded the same as J.
2.
52
Highest yields are generally further away from the river (not being a geographer I
will not suggest any reasons for this!).
OUTCOME 1/2: NUMERACY/HIGHER
ANSWERS TO SAQs
?3.3B: Answers
Apart from the Madrid area (the black splodge in the middle) the most densely populated
parts are nearest the coasts. Presumably the more fertile and flatter land is there.
?3.3C: Answers
Lisbon – total employment around 750,000, more than half of whom are in tertiary
employment.
Oporto – employment 500,000 but more than half are in the secondary sector.
No one expects any protractor work with pie charts like this.
?3.3D: Answers
In general, those areas with highest income are the same as those with highest house
prices. The same applies to lowest income/lowest house prices.
?3.4A: Answers
Braemar – over 50cm of snow.
Dingwall – between 25cm and 50cm of snow.
Glasgow – less than 25cm of snow.
?3.4B: Answers
Staines – less than 20 units.
Hounslow – between 20 and 30 units.
?3.5A: Answers
1.
Germany had most (thickest line).
2.
4mm thick so 800 people on the mailing list. This assumes, of course, that the map is
photocopied exactly the right size as originally drawn! Any enlargement or reduction
will affect any numerical answers.
Clearly, they are not designed for numerical accuracy!
(NB: A map like 3.3C will not suffer in the same way from a change of scale. Angles
in pie charts will remain the same.)
OUTCOME 1/2: NUMERACY/HIGHER
53
ANSWERS TO SAQs
3.
586 ÷ 200 = 2.93 so a 3mm wide line will be in scale with the others.
?3.5A: Answers
The little square representing 40 thousand million dollars is 3mm by 3mm. The UK is
11½mm by 6mm. So just over 7½ squares will fit into the UK. This equates to about 300
thousand million dollars (or 300 billion as we tend to call it now).
?3.6B: Answers
1.
Japan, ‘Rest of Europe’.
2.
Iran, Saudi Arabia.
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OUTCOME 1/2: NUMERACY/HIGHER