CIMA - C3 Fundamentals Of Business Mathematics
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90 3: Data presentation ⏐ Part B Summarising and analysing data
2.2.4 Example: A percentage component bar chart
The information in the previous example of sales of Charbart could have been shown in a percentage component
bar chart as follows.
Working
20X7
20X8
20X9
$'000
%
$'000
%
$'000
%
Product A
1,000
42
1,200
43
1,700
50
Product B
900
37
1,000
36
1,000
29
Product C
500
21
600
21
700
21
Total
2,400
100
2,800
100
3,400
100
This chart shows that sales of C have remained a steady proportion of total sales, but the proportion of A in total
sales has gone up quite considerably, while the proportion of B has fallen correspondingly.
A multiple bar chart (or compound bar chart) is a bar chart in which two or more separate bars are used to
present sub-divisions of data.
2.2.5 Example: A multiple bar chart
The data on Charbart's sales could be shown in a multiple bar chart as follows.
Key term
Part B Summarising and analysing data ⏐ 3: Data presentation 91
A multiple bar chart uses several bars for each total. In this multiple bar chart, the sales in each year are shown as
three separate bars, one for each product, A, B and C.
2.2.6 Information presented by multiple bar charts
Multiple bar charts present similar information to component bar charts, except for the following.
(a) Multiple bar charts do not show the grand total whereas component bar charts do.
(b) Multiple bar charts illustrate the comparative magnitudes of the components more clearly than
component bar charts.
Multiple bar charts are sometimes drawn with the bars horizontal instead of vertical.
Question
Multiple bar charts
Income for Canary Bank in 20X0, 20X1 and 20X2 is made up as follows.
20X0 20X1 20X2
$'000 $'000 $'000
Interest income 3,579 2,961 2,192
Commission income 857 893 917
Other income 62 59 70
Using the above data complete the following graphs.
(a) A simple bar chart
92 3: Data presentation ⏐ Part B Summarising and analysing data
(b) A multiple bar chart
Answe
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Workings
20X0
20X1
20X2
$'000
$'000
$'000
3,579
2,961
2,192
857
893
917
62
59
70
4,498
3,913
3,179
(a) A simple bar chart
Part B Summarising and analysing data ⏐ 3: Data presentation 93
(b) A multiple bar chart
2.3 Pie charts
A pie chart is a chart which is used to show pictorially the relative size of component elements of a total.
It is called a pie chart because it is circular, and so has the shape of a pie in a round pie dish. The 'pie' is then cut
into slices with each slice representing part of the total.
Pie charts have sectors of varying sizes, and you need to be able to draw sectors fairly accurately. To do this, you
need a protractor. Working out sector sizes involves converting parts of the total into equivalent degrees of a
circle. A complete 'pie' = 360°: the number of degrees in a circle = 100% of whatever you are showing. An element
which is 50% of your total will therefore occupy a segment of 180°, and so on.
2.3.1 Using shading and colour
Two pie charts are shown as follows.
Key term
94 3: Data presentation ⏐ Part B Summarising and analysing data
• Shading distinguishes the segments from each other
• Colour can also be used to distinguish segments
2.3.2 Example: Pie charts
The costs of materials at the Cardiff Factory and the Swansea Factory during January 20X0 were as follows.
Cardiff factory
Swansea factory
$'000
%
$'000
%
Material W
70
35
50
20
Material A
30
15
125
50
Material L
90
45
50
20
Material E
10
5
25
10
200
100
250
100
Show the costs for the factories in pie charts.
Solution
To convert the components into degrees of a circle, we can use either the percentage figures or the actual cost
figures.
Using the percentage figures
The total percentage is 100%, and the total number of degrees in a circle is 360°. To convert from one to the other,
we multiply each percentage value by 360/100% = 3.6.
Cardiff factory
Swansea factory
%
Degrees
%
Degrees
Material W
35
126
20
72
Material A
15
54
50
180
Material L
45
162
20
72
Material E
5
18
10
36
100
360
100
360
Using the actual cost figures
Cardiff factory
Swansea factory
$'000
Degrees
$'000
Degrees
Material W (70/200 × 360°)
70
126
50
72
Material A
30
54
125
180
Material L
90
162
50
72
Material E
10
18
25
36
200
360
250
360
A pie chart could be drawn for each factory.
Part B Summarising and analysing data ⏐ 3: Data presentation 95
(a) If the pie chart is drawn manually, a protractor must be used to measure the degrees accurately to
obtain the correct sector sizes.
(b) Using a computer makes the process much simpler, especially using a spreadsheet. You just draw
up the data in a spreadsheet and click on the chart button to create a visual representation of what
you want. Note that you can only use colour effectively if you have a colour printer!
2.3.3 Advantages of pie charts
• They give a simple pictorial display of the relative sizes of elements of a total
• They show clearly when one element is much bigger than others
• They can clearly show differences in the elements of two different totals
2.3.4 Disadvantages of pie charts
(a) They only show the relative sizes of elements. In the example of the two factories, for instance, the
pie charts do not show that costs at the Swansea factory were $50,000 higher in total than at the
Cardiff factory.
(b) They involve calculating degrees of a circle and drawing sectors accurately, and this can be time
consuming unless computer software is used.
(c) It is often difficult to compare sector sizes easily. For example, suppose that the following two pie
charts are used to show the elements of a company's sales.
Without the percentage figures, it would not be easy to see how the distribution of sales had changed between
20X0 and 20X1.
96 3: Data presentation ⏐ Part B Summarising and analysing data
Question
Pie charts
The European division of Scent to You, a flower delivery service has just published its accounts for the year ended
30 June 20X0. The sales director made the following comments.
'Our total sales for the year were $1,751,000, of which $787,000 were made in the United Kingdom, $219,000 in
Italy, $285,000 in France and $92,000 in Germany. Sales in Spain and Holland amounted to $189,000 and $34,000
respectively, whilst the rest of Europe collectively had sales of $145,000 in the twelve months to 30 June 20X0.'
Required
Present the above information in the form of a pie chart. Show all of your workings.
Answe
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Workings
Sales
$'000
Degrees
United Kingdom
787
(787/1,751 × 360)
162
Italy
219
45
France
285
58
Germany
92
19
Spain
189
39
Rest of Europe
145
30
Holland
34
7
1,751
360
Scent to You
Sales for the year ended 30 June 20X0
A computer based assessment cannot require you to draw charts so questions will focus on labelling, calculating
values, choosing an appropriate chart and coming to conclusions using charts.
Assessment
focus point
Part B Summarising and analysing data ⏐ 3: Data presentation 97
3 Frequency distributions
3.1 Introduction to frequency distributions
Frequency distributions are used if values of particular variables occur more than once.
Frequently the data collected from a statistical survey or investigation is simply a mass of numbers.
65 69 70 71 70 68 69 67 70 68
72 71 69 74 70 73 71 67 69 70
The raw data above yields little information as it stands; imagine how much more difficult it would be if there were
hundreds or even thousands of data items. The data could, of course, be arranged in order size (an array) and the
lowest and highest data items, as well as typical items, could be identified.
3.2 Example: Frequency distribution
Many sets of data, however, contain a limited number of data values, even though there may be many occurrences
of each value. It can therefore be useful to organise the data into what is known as a frequency distribution (or
frequency table) which records the number of times each value occurs (the frequency). A frequency distribution
for the data in Paragraph 3.1 (the output in units of 20 employees during one week) is as follows.
Output of employees in one week in units
Output Number of employees (frequency)
Units
65 1
66 0
67 2
68 2
69 4
70 5
71 3
72 1
73 1
74
1
20
When the data are arranged in this way it is immediately obvious that 69 and 70 units are the most common
volumes of output per employee per week.
3.3 Grouped frequency distributions
If there is a large set of data or if every (or nearly every) data item is different, it is often convenient to group
frequencies together into bands or classes. For example, suppose that the output produced by another group of 20
employees during one week was as follows, in units.
1,087 850 1,084 792
924 1,226 1,012 1,205
1,265 1,028 1,230 1,182
1,086 1,130 989 1,155
1,134 1,166 1,129 1,160
FA
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T F
O
RWAR
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98 3: Data presentation ⏐ Part B Summarising and analysing data
3.4 Class intervals
The range of output from the lowest to the highest producer is 792 to 1,265, a range of 473 units. This range could
be divided into classes of say, 100 units (the class width or class interval), and the number of employees
producing output within each class could then be grouped into a single frequency, as follows.
Output
Number of employees (frequency)
Units
700 – 799
1
800 – 899
1
900 – 999
2
1,000 – 1,099
5
1,100 – 1,199
7
1,200 – 1,299
4
20
Note, however, that once items have been 'grouped' in this way their individual values are lost.
As well as being used for discrete variables (as above), grouped frequency distributions (or grouped frequency
tables) can be used to present data for continuous variables.
3.5 Example: A grouped frequency distribution for a continuous variable
Suppose we wish to record the heights of 50 different individuals. The information might be presented as a
grouped frequency distribution, as follows.
Height
Number of individuals
cm
(frequency)
Up to and including 154
1
Over 154, up to and including 163
3
Over 163, up to and including 172
8
Over 172, up to and including 181
16
Over 181, up to and including 190
18
Over 190
4
50
Note the following points.
(a) It would be wrong to show the ranges as 0 – 154, 154 – 163, 163 – 172 and so on, because 154 cm and
163 cm would then be values in two classes, which is not permissible. Although each value should only be
in one class, we have to make sure that each possible value can be included. Classes such as 154–162,
163–172 would not be suitable since a height of 162.5 cm would not belong in either class. Such classes
could be used for discrete variables, however.
(b) There is an open ended class at each end of the range. This is because heights up to 154 cm and over
190 cm are thought to be uncommon, so that a single 'open ended' class is used to group all the
frequencies together.
3.6 Guidelines for preparing grouped frequency distributions
To prepare a grouped frequency distribution, a decision must be made about how wide each class should be.
You should observe the following guidelines if you are not told how many classes to use or what the class interval
should be.
Part B Summarising and analysing data ⏐ 3: Data presentation 99
(a) The size of each class should be appropriate to the nature of the data being recorded, and the most
appropriate class interval varies according to circumstances.
(b) The upper and lower limits of each class interval should be suitable 'round' numbers for class
intervals which are in multiples of 5, 10, 100, 1,000 and so on. For example, if the class interval is
10, and data items range in value from 23 to 62 (discrete values), the class intervals should be 20–
29, 30–39, 40–49, 50–59 and 60–69, rather than 23–32, 33–42, 43–52 and 53–62.
(c) With continuous variables, either:
(i) the upper limit of a class should be 'up to and including ...' and the lower limit of the next
class should be 'over ...'
(ii) the upper limit of a class should be 'less than...', and the lower limit of the next class
should be 'at least ...'
Question
Grouped frequency distributions
The commission earnings for May 20X0 of the assistants in a department store were as follows (in dollars).
60 35 53 47 25 44
55 58 47 71
63 67 57 44 61 48
50 56 61 42
43 38 41 39 61 51
27 56 57 50
55 68 55 50 25 48
44 43 49 73
53 35 36 41 45 71
56 40 69 52
36 47 66 52 32 46
44 32 52 58
49 41 45 45 48 36
46 42 52 33
31 36 40 66 53 58
60 52 66 51
51 44 59 53 51 57
35 45 46 54
46 54 51 39 64 43
54 47 60 45
Required
Prepare a grouped frequency distribution classifying the commission earnings into categories of $5 commencing
with '$25 and under $30'
Answe
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We are told what classes to use, so the first step is to identify the lowest and highest values in the data. The lowest
value is $25 (in the first row) and the highest value is $73 (in the fourth row). This means that the class intervals
must go up to '$70 and under $75'.
We can now set out the classes in a column, and then count the number of items in each class using tally marks.