© 2014, John Bird
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CHAPTER 85 PRESENTATION OF STATISTICAL DATA
EXERCISE 323 Page 906
1. State whether data relating to the topics given are discrete or continuous:
(a) The amount of petrol produced daily, for each of 31 days, by a refinery.
(b) The amount of coal produced daily by each of 15 miners.
(c) The number of bottles of milk delivered daily by each of 20 milkmen.
(d) The size of 10 samples of rivets produced by a machine.
(a) Continuous – could be any amount of petrol
(b) Continuous – could be any amount of coal
(c) Discrete – can only be a whole number of bottles of milk
(d) Continuous – could be any size of rivet
2. State whether data relating to the topics given are discrete or continuous:
(a) The number of people visiting an exhibition on each of 5 days.
(b) The time taken by each of 12 athletes to run 100 metres.
(c) The value of stamps sold in a day by each of 20 post offices.
(d) The number of defective items produced in each of 10 one-hour periods by a machine.
(a) Discrete – can only be a whole number of people
(b) Continuous – could be any time taken
(c) Discrete – can only be a whole number of stamps
(d) Discrete – can only be a whole number of defective items
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EXERCISE 324 Page 909
1. The number of vehicles passing a stationary observer on a road in 6 ten-minute intervals is as
shown. Draw a pictogram to represent these data.
Period of time 1 2 3 4 5 6
Number of vehicles 35 44 62 68 49 41
If one symbol is used to represent 10 vehicles, working correct to the nearest 5 vehicles gives 3.5,
4.5, 6, 7, 5 and 4 symbols, respectively, as shown below.
2. The number of components produced by a factory in a week is as shown below:
Day Mon Tues Wed Thur Fri
Number of components 1580 2190 1840 2385 1280
Show these data on a pictogram.
If one symbol represents 200 components, working correct to the nearest 100 components gives:
Mon 8, Tues 11, Wed 9, Thurs 12 and Fri 6.5
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3. For the data given in Problem 1, draw a horizontal bar chart.
A horizontal bar chart is shown below
4. Present the data given in Problem 2 on a horizontal bar chart.
A horizontal bar chart is shown below
5. For the data given in Problem 1, construct a vertical bar chart.
A vertical bar chart is shown below
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6. Depict the data given in Problem 2 on a vertical bar chart.
A vertical bar chart is shown below
7. A factory produces three different types of components. The percentages of each of these
components produced for 3 one-month periods are as shown below. Show this information on
percentage component bar charts and comment on the changing trend in the percentages of the
types of component produced.
Month 1 2 3
Component P 20 35 40
Component Q 45 40 35
Component R 35 25 25
© 2014, John Bird
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The information is shown on the percentage component bar chart below
It is seen that P increases by 20% at the expense of Q and R
8. A company has five distribution centres and the mass of goods in tonnes sent to each centre
during 4 one-week periods is as shown.
Week 1 2 3 4
Centre A 147 160 174 158
Centre B 54 63 77 69
Centre C 283 251 237 211
Centre D 97 104 117 144
Centre E 224 218 203 194
Use a percentage component bar chart to present these data and comment on any trends.
Week 1: Total = 147 + 54 + 283 + 97 + 224 = 805
A = 147 100% 18%805
× ≈ , B = 54 100% 7%805
× ≈ , C ≈ 35%, D ≈ 12%, E ≈ 28%
Week 2: Total = 160 + 63 + 251 + 104 + 218 = 796
A = 160 100% 20%796
× ≈ , B = 63 100% 8%796
× ≈ , C ≈ 32%, D ≈ 13%, E ≈ 27%
© 2014, John Bird
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Week 3: Total = 174 + 77 + 237 + 117 + 203 = 808
A = 174 100% 22%808
× ≈ , B = 77 100% 10%808
× ≈ , C ≈ 29%, D ≈ 14%, E ≈ 25%
Week 4: Total = 158 + 69 + 211 + 144 + 194 = 776
A = 158 100% 20%776
× ≈ , B = 69 100% 9%776
× ≈ , C ≈ 27%, D ≈ 19%, E ≈ 25%
A percentage component bar chart is shown below
From the above percentage component bar chart, it is seen that there is little change in centres A
and B, there is a reduction of around 8% in centre C, an increase of around 7% in centre D and a
reduction of about 3% in centre E
9. The employees in a company can be split into the following categories:
managerial 3, supervisory 9, craftsmen 21, semi-skilled 67, others 44
Show these data on a pie diagram.
Number of employees = 3 + 9 + 21 + 67 + 44 = 144
1 employee corresponds to 1360 2.5144
× = °
Hence, 3 employees corresponds to 3 × 2.5 = 7.5°, 9 employees corresponds to 9 × 2.5 = 22.5°
Similarly, 21, 67 and 44 employees correspond to 52.5°, 167.5° and 110°, respectively
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A pie diagram is therefore a circle of any radius, subdivided into sectors having angles of 7.5°,
22.5°, 52.5°, 167.5° and 110°, respectively, as shown below
10. The way in which an apprentice spent his time over a one-month period is as follows:
drawing office 44 hours, production 64 hours, training 12 hours, at college 28 hours
Use a pie diagram to depict this information.
Total hours = 44 + 64 + 12 + 28 = 148
Drawing office, D = 44 360 107148
× ° ≈ ° , Production, P = 64 360 156148
× ° ≈ ° ,
Training, T = 12 360 29148
× ° ≈ ° , College, C = 28 360 68148
× ° ≈ °
A pie chart to depict this information is shown below.
11. (a) With reference to Figure 85.5, determine the amount spent on labour and materials to produce
1650 units of the product.
(b) If in year 2 of Figure 85.4, 1% corresponds to 2.5 dwellings, how many bungalows are sold in
that year
© 2014, John Bird
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(a) Each product costs £2. Hence, 1650 units will cost £2 × 1650 = £3300
(Labour + materials) represents (36° + 18° = 44°) of 360°, i.e. 54360
of total
Hence, labour and material costs = 54360
of £3300 = 54 £3300360
× = £495
(b) In year 2 bungalows account for 7 + 28 = 35% of annual sales
If 1% corresponds to 2.5 dwellings, then
number of bungalows sold = 35 × 2.5 = 87.5 = 88 correct to nearest whole number
12. (a) If the company sell 23 500 units per annum of the product depicted in Figure 85.5, determine
the cost of their overheads per annum.
(b) If 1% of the dwellings represented in year 1 of Figure 85.4 corresponds to two dwellings, find
the total number of houses sold in that year.
(a) Overheads = 126 100% 35%360
× = of total costs
Cost per unit = £2, hence total income per annum = 23 500 × 2 = £47 000
Cost of overheads per annum = 35% of £47 000 = 35 47 000100
× = £16 450
(b) Percentage of houses sold in year 1 = 22 + 32 + 15 = 69%
If 1% corresponds to two dwellings then the number of houses sold = 69 × 2 = 138 houses
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EXERCISE 325 Page 915
1. The mass in kilograms, correct to the nearest one-tenth of a kilogram, of 60 bars of metal are as
shown. Form a frequency distribution of about eight classes for these data.
39.8 40.1 40.3 40.0 40.6 39.7 40.0 40.4 39.6 39.3
39.6 40.7 40.2 39.9 40.3 40.2 40.4 39.9 39.8 40.0
40.2 40.1 40.3 39.7 39.9 40.5 39.9 40.5 40.0 39.9
40.1 40.8 40.0 40.0 40.1 40.2 40.1 40.0 40.2 39.9
39.7 39.8 40.4 39.7 39.9 39.5 40.1 40.1 39.9 40.2
39.5 40.6 40.0 40.1 39.8 39.7 39.5 40.2 39.9 40.3
The range of values is 39.3–40.8. With eight classes therefore the classes chosen are 39.3–39.4,
39.5–39 6, and so on. A tally diagram is shown below with eight classes
A frequency distribution is shown below
Class Class mid-point Frequency
39.3–39.4 39.35 1
39.5–39.6 39.55 5
39.7–39.8 39.75 9
39.9–40.0 35.95 17
40.1–40.2 40.15 15
40.3–40.4 40.35 7
40.5–40.6 40.55 4
40.7–40.8 40.75 2
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2. Draw a histogram for the frequency distribution given in the solution of Problem 1.
A histogram for the frequency distribution given in the solution of Problem 1 is shown below
3. The information given below refers to the value of resistance in ohms of a batch of 48 resistors of
similar value. Form a frequency distribution for the data, having about six classes, and draw a
frequency polygon and histogram to represent these data diagramatically.
21.0 22.4 22.8 21.5 22.6 21.1 21.6 22.3
22.9 20.5 21.8 22.2 21.0 21.7 22.5 20.7
23.2 22.9 21.7 21.4 22.1 22.2 22.3 21.3
22.1 21.8 22.0 22.7 21.7 21.9 21.1 22.6
21.4 22.4 22.3 20.9 22.8 21.2 22.7 21.6
22.2 21.6 21.3 22.1 21.5 22.0 23.4 21.2
The range is from 20.5 to 23.4, i.e. range = 23.4–20.5 = 2.9
2.9 ÷ 6 ≈ 0.5 hence, classes of 20.5–20.9, 21.0–21.4, and so on are chosen, as shown in the frequency distribution below
© 2014, John Bird
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A frequency polygon is shown below where class mid-point values are plotted against frequency
values. Class mid-points occur at 20.7, 21.2, 21.7, and so on.
The histogram for the above frequency distribution is shown below
© 2014, John Bird
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4. The time taken in hours to the failure of 50 specimens of a metal subjected to fatigue failure tests
are as shown. Form a frequency distribution having about seven classes and unequal class
intervals for these data.
28 22 23 20 12 24 37 28 21 25
21 14 30 23 27 13 23 7 26 19
24 22 26 3 21 24 28 40 27 24
20 25 23 26 47 21 29 26 22 33
27 9 13 35 20 16 20 25 18 22
There is no unique solution, but one solution is:
The range of values is 3–47. The seven classes chosen are shown in the tally diagram below
A frequency distribution is shown below
Class Frequency Upper class boundary
Lower class boundary
Class range Height of rectangle
1–10 3 10.5 0.5 10 310
= 0.3
11–19 7 19.5 10.5 9 79
= 0.78
20–22 12 22.5 19.5 3 123
= 4
23–25 11 25.5 22.5 3 113
= 3.67
26–28 10 28.5 25.5 3 103
= 3.33
29–38 5 38.5 28.5 10 510
= 0.5
39–48 2 48.5 38.5 10 210
= 0.2
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5. Form a cumulative frequency distribution and hence draw the ogive for the frequency distribution
given in the solution to Problem 3.
A cumulative frequency distribution is shown below
Class Frequency Upper class boundary
Cumulative frequency
20.5–20.9
21.0–21.4
21.5–21.9
22.0–22.4
22.5–22.9
23.0–23.4
3
10
11
13
9
2
Less than
20.95
21.45
21.95
22.45
22.95
23.45
3
13
24
37
46
48
An ogive for the above frequency distribution is shown below
6. Draw a histogram for the frequency distribution given in the solution to Problem 4.
From the frequency distribution in Problem 4, the histogram is shown below
© 2014, John Bird
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7. The frequency distribution for a batch of 50 capacitors of similar value, measured in microfarads,
is:
10.5–10.9 2, 11.0–11.4 7, 11.5–11.9 10,
12.0–12.4 12, 12.5–12.9 11, 13.0–13.4 8
Form a cumulative frequency distribution for these data.
A cumulative frequency distribution for the data is shown in the table below
Class Frequency Upper class boundary less than
Cumulative frequency
10.5–10.9 2 10.95 2
11.0–11.4 7 11.45 9
11.5–11.9 10 11.95 19
12.0–12.4 12 12.45 31
12.5–12.9 11 12.95 42
13.0–13.4 8 13.45 50
8. Draw an ogive for the data given in the solution of Problem 7.
© 2014, John Bird
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An ogive, i.e. a graph of cumulative frequency against upper class boundary values, having
coordinates given in the above answer to Problem 7, is shown below
9. The diameter in millimetres of a reel of wire is measured in 48 places and the results are as shown.
2.10 2.29 2.32 2.21 2.14 2.22
2.28 2.18 2.17 2.20 2.23 2.13
2.26 2.10 2.21 2.17 2.28 2.15
2.16 2.25 2.23 2.11 2.27 2.34
2.24 2.05 2.29 2.18 2.24 2.16
2.15 2.22 2.14 2.27 2.09 2.21
2.11 2.17 2.22 2.19 2.12 2.20
2.23 2.07 2.13 2.26 2.16 2.12
(a) Form a frequency distribution of diameters having about 6 classes.
(b) Draw a histogram depicting the data.
(c) Form a cumulative frequency distribution.
(d) Draw an ogive for the data.
© 2014, John Bird
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(a) Range = 2.34–2.05 = 0.29
0.29 ÷ 6 ≈ 0.5, hence classes of 2.05–2.09, 2.10–2.14 and so on are chosen, as shown in the
frequency distribution below
(b) A histogram depicting the data is shown below
(c) A cumulative frequency distribution is shown below
Class Frequency Upper class boundary Cumulative frequency
2.05–2.09
2.10–2.14
2.15–2.19
2.20–2.24
2.25–2.29
2.30–2.34
3
10
11
13
9
2
Less than
2.095
2.145
2.195
2.245
2.295
2.345
3
13
24
37
46
48
(d) An ogive for the above data is shown below
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