Why do we collect, display and analyse data?
1.5 Comparing different data representations
Pictograms, vertical line graphs, bar charts and pie charts are useful ways of displaying qualitative data and ungrouped quantitative data, and people generally find them easy to understand. Nevertheless, it may be of benefit to group a set of raw data so that we can see how the values are distributed. Knowing the proportion of small, medium and large values, for example, may prove to be useful. For small datasets we can do this by constructing stem-and-leaf diagrams, which have the advantage that raw values can still be seen after grouping.
In large datasets individual values lose their significance and a picture of the whole is more informative. We can use frequency tables to make a compact summary by grouping but most people find the information easier to grasp when it is shown in a graphical format, which allows absolute, relative or cumulative frequencies to be seen.
Although some data are lost by grouping, histograms and cumulative frequency graphs have the advantage that data can be grouped into classes of any and varied widths.
The choice of which representation to use will depend on the type and quantity of data, the audience and the objectives behind making the representation. Most importantly, the representation must show the data clearly and should not be misleading in any way.
The following chart is a guide to some of the most commonly used methods of data representation.
Qualitative data
Ungrouped Discrete
quantitative data
Grouped
Continuous data
Small amount
Pictogram, vertical line graph, bar chart, pie chart, sectional bar chart
Stem-and-leaf diagram
Histogram
Cumulative frequency graph Large amount
In Chapter 2 and Chapter 3, we will see how grouping effects the methods we use to find measures of central tendency and measures of variation.
FAST FORWARD EXPLORE 1.3
The following table shows data on the masses, m grams, of 150 objects.
Mass (mg) m<0 m<12 m<30 m<53 m<70 m<80
No. objects (cf) 0 24 60 106 138 150
By drawing a cumulative frequency polygon, the following estimates will be obtained:
a Number of objects with masses less than 20 g=40 objects.
b Arranged in ascending order, the mass of the 100th object=50 g.
However, we can calculate these estimates from the information given in the table without drawing the polygon.
Investigate the possible methods that we can use to calculate these two estimates.
The six plotted points, whose coordinates you know, are joined by straight lines.
TIP
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EXERCISE 1D
1 Jamila noted each student’s answer when her year group was asked to name their favourite colour.
a List the methods of representation that would be suitable for displaying Jamila’s data.
b Jamila wishes to emphasise that the favourite colour of exactly three-quarters of the students is blue.
Which type of representation from your list do you think would be the most effective for Jamila to use?
Explain why you have chosen this particular type of representation.
2 A large number of chickens’ eggs are individually weighed. The masses are grouped into nine classes, each of width 2 grams, from 48 to 66 g.
Name a type of representation in which the fact could be seen that the majority of the eggs have masses from 54 to 60 g. Explain how the representation would show this.
3 Boxes of floor tiles are to be offered for sale at a special price of $75. The boxes claim to contain at least 100 tiles each.
a Why would it be preferable to use a stem-and-leaf diagram rather than a bar chart to represent the numbers of tiles, which are 112, 116, 107, 108, 121, 104, 111, 106, 105 and 110?
b How may the seller benefit if the numbers 12, 16, 7, 8, 21, 4, 11, 6, 5 and 10 are used to draw the stem-and- leaf diagram instead of the actual numbers of tiles?
4 A charity group’s target is to raise a certain amount of money in a year. At the end of the first month the group raised 36% of the target amount, and at the end of each subsequent month they manage to raise exactly half of the amount outstanding.
a How many months will it take the group to raise 99% of the money?
b Name a type of representation that will show that the group fails to reach its target by the end of the year.
Explain how this fact would be shown in the representation.
5 University students measured the heights of the 54 trees in the grounds of a primary school. As part of a talk on conservation at a school assembly, the students have decided to present their data using one of the following diagrams.
Heights of 54 trees
2 to 3 m 3 to 5 m 5 to 8 m
0 4 8 12
2 3 4 5 6 7
Heights of trees (m)
Frequency density
8
tall
short
medium
a Give one disadvantage of using each of the representations shown.
b Name and describe a different type of representation that would be appropriate for the audience, and that has none of the disadvantages given in part a.
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6 The percentage scores of 40 candidates who took a Health and Safety test are given:
77 44 65 84 52 60 35 83 68 66 50 68 65 57 60 50 93 38 46 55 45 69 61 64 40 66 91 59 61 74 70 75 42 65 85 63 73 84 68 30
a Construct a frequency table by grouping the data into seven classes with equal-width intervals, where the first class is 30–39. Label this as Table 1.
b It is proposed that each of the 40 candidates is awarded one of three grades, A, B or C. Construct a new frequency table that matches with this proposal. Label this as Table 2.
c A student plans to display all three versions of the data (i.e. the raw data, the data in Table 1 and the data in Table 2) in separate stem-and-leaf diagrams.
For which version(s) of the data would this not be appropriate? Suggest an alternative type of representation in each case.
7 Last year Tom renovated an old building during which he worked for at least 9 hours each week. By plotting four points in a graph, he has represented the time he spent working.
0 49 50 51 52
34 36 38 40 44
Time spent working (hours) 42
cf (No. weeks)
a What can you say about the time that Tom spent working on the basis of this graph?
b Explain why Tom’s graph might be considered to be misleading.
c Name the different types of representation that are suitable for displaying the amount of time that Tom worked each week throughout the year.
Consider the benefits of each type of representation and then fully describe (but do not draw) the one you believe to be the most suitable.
8 The following table shows the focal lengths, lmm, of the 84 zoom lenses sold by a shop. For example, there are 18 zoom lenses that can be set to any focal length between 24 and 50 mm.
Focal length ( mm)l 24 – 50 50–108 100–200 150–300 250–400
No. lenses ( )f 18 30 18 12 6
a What feature of the data does not allow them to be displayed in a histogram?
b What type of diagram could you use to illustrate the data? Explain clearly how you would do this.
M
PS
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Past, present and predicted world population figures by age group, sex and other categories can be found on government census websites.
You may be interested, for example, in the population changes for your own age group during your lifetime. This is something that can be represented in a diagram, either manually, using spreadsheet software or an application such as GeoGebra, and for which you may like to try making predictions by looking for trends shown in the raw data or in any diagrams you create.
EXPLORE 1.4