3.7
In Chapter 3.1 we saw that there are three main skills involved in solving problems. We have already dealt with the first two of these (identifying important information and combining pieces of information). This chapter deals with the third skill, that of identifying pieces or sets of information in different forms which are equivalent. In particular, this chapter deals with graphical, verbal and tabular information.
Activity
The table shows the results of a survey into ownership of various household appliances by families who live in a town.
Which of the bar charts accurately represents the data shown below?
Appliance Dishwasher Vacuum
cleaner Washing
machine Microwave
oven Food processor Toaster
% ownership 68 98 77 54 34 92
A100
50 0
Vacuum cleaner Toaster Washing machine Dishwasher Microwave oven Food processor
% ownership
B100
50 0
Dishwasher Vacuum cleaner Washing machine Microwave oven ToasterFood processor
% ownership
C100
50 0
Toaster Food processor Microwave oven Washing machine Vacuum cleaner Dishwasher
% ownership
D100
50 0
Vacuum cleaner Toaster Washing machine Dishwasher Microwave oven Food processor
% ownership
E100
50 0
Vacuum cleaner Toaster Washing machine Dishwasher Microwave oven Food processor
% ownership
Commentary
This question is actually quite easy. It is only a matter of being careful and matching
appliance to bar length correctly. The main complication (and this is a potential trap for those who don’t look at the question and the graphs carefully) is that the order of the appliances in the graphs is different in some cases from the order in the table. Also, the exact heights of the bars cannot always be read accurately enough at the scale on which the graphs are drawn, so it is necessary to look at the relative heights of the different bars.
In fact D is the correct graph. The appliances have been put into order by their percentage ownership. A has the appliances ordered as for D but the bars are in the order of the table. The other graphs have similar errors – you might
like to identify the error in each case. Commentary
The data expressed in local currency is not very useful for a direct comparison. It is easier if the costs are all expressed as percentages so that the appropriate components can be compared.
The table is repeated below with the costs as percentages of the totals for each country.
Sudaria Idani Anguda Boralia Crude oil 51.09 36.00 30.10 47.62 Refining 1.46 9.00 1.50 1.90 Wholesale 6.57 7.00 3.76 13.33
Retail 4.38 4.00 4.36 4.76
Tax 36.50 44.00 60.27 32.38
In the pie chart, the largest segment is just under half the area. It could, therefore, only be tax in Idani or crude oil in Boralia. We cannot easily distinguish which as the difference in the second-largest segments is not very great.
We must, therefore, look at the smallest three segments. Boralia has one (wholesale) three times the size of either of the other two, but in the pie chart they are much closer than this, so the answer must be Idani.
This activity reverses the skill shown above:
the graph is given (a pie chart in this case) and the cost structure it represents has to be identified.
A student is drawing pie charts to show how the various elements of the cost of fuel contribute to the total price in various countries. The data she is using is shown below, with the prices in local currencies.
Sudaria Idani Anguda Boralia
Crude oil 0.70 18.68 0.40 0.50
Refining 0.02 4.67 0.02 0.02
Wholesale 0.09 3.63 0.05 0.14
Retail 0.06 2.08 0.06 0.05
Tax 0.50 22.84 0.80 0.34
Total 1.37 51.90 1.33 1.05
Activity
She drew one pie chart last night, but has not labelled the segments and cannot remember which country it represents. The pie chart is shown below.
Which country is it?
This could, in fact, have been solved without going to percentages by looking at the relative sizes of the components in the table for each country. It would have been quicker to do this, but would have taken more mental arithmetic.
Commentary
This question is a little more difficult than some we have seen so far. There are several ways to approach it. We can note that if we knew the actual averages for the four colleges the newspaper did include, it might be possible to see if these averages disagreed with an estimated average for the five colleges, and the direction of the error would give some
indication of which one was forgotten.
Looking at the ‘averages’, the approximate values (we have to estimate these from the graph) are 9, 19, 35 and 16 respectively for 1, 2, 3 and 4 A Levels. Multiplying these by 54 (to correct for the fact that they were divided by 5 instead of being divided by 4), we get
(approximately): 11, 24, 44 and 20.
If we were being very systematic, we could now compare these with all sets of four averages, but it would take a long time.
Instead, let us note that the 11 looks a little low for the average of 1 A Level, as does 24 for the average of 2. 44 for the average of 3 looks very low and 20 for the average of 4 looks far too high. From this, we may suspect that
Danbridge has been missed as it is higher than the others for 3 A Levels and lower for 4.
We can check this by averaging one of the columns for the other four colleges (preferably use 3 or 4 A Levels as they look to have the biggest discrepancy) and comparing the results – try this for yourself and see whether you can confirm that Danbridge is the college whose results are missing.
The table shows the results of a questionnaire, asking the five colleges in a town the proportion of students taking 1–4 A Level subjects.
Percentage of students taking number of A Levels shown
College 1 2 3 4
Abbey Road 13 25 42 20
Barnfield 5 18 55 22
Colegate 24 36 28 12
Danbridge 16 18 61 5
Eden House 10 14 48 28
The local newspaper (forgetting that there might be different total numbers of students in the five colleges) just added the numbers together and divided by five to produce a percentage graph for the town as a whole. However, they forgot to add in the data for one college so their percentages did not add up to 100.
30
20
10
0 1 2 3
Number of A Levels
Percentage of students
4
Which one did they forget?
Activity
Summary
• We have learned how data may be
represented in more than one way and the importance of systematic comparisons between two sets of data in ascertaining that they are the same.
• We saw that reading graphs and tables carefully is necessary in order not to make errors in identifying similarities.
played? (Hint: you first need to decide which games have already been played, so you know what is left.)
A B C D E
Britons 5 3 5 5 2
Danes 5 4 7 5 5
Normans 3 4 3 4 6
Saxons 2 4 1 1 2
4 The graph shows the charges made by a printing company for making various numbers of posters.
1 2 3 4 5 6 7
0 50 100 150 200 250
Number of posters
Total cost ($)
Which of the following pricing structures would give the graph shown?
A $30 per poster
B $50 set-up charge + $20 per poster C $40 per poster for the first four, any
extra $20 each
D $30 set-up charge, $30 per poster for the first four, any extra $20 each Draw the graphs for the other price
structures.
Answers and comments are on page 319.
1 Look in newspapers (business pages are often useful) or on the internet to find examples of numerical data in various forms (verbal, graphical, tabular). Express the data in a different form. Consider which form makes the data clearest to understand.
2 Four-digit personal identification numbers (PINs) are used to withdraw cash from banks’ machines using plastic cards. It can be very difficult to remember your personal number. I have a method of remembering mine. It is the two numbers of my birth date (i.e. the date in the month) reversed, followed by the two digits of my month of birth reversed (using a zero in front if either is a single number so, for example, May would be 05).
Which of the following could not be my PIN?
A 3221 B 5060 C 1141 D 2121 E 1290
3 Four house teams play each other in a school basketball league. The scoring system gives three points for a win, one for a draw and none for losing.
They all play each other once, and the league table before the last round of matches is as follows:
Played Won Drawn Lost Points
Britons 2 0 2 0 2
Danes 2 1 1 0 4
Normans 2 1 0 1 3
Saxons 2 0 1 1 1
Which of the following points columns are possible after the last two matches are
End-of-chapter assignments
Commentary
Looking at the statements in turn:
A This statement explains the initial increase in growth rate – the increase looks exponential (increasing in size at a constantly growing rate).
B This statement would not explain the initial growth – it would start at a higher growth rate, which would then decrease all the time.
C This statement would explain the drop to zero growth after a time, linked to a lack of nutrient.
D There is no indication of death; in that case the population would fall.
E If both processes were linear (resulting in straight-line graphs), a combination of them would also result in a straight-line graph.