Brief Description

Pupils are introduced to various abuses of statistics, including errors in representation and presentation and errors of omission.

Design time: About 4 hours

Aims and Objectives

On completing this unit pupils should be able to criticize constructively any of the more common misuses of data in advertising or argument. Specically, they should be able to identify bar charts using bars of different wid ths, the omission o f scales, the use of a false origin, percentages calculated on the wrong base, the use of meaningless phrases (such as 'up to 20%'), the use of exaggerated scales, false prediction lines, and true statements selected only because they support an argument. They should be aware of the effect of these abuses on the general impression given. They should know the true measures of 'average' and the differences between them.

They will have practised drawing bar charts, drawing graphs of time series, making up their own misleading representations, calculating a median and a mean, reading tables, and using data in an argument.

They will be more aware of the ways statistics can be used and abused, of the need to look back at the original data, of the need to look at all the appropriate data, of the effect of variability and of the fact that many items may be better than the average, but they are not necessarily the best.

Prerequisites

Pupils need to be able to draw a bar chart, plot points on a graph and be able to calculate a simple mean and find the mode.

Equipment and Planning

It is most desirable that, before beginning this unit, pupils should collect examples of advertisements from newspapers, magazines and television that use statistical terms or ideas. This collection then forms the basis of the initial discussion and some of the later work. Graph paper is needed, but no other extra material is essential.

Section A introduces some of the better-known techniques of misleading statistical representation. Section B looks briefly at some of the problems connected with the word 'average'. Section C shows how even the truth can be misleading if only selected truths are quoted. It is set in the context of a debate on extending Birmingham Airport; an alternative in these teachers' notes uses immigration as the subject under discussion.

Detailed Notes

'There are lies, damned lies and statistics'. 'You can prove anything with statistics'. 'He uses statistics like a drunkard uses a lamp-post: for support rather than illumination.'

These are just a few quotations warning of the dangers of abuses in statistics. It is highly desirable that the general citizen be aware of these limitations. For often the misrepresentations are not sophisticated, yet they can still be damaging and misleading. It is hoped that a careful examination of a few commoner abuses will lead pupils to a greater critical awareness of dealing with statistical data.

Below is presented a list of different types of abuses of statistics and some general principles in helping to spot them. It is not an exhaustive list. Some of these could be investigated through suitable examples from newspapers, magazines or television.

It would be worthwhile giving the pupils a list of these abuses and discussing them at the end of the unit.

Abuses

1. Eye compares volumes or areas rather than lengths in diagrams (A1, B4).
2. Different size pie charts can distort differences unfairly.
3. Spurious accuracy (8.387% of people, etc.)
4. Quoting figures out of context (A1, B2, C1).
5. Lack of scale or curtailed scale (A1).
6. Prediction and difficulties entailed (A1, C1).
7. Using the particular 'average' which sounds best (B2, B4).
8. Bad or misleading questions on a questionnaire.
9. Self-select sample answering a questionnaire.
10. Inferring individual results from an average, and ignoring spread (B4).
11. Spurious or irrelevant connections (A1, B5, C1).
12. Quoting meaningless or irrelevant figures (A1, C1).
13. Selecting only the data which support an argument (B2, C1).

Almost any statistical technique can be abused.

The sort of questions one should ask when dealing with possible abuses are: Who says so? On what evidence? What is missing? Does it make sense?

Section A

It is valuable and instructive for pupils to make up a scrapbook or collage of advertisements which misuse statistics. Advertisements are designed to sell, and the temptation to overstate one's case is often very strong. It is interesting to see how often it is difficult to find the price of a product. The initial discussion in a is to set the scene for a more detailed study in the next sections. Try to get the pupils to identify themselves with the detectives of Fact Ferrets.

A1
The widening of the bars and their shading should be apparent to the pupils in a. They may also notice the fat happy cat and the thin miserable cat. The possibility of using a false zero, and its effect, is seen in part 2 of Figure 2, and this is perhaps the most misleading representation. Since the bars are different in part 1 it follows that the answer to c is 3. An underlying assumption is that 'more means better'; it is possible to have too much of some of these ingrediants.

A2
The 'save 50%' is a common error. The saving should be based on the original price not the sale price, as is brought out in a and b. 'Up to 20% extra' could, of course, include 'down to 100% less'. There is no indication what the 20% is of. Does it mean that the tin of Catchunks weighing 480g contains 20% more goodness than the one weighing 400g, or is it making a comparison with some other cat food? This supposes that we know what is meant by 'goodness', and this is followed up in A4.

A3
Questions a, b and c show how the graph has a false zero and an exaggerated scale. The figures on the vertical axis have then been omitted to give a distorted picture of a fairly minimal increase in sales.

A4
The points made here are more subtle, and pupils may benefit from a class discussion before answering the questions.

A5
This section gives pupils a chance to put into practice the faults they have seen in the previous sections and criticize (ferret out) the attempts of their friends. This could be done for homework. To draw together all the points from Section A it is suggested that pupils should in c look again at all the collected advertisements.

Section B

This section deals mainly with various abuses of the term 'average', either by not indicating whether it is the mean, the median or the mode that is being quoted, or by quoting an 'average' value when it is the variability that is more important. B4 is optional and looks at some fallacies connected with proportions. In the text the fuller phrase 'arithmetic mean' has been used. In practice this is often just abbreviated to 'mean', especially when there is no danger of confusion with other 'means', such as the harmonic mean and the geometric mean.

B1
Pupils may need some guidance in drawing the bar chart (or vertical line chart). A linear scale from £0 to £500 should be used on the horizontal axis to show the distribution, with the large gap between £100 and £490.

B2
With a skewed distribution like this the mean is affected by the single payment of £490, and the mode takes no account of figures above the lowest category of £50. Putting the wages in order gives the median £60, the wage of the 13th person. If a single figure has to be given for wages, the median is the least misleading here.

B3
The first three statements all make the point that a single figure is sometimes useless, and that the distribution as a whole is important. This point is also made in the fourth statement, with house prices ranging so widely and being different in different parts of the country. The illustration to the fourth statement also shows another form of visual misrepresentation. Does the eye see ratios of lengths, areas or volumes? The ratio of the lengths is 2:1, areas 4:1 and volumes 8:1.

*B4
These inferences are incorrect because of the proportions involved. More people drive in the daytime so, of course, there are more accidents. Suppose, for example, in statement 3, that 95% of drivers wear seat belts (it is compulsory by law) and that 1 driver in 100 is injured. Then out of every 10000 drivers there will be 100 injured, and 90 of these will be wearing seat belts. The figures are shown in Table T1.

Table - T1 Seat belts and injuries (fictitious)

From this it will be seen that the probability of injury when wearing a seat belt is ⁹⁰/₉₅₀₀ = 0.009, whereas the probability of injury when not wearing a seat belt is ¹⁰/₅₀₀ = 0.02, which is much higher;

This method is explained more fully in the Level Four unit Testing Testing.

Section C

This is an important section because statistics are often misquoted. Errors through omission or careful selection are hardest to spot, yet unfortunately are widely prevalent. It is relatively easy to make emotive and biased speeches as quoted. It is much harder to make a clear analysis, and this is not usually as interesting to listen to or read.

The local resident has carefully chosen his years. 1974 was the first year for many years that passenger figures dropped. The aircraft figures in 1973 were higher than neighbouring years, and those in 1958 were lower than neighbouring years, so the ratio is artificially high. It is arguable whether modern aircraft all make more noise than older ones. The airport spokesman chooses his figures just as carefully to achieve the opposite effect.

Table T2 gives the full version of Table 6 in the pupils' notes.

Some other points that may come up in discussion are as follows.

1. Aircraft are larger and therefore carry more passengers.
2. Therefore the number of passengers can increase somewhat independently of aircraft numbers.
3. As aircraft are larger, longer runways may well be necessary.
4. More passenger facilities will be needed as numbers increase.
5. Existing facilities should be taken into account.
6. Some of the aircraft are privately or company owned and therefore carry small numbers of passengers.
7. Other aircraft just stop to refuel. Some aircraft carry freight (see Table T2, the full version of Table 6).
8. Access to the airport has been improved (extra bus routes, the international railway station)
9. National Exhibition Centre: this will attract passengers who would previously have used Heathrow or Gatwick.
10. Bearing point 10in mind, increased international business would seem to be the result of the NEC development leading to more air traffic at Birmingham Airport.
11. Larger aircraft can be noiser.

Table T2 - Traffic statistics, 1955-1977
(Source: Birmingham Airport Handbook 1976, 1977, 1978)

In the original tested version of this unit there was a discussion on immigration, with a prejudiced speaker using selected statistics to make his point. In areas where race was not a sensitive issue this was well received, you may like to use it for reinforcement of the ideas of Section C.

Table T3 gives the migration figures for 1966, 1972 and 1974.

Table T3 - Migration to and from Great Britain (thousands per year)
(Source: Population Trends)
¹New Commonwealth excludes Australia, New Zealand and Canada.

The pupils can then be asked the following questions.

1. How many people left Britain in 1972?
2. How many people entered Britain in 1974?
3. In 1966 were there more immigrants to Britain than emigrants from Britain?
4. By how many did the population change? Which way?
5. Describe the trend in immigration.

After being told the definitions that an immigrant is someone who comes to Britain to settle and an emigrant is someone who leaves Britain to settle elsewhere they can then be shown this emotive speech.

'In 1966 over 75000 black immigrants came to settle in England. This swelled to a massive 84000 in 1972. At this rate over 100000 will come each year in the 1980s. We cannot continue to absorb such large numbers. Our country is getting more and more crowded. We must stop immigration now, let us make England great again.'

They can then be asked:

6. Take each of the sentences in the speech, your answers to a to e, and the figures in the Table T3 to criticize the speech. Is the speech fair?

Points that might be made on each of the sentences in the speech are:

Sentence 1
Not all people coming from the New Commonwealth are black: some are British returning home. In 1966 42000 people went out to those countries. There have always been more white immigrants than black immigrants into Britain.

Sentence 2
In 1972 General Amin declared 'Uganda for Ugandans'. About 26000 people migrated to Britain because of this. So the figure for 1972 is higher for special reasons. 84000 is small compared with Britain's population of 54000000.

Sentence 3
In fact the trend of immigration from the New Commonwealth has been 'downwards. The figure dropped to 52600 in 1973 and 51500 in 1974. There is no evidence to support the figure of 100000.

Sentences 4 and 5
For rnany years more people have left Britain than have come here to settle. There have been more emigrants than immigrants. Britain is not getting more crowded because of migration.

Sentence 6
This links immigration with 'greatness' without explanation. There is no evidence given of any connection.

It might also be worth emphasizing that matters are rarely as simple as they appear. Britons are hardly the ones to criticize migration. The following points could be made as background.

1. A large proportion, if not a majority, of Americans, Canadians, Australians, New Zealanders, etc., are descendants of Britons. Is Great Britain their home? Could they all return if they wished to?
2. African Asians
   1. These were taken to Africa as part of the development of the British Empire and have been there for several generations.
   2. India has no more obligation to them than the UK has to Americans wishing to migrate here. It is a poorer country than the UK with a major population problem. In fact, however, India did accept large numbers of refugees.
   3. An obligation to these people had been entered into by succeeding British Governments. But it is fair to say that at the time it was not realized that the numbers wishing to migrate would be so large.
3. West Indians and Moslem Asians

   Many of these were drawn here by British organizations, with Government agreement, to help run the London Underground, railways and hospital services, to do jobs that, at the wages offered at that time, Britons were not interested in doing.

4. Immigration of other nationalities to run restaurants, etc. Fish and chips are no longer the only take-away food available.

Pupils can then be asked to write a fair speech on immigration.

Answers

A1	a	See detailed notes.
	b	Probably (ii)
	c	See detailed notes
A2	a	75p, 50p, 25p, 1/3
	b	331/3%
	c	See detailed notes.
A3	c	Similar to that in b
	d	That in a
	e	See detailed notes.
A4		See detailed notes.
B1	a	Last column of table is (£550), (£360), £400, £200, £490. Total £2000
	b	25
B2	a	The mean and the mode
	b	80, 60, 50
	c	£490 - £50 = £440
	e	The median
B3	a	See detailed notes.
*B4	a	See detailed notes.
C	a	See detailed notes.
	b	See detailed notes.
	c	1113745 - 572172 = 541573
	d	66084 - 54368 = 11716
	e	See detailed notes.

Test Questions

1. An advertisement for Bark, 'the dog food made from trees,' says:
   
   The true figures (in standard units) are shown in Table 1.
   

   	Protein	Vitamins	Calcium
   Bark	24	20	21
   Average dog food	23	18	20

   Table 1 - Bark and the average of other dog foods

   1. Write down four ways in which the advertisement is misleading.
   2. Draw a fair bar chart showing the figures of Table l.
2. Bark also advertises:
   
   What is the true saving?
3. Bark's sales for the years 1976 to 1980 are shown in Table 2.
   

   Year	1976	1977	1978	1979	1980
   Sales (thousands of tins)	530	533	534	540	550

   Table 2 - Sales of Bark

   Draw a misleading graph of these figures so that Bark's sales appear to be rising rapidly.

4. On average it costs £20 to repair my motorbike. I have £25 so I can afford to have it repaired'.
   What is wrong with this argument?
5. Table 3 shows, to the nearest thousand, the number of votes cast at three successive elections for the candidates of the three main political parties in one constituency.
   

   Party	1st election		2nd election		3rd election
   	Votes	% of those voting	Votes	% of those voting	Votes	% of those voting
   Conservatives	8000	40%	10000	33%	12000	30%
   Labour	10000	50%	15000	50%	18000	45%
   Liberal	2000	10%	5000	17%	10000	25%
   Total votes	20000		30000		40000

   Table 3
   Total number of people eligible to vote: 10000

   1. Write down: a Two facts that the Conservatives could state to show they were doing well.
   2. Two facts that Labour supporters could state to show they were doing well.
   3. Two facts that Liberal supporters could state to show they were doing well.

Answers

1	a	Bars different width; no numbers on vertical axis; vertical scale does not start from zero; better than average does not mean better than all; choosing three particular contents, why not others?
2		50%
4		Does not allow much for variability, which is not quoted. Average £20 could give a single bill well over £25.
5		There are many answers. Some possibilities are given below:
	a	Every election we gained more votes. The percentage of people voting Labour has been dropping. Liberals have never been able to get more than a quarter of the votes.
	b	We have won all the elections. Every time we get increasingly more votes than the Conservatives. The percentage of people voting Conservative is dropping steadily.
	c	Every time we have had an increase in the number of our voters. We have gained a bigger percentage of the votes cast at each election. Conservatives and Labour are each getting a decreasing percentage of the votes at each election. We had five times as many votes in the third election as we did in the first.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 3)

Car Careers
Net Catch
Multiplying People
Pupil Poll
Cutting it Fine

Units at Other Levels in the Same or Allied Areas of the Curriculum

Level 1

Leisure for Pleasure

Level 2

Authors Anonymous
On the Ball
Seeing is Believing

Level 4

Figuring the Future
Sampling the Census

This unit is particularly relevant to: General Knowledge, Humanities, Mathematics, Commerce.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

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