Statistics in Everyday LifeAuthor(s): J. GaniSource: Mathematics in School, Vol. 2, No. 5 (Sep., 1973), pp. 2-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211066 .

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by Professor J. Gani, Manchester-Sheffield School of Probability and Statistics

1. INTRODUCTION: Statistics is all around us A few years ago, I was asked to make a short Television film to introduce Statistics to schoolchildren. Part of the purpose of the film was to inform young people of the careers they could make for themselves in this developing field.

I had always felt that we lived in a world which surrounded us with Statistics in one form or another. But in order to convince myself that this feeling was not entirely subjective, I picked up The Times news- paper on the morning of 10 February 1968 and began to read it. On page 4 there was an article entitled "Politicians feud while garbage stays". This described the dustmen's strike in New York which had started on 1 February that year. The article was full of Statistics. It stated that the 10,000 dustmen wanted another 425 dollars a year on top of their current wage scale of 6,424 to 7,956 dollars; it went on to discuss the 400 dollar increase a year which had been offered to the Union as against the annual increase of 600 dollars which it demanded.

Such figures constitute raw statistical data. In this case, we would refer to them as Social Statistics, because they provide numerical information of a social nature on a certain group of workers in American society.

Looking through the rest of the paper I found on page 3 an article on "Mr. Powell urges immigrant curb", which gave a variety of figures on the numbers of immi- grants entering Britain at the time. On page 4 (of the early edition of The Times sold in Sheffield), there was an article on Russian doubts on the affluent society, which again gave a large number of figures on produc- tion and consumption in the U.S.S.R. The average annual meat consumption in the Soviet Union was given as 90 pounds a person, as against 237 pounds a person in the United States. An American, it was reported, ate 304 eggs a year, while a Russian consumed only 124. These were more Social Statistics, though they would qualify more appropriately as gastronomic statistics.

On page 6, in the sports news, there was an article on Cowdrey hitting a century, and the score card gave

the individual runs scored by the players. Perhaps these could be called sporting statistics.

On page 7, Parliament was reported to have voted by 159 to 63 votes for the divorce law based on the break- down of marriage. Once again these were Social Statistics. And on page 10, there was a list of the latest wills, which I shall use later in an example on statistical procedures.

There was a great deal more in the paper: I found, for example, two articles which involved Operational Research. This is the name given to that method of solving industrial, business and other practical problems which makes use of Mathematics and Statistics. The first article, on page 1, was on cars and their effect on London transport. The second in The Times Diary column, concerned the popularity of various news- papers in a poll carried out at Balliol College, Oxford. As the writer of this column noted with gratification, The Times topped the vote! I shall later use these poll results to illustrate some Statistical Methods.

The Weather Forecast, listing temperatures in approximately 40 cities in Europe, provided a further example of what might be called general statistics. There were 4 more articles in Actuarial and Business Statistics. Actuaries are the mathematically and statis- tically trained persons who decide on the premiums to be paid to insurance companies by persons requiring life, motor car, or other types of insurance. Any statistical data used by them is referred to as actuarial statistics. One of the articles in the paper was on car insurance, another on banks and their computer plans, and another on the latest company dividerids. Finally, there were lists of the Stock Exchange closing prices.

I was frankly surprised to find how much Statistics was contained in a single issue of The Times newspaper. I reminded myself with some scepticism that every expert likes to think that his speciality is what makes the world go round. However, the fact that in a single copy of The Times, there were 13 items of statistical information, 6 in Social Statistics, 3 in General Statis- tics and Operational Research, and 4 in Actuarial and Business Statistics, confirmed my belief that Statistics must be regarded as an integral part of everyday life. This was the main point I was to make in my Television film.

Thursday July 12 1973

No 58,832 Price 5p

Gobvernment resists Labour move to

The Gov eruet lsS nigh MarSicr' YMgY

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What I shall try to illustrate in the rest of this article is the uses to which Statistics can be put; I shall also try to outline some Elements of Statistical Method.

2. GRAPHICAL REPRESENTATION of Statistics Statistics does not consist of just any collection of figures. One has to set them out properly and analyse them in such a way that they make sense. One of the simplest ways of making sense of numerical data is to try to draw pictures of them. This is what we mean by the graphical representation of Statistics.

Suppose that we consider the item on wills which appeared on page 10 of The Times. Reading through the article, one would note for example, that a Mr. Joseph left a8,545 net; Mrs. Gustard left a98,525; Mrs. Maguire left a55,944; Mr. Milne left a148,408; Mr. Richardson left a48,439; Mr. Wilson left a58,140; and Miss Hendry a561,202.

So far, all we have is a rather uninspiring list of figures. But suppose we now try to make a graph of these, as an illustration of the amounts of money left in their wills by people dying in Britain. If we draw an ordinary frequency diagram, as we call it, we would have to put down that the number of people leaving wills between a0 and a10,000 was just 1, between 40 and 50 thousand again 1, between 50 and 60 thousand 2, between 90 and 100 thousand 1, between 140 and 150 thousand another 1, and finally between 560 and 570 thousand, 1 again.

2 2 a)

07 a, IL U-

Q 10 40 50 60 90 100 140 150

Value of wills in a1,ooo000s

Figure 1I

I 560 570

The frequency diagram looks rather scattered, and does not really give us a particularly striking represen- tation of the wealth of the people who died. If we think more about the matter, we may decide that the scale for the money is wrongly represented, and that we should perhaps graph it on a modified scale which bunches together all values between a0 and a10,000, all values between a10,000 and a100,000, and all values between a100,000 and a million pounds. This means that one unit of scale will be used for values

between 0 and 10 thousand, another unit for values between 10 and 102 thousand, and a third unit for values between 102 and 103 thousand. In this case we obtain a second, much more compact, frequency diagram, and this gives us a much simpler and more pleasing summary of the wills than the one we had previously drawn.

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3

2

1

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Figure 2

10 102 103 Value of wills in a1,000s

It would appear from this, that there were 4 times as many people leaving between a10,000 and a100,000, and twice as many people leaving between a100,000 and a million pounds as there were leaving wills between a0 and a10,000.

Of course, the information we have obtained comes from only a small sample, that is, a small selection from the population. One would have to be very careful about drawing conclusions from this for the whole population. We can, however, carry out some elemen- tary calculations on our data. For example, suppose we try to find the average amount left in a will. This can be done quite simply by adding up the previous figures:

8,545 98,525 55,944

148,408 48,439 58,140

561,202

979,203

Having done this, we find that the sum total of the amount left was a979,203. If we divide this by the 7

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people who left the wills, this indicates that the average amount left was a139,886 per person, since

139,886" 7) 979,203

Now, although Figure 2 is a much rougher though more compact representation of the same information, we can obtain from it another kind of average. Because we shall now group our data into class intervals, we may call this average a grouped average; it will be an approximation to the exact result we have previously obtained. Suppose that we decide to take all wills in the class interval between a0 and a10,000 as having a value of a5,000, this being the average of 0 and 10,000. All those in the class interval between a10,000 and a100,000 will be assumed to have a value of a55,000, this being the average of 10 and 100 thousand. Finally, all those between a100,000 and a million pounds will be taken to have a value of a550,000, this again being the average between 100,000 and a million. If we take the average on the basis of this condensed frequency diagram we find that we have one person at 5,000, 4 at 55,000, 2 at 550,000, and if we take the average in this case we obtain a value of a189,286:

1 x 5,000 = 5,000 4 x 55,000 = 220,000 2 x 550,000 = 1,100,000

Total 1,325,000

189, 285- 7) 1,325,000

We know that this method is not very accurate because of the way we have grouped wills into wide class intervals. But you will see that it is very much quicker than the first method, and we finally obtain a value which is only about a50,000 different. Although this may appear a large sum, in terms of the sample we have been dealing with, the grouped average is not vastly different from the exact average. You will note from this that a problem of judgement is involved in deciding on suitable class intervals for observed data.

What have we done so far? We have learnt how to draw a frequency diagram, that is, a picture to sum- marise as clearly as possible the information we have collected. We have obtained an exact average, and from the frequency diagram, we have also been able to find what we have called a grouped average.

3. An example of MARKET RESEARCH So much for Social Statistics. Let us now turn to a problem of Operational Research; this is the term used to describe any investigation into business, industrial or practical problems, which makes use of Mathe- matics and Statistics. An example of such an investiga- tion is Market Research, where numerical data is obtained about the response of customers to particular questions on political or social issues or commercial products. One of the most interesting forms of Market Research is the political vote, where the voter indicates by his decision his approval or disapproval of the policies of contending candidates.

It would appear that early in 1968, Mr. Christopher Dawkins, a Balliol undergraduate, was asked to select the reading matter for the College Junior Common Room. In order to arrive at a fair estimate of student reading tastes, he organised a poll of newspapers and periodicals among the undergraduates at Balliol College, Oxford, and obtained the following list of figures.

The Times came first with 204 votes, and was followed by other papers as follows:

4

The Times Punch Sunday Times Private Eye The Guardian The Observer Cherwell Oxford Mail Daily Mirror New Statesman Paris Match Isis Which The Listener The Economist Daily Mail Daily Telegraph Times Literary Supplement Playboy News of the World Total

204 199 190 179 178 166 156 148

137 134 131 126 123 123 122 117 117 114 114 106

2,884

The information as it stands, indicates which 20 newspapers topped the poll, but it gives very little idea of the way in which the readers were distributed. If once again we attempt to draw a frequency diagram, we shall obtain Figure 3 with a hump at the beginning, and a long tail at the end. This indicates that there was one newspaper with between 100 and 110 readers; 4 with between 110 and 120 readers, and so on right up to a single paper with between 200 and 210 readers, namely The Times.

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Figure 3

5'0 1o0 10 200 Number of readers

This diagram summarises the information available much more clearly than a list of the number of readers for each newspaper. Once again we can use the diagram to obtain the grouped average of readers of a news- paper, and we can compare it with the average which we would have worked out by simply adding the total column of readers previously presented.

The exact average obtained from the column is 144.2, while the grouped average obtained from the frequency diagram is 143.5. You will see that since we have a great deal more information this time than in the case of the wills, we are able to get a much closer agreement between these two types of average. Since the grouped average is quicker to find, one usually obtains it in preference to the other, in the following manner:

1 x 105 = 105 4 x 115 = 460 4 x 125 = 500 3 x 135 = 405 1 x 145 = 145 1 x 155 = 155 1 x 165 = 165 2 x 175 = 350 1 x 185 = 185 1 x 195 = 195 1 x 205 = 205

Total 2,870

143.5

20 )2,870

4. Caution in BUSINESS STATISTICS Let me now turn to my last example in Business Statistics; here I want to indicate how diagrams can sometimes mislead as well as assist you. An article on page 11 of The Times discusses banks and computer This content downloaded from 86.132.102.195 on Tue, 22 Apr 2014 13:20:04 PM

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plans, and gives a set of figures for Barclays, Lloyds and Martins banks and their computerisation. As you know, electronic computers are able to store a great deal of numerical and other information, and carry out a large number of arithmetical operations at immense speeds. Banks are now replacing ledger methods of storing bank account information by computer methods. Computers then carry out such operations as the work- ing out of interest, or the transfer of sums between stored accounts, much more rapidly than could pre- viously be done.

We are told in the article that in 1968 Barclays had 3,200,000 accounts, of which 700,000 were com- puterised. Lloyds had 2,500,000 accounts, of which a million were computerised. Also, that in the a5,000 million merger then proposed by Barclays, Lloyds and Martins banks, approximately a100 million would be spent on computerisation by 1970-71.

Figure 4

0

Barclays: 3,200,000

::::::::::::::::::::::::::::::::::::: jii::::::: ::::: :::::: ::::: ::::: :::::: :::: : ::::

Lloyds: 2,500,000

One way in which one can represent such informa- tion is by so-called pie-diagrams; if we draw a pie of a certain size to indicate the total number of accounts at Barclays, and a smaller pie to indicate the smaller number of total accounts at Lloyds, we can see fairly rapidly that the proportion of computerised accounts at Barclays is smaller than the proportion of computer- ised accounts at Lloyds.

Now one would be tempted to argue that the position is that Barclays is perhaps less computerised than Lloyds. But, in fact, a moment's thought would indicate that the evidence is not sufficient to argue in this manner. In order to have full knowledge of the position, one would need to know the exact size of each computerised account at Lloyds and at Barclays, as well as the number of transactions involved. After all, it is possible that the 700,000 computerised accounts at Barclays were larger accounts, and that there were many more transactions connected with them than the accounts at Lloyds. We simply do not know, and this is a point to remember when presented with Statistics. It is precisely this possible inaccuracy of representation which has given rise to the old unmerited adage about "lies, damned lies, and Statistics"; to avoid such errors in judgement, one must reason carefully and use great caution in drawing one's conclusions.

The development of Statistics rests heavily on a mathematical discipline known as Probability Theory. I cannot spend too much time on this as the subject, though fascinating, can become quite complicated. All that needs to be said is that Probability Theory began as an extension of games of chance, such as dicing. The sort of question that a gambler might ask himself is "If I throw a die 20 times, what is the chance that I will obtain 2 sixes?" Probability Theory is able to pro- vide answers to such questions, and in this case the answer is 20)(1)2(518

2 6) =0.19

On the basis of Probability Theory it has been possible to erect an imposing structure of statistical thought. This enables us to answer all sorts of practical questions; for example, does a particular observation obtained from a sample differ significantly from what might have been expected?

5. Opportunities for STATISTICIANS I should like to conclude this article with a brief account of job opportunities for trained statisticians. To train as a statistician at a British University, it is usually necessary to have completed one's matricula- tion with two 'A' levels in mathematical subjects of suitably high standards. Many universities in Britain have excellent Statistics Departments, and after 3 years' training in one of these, a student can obtain a Bachelor's degree in Mathematics with Statistics, or in some cases, even a specialised Bachelor's degree in Statistics.

What sort of jobs are open to a graduate in Statistics? He may become an actuary; that is, a person who determines the premiums for various types of insurance policies. We all know that when a person wishes to take out life insurance, he must pay certain premiums to an insurance company against the possibility of his death. If he wishes to buy car insurance, he must pay a certain premium against the possibility of car accidents. It is the job of the actuary to work out the size of the premiums to ensure that the customer gets a fair deal, while also avoiding the danger that the insurance com- pany goes bankrupt.

A second type of job for the statistician is in Operational Research in an industrial or business con- cern. In a business concern, the Operational Research specialist may have to deal with such problems as stock control. For example, he might have to decide on the best policy of ordering replacements for certain parts needed by the business. He may also have to decide on policies of shipment and delivery of products through- out the country. Usually, statisticians working in Operational Research may, at a later stage in their careers, become managers.

Another job that a trained statistician can undertake is with the Central Statistical Office. This Office collects all sorts of Statistics about people, trade, and production in Britain; it issues detailed statistical reports at regular intervals. This is a rapidly developing area of work, which holds a great future for those who wish to enter government service.

Some statistical graduates may wish to become teachers of statistics in schools or mathematical statisticians working in universities. There are a number of opportunities for graduates in mathematics and statistics in schools, where courses in statistics are becoming increasingly popular with schoolchildren. At universities, statisticians will give instruction to under- graduate and graduate students, carry out statistical research, and often consult on scientific problems both within the university and with industrial concerns outside. Those wishing to pursue an academic career of this type need further training, and must usually take their Master's and Doctor's degrees, after completing their first Bachelor's degree.

This brief survey of possible jobs should be enough to indicate the scope, interest, and variety of the work which can be done by statisticians after they have completed their training.

Though I have barely skimmed the top of the sub-

ject, I hope I have succeeded in interesting you in it. If the subject has any appeal to you, you may wish to obtain further information on it in the references listed below.

1. Careers in Statistics-American Statistical Association, 810 18th Street NW, Washington DC 20006, USA. 2. Statistics-The Field, by W. Kruskal, International Encyclopedia of the Social Sciences XV, pp. 206-224 (1968) Crowell-Collier and Macmillan. 3. Mathematics in Actuarial Work-Bull. Inst. Math. Applics. 8 No. 1 (1972).

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