| Year | SRP |
|---|---|
| 1974 | 26.2 |
| 1975 | 22.8 |
| 1976 | 37.2 |
| 1983 | 54.7 |
| 1984 | 37.7 |
| 1987 | 54.3 |
| 1989 | 35.7 |
| 1991 | 72.0 |
| 1992 | 85.1 |
| 1993 | 86.7 |
| 1994 | 93.3 |
| 1995 | 107.2 |
| 1996 | 80.3 |
| 1997 | 70.7 |
1 Introduction
1.1 Uses of Probability and Statistics
This chapter introduces you to probability and statistics. First come examples of the kinds of practical problems that this knowledge can solve for us. One reason that the term “statistic” often scares and confuses people is that the term has several sorts of meanings. We discuss the meanings of “statistics” in the section “Types of statistics”. Then comes a discussion on the relationship of probabilities to decisions. Following this we talk about the limitations of probability and statistics. And last is a discussion of why statistics can be such a difficult subject. Most important, this chapter describes the types of problems the book will tackle.
At the foundation of sound decision-making lies the ability to make accurate estimates of the probabilities of future events. Probabilistic problems confront everyone — a company owner considering whether to expand their business, to the scientist testing a vaccine, to the individual deciding whether to buy insurance.
1.2 What kinds of problems shall we solve?
These are some examples of the kinds of problems that we can handle with the methods described in this book:
You are a doctor trying to develop a treatment for Covid. Currently you are working on a medicine labeled AntiAnyVir. You have data from patients to whom medicine AntiAnyVir was given. You want to judge on the basis of those results whether AntiAnyVir really improves survival or whether it is no better than a sugar pill.
You are the campaign manager for the Republicrat candidate for President of the United States. You have the results from a recent poll taken in New Hampshire. You want to know the chance that your candidate would win in New Hampshire if the election were held today.
You are the manager and part owner of one of several contractors providing ambulances to a hospital. You own 16 ambulances. Based on past experience, the chance that any one ambulance will be unfit for service on any given day is about one in ten. You want to know the chance on a particular day — tomorrow — that three or more of them will be out of action.
You are an environmental scientist monitoring levels of phosphorus pollution in a lake. The phosphorus levels have been fluctuated around a relatively low level until recently, but they have been higher in the last few years. Do these recent higher levels indicate some important change or can we put them down to ordinary variation from year to year?
The core of all these problems, and of the others that we will deal with in this book, is that you want to know the “chance” or “probability” — different words for the same idea — that some event will or will not happen, or that something is true or false. To put it another way, we want to answer questions about “What is the probability that…?”, given the body of information that you have in hand.
The question “What is the probability that…?” is usually not the ultimate question that interests us at a given moment.
Eventually, a person wants to use the estimated probability to help make a decision concerning some action one might take. These are the kinds of decisions, related to the questions about probability stated above, that ultimately we would like to make:
Should you (the researcher) advise doctors to prescribe medicine AntiAnyVir for Covid patients, or, should you (the researcher) continue to study AntiAnyVir before releasing it for use? A related matter: should you and other research workers feel sufficiently encouraged by the results of medicine AntiAnyVir so that you should continue research in this general direction rather than turning to some other promising line of research? These are just two of the possible decisions that might be influenced by the answer to the question about the probability that medicine AntiAnyVir is effective in treating Covid.
Should you advise the Republicrat presidential candidate to go to New Hampshire to campaign? If the poll tells you conclusively that she or he will not win in New Hampshire, you might decide that it is not worthwhile investing effort to campaign there. Similarly, if the poll tells you conclusively that they surely will win in New Hampshire, you probably would not want to campaign further there. But if the poll is not conclusive in one direction or the other, you might choose to invest the effort to campaign in New Hampshire. Analysis of the chances of winning in New Hampshire based on the poll data can help you make this decision sensibly.
Should your company buy more ambulances? Clearly the answer to this question is affected by the probability that a given number of your ambulances will be out of action on a given day. But of course this estimated probability will be only one part of the decision.
Should we search for new causes of phosphorus pollution as a result of the recent measurements from the lake? If the causes have not changed, and the recent higher values were just the result of ordinary variation, our search will end up wasting time and money that could have been better spent elsewhere.
The kinds of questions to which we wish to find probabilistic and statistical answers may be found throughout the social, biological and physical sciences; in business; in politics; in engineering; and in most other forms of human endeavor.
1.3 Types of statistics
The term statistics sometimes causes confusion and therefore needs explanation.
Statistics can mean two related things. It can refer to a certain sort of number — of which more below. Or it can refer to the field of inquiry that studies these numbers.
A statistic is a number that we can calculate from a larger collection of numbers we are interested in. For example, table Table 1.1 has some yearly measures of “soluble reactive phosphorus” (SRP) from Lough Erne — a lake in Ireland (Zhou, Gibson, and Foy 2000).
We may want to summarize this set of SRP measurements. For example, we could add up all the SRP values to give the total. We could also divide the total by the number of measurements, to give the average. Or we could measure the spread of the values by finding the minimum and the maximum — see table Table 1.2). All these numbers are descriptive statistics, because they are numbers that summarize and therefore describe the collection of SRP measurements.
| Descriptive statistics for SRP | |
|---|---|
| Total | 863.9 |
| Mean | 61.7 |
| Minimum | 22.8 |
| Maximum | 107.2 |
Descriptive statistics are nothing new to you; you have been using many of them all your life.
We can calculate other numbers that can be useful for drawing conclusions or inferences from a collection of numbers; these are inferential statistics. Inferential statistics are often probability values that give the answer to questions like “What are the chances that …”.
For example, imagine we suspect there was some environmental change in 1990. We see that the average SRP value before 1990 was 38.4 and the average SRP value after 1990 was 85. That gives us a difference in the average of 46.6. But, could this difference be due to chance fluctuations from year to year? Were we just unlucky in getting a few larger measurements in later years? We could use methods that you will see in this book to calculate a probability to answer that question. The probability value is an inferential statistic, because we can use it to draw an inference about the measures.
Inferential statistics use descriptive statistics as their input. Inferential statistics can be used for two purposes: to aid scientific understanding by estimating the probability that a statement is true or not, and to aid in making sound decisions by estimating which alternative among a range of possibilities is most desirable.
1.4 Probabilities and decisions
There are two differences between questions about probabilities and the ultimate decision problems:
Decision problems always involve evaluation of the consequences — that is, taking into account the benefits and the costs of the consequences — whereas pure questions about probabilities are estimated without evaluations of the consequences.
Decision problems often involve a complex combination of sets of probabilities and consequences, together with their evaluations. For example: In the case of the contractor’s ambulances, it is clear that there will be a monetary loss to the contractor if she makes a commitment to have 14 ambulances available for tomorrow and then cannot produce that many. Furthermore, the contractor must take into account the further consequence that there may be a loss of goodwill for the future if she fails to meet her obligations tomorrow — and then again there may not be any such loss; and if there is such loss of goodwill it might be a loss worth $10,000 or $20,000 or $30,000. Here the decision problem involves not only the probability that there will be fewer than 14 ambulances tomorrow but also the immediate monetary loss and the subsequent possible losses of goodwill, and the valuation of all these consequences.
Continuing with the decision concerning whether to do more research on medicine AntiAnyVir: If you do decide to continue research on AntiAnyVir, (a) you may, or (b) you may not, come up with an important general treatment for viral infections within, say, the next 3 years. If you do come up with such a general treatment, of course it will have very great social benefits. Furthermore, (c) if you decide not to do further research on AntiAnyVir now, you can direct your time and that of other people to research in other directions, with some chance that the other research will produce a less-general but nevertheless useful treatment for some relatively infrequent viral infections. Those three possibilities have different social benefits. The probability that medicine AntiAnyVir really has some benefit in treating Covid, as judged by your prior research, obviously will influence your decision on whether or not to do more research on medicine AntiAnyVir. But that judgment about the probability is only one part of the overall web of consequences and evaluations that must be taken into account when making your decision whether or not to do further research on medicine AntiAnyVir.
Why does this book limit itself to the specific probability questions when ultimately we are interested in decisions? A first reason is division of labor. The more general aspects of the decision-making process in the face of uncertainty are treated well in other books. This book’s special contribution is its new approach to the crucial process of estimating the chances that an event will occur.
Second, the specific elements of the overall decision-making process taught in this book belong to the interrelated subjects of probability theory and statistics. Though probabilistic and statistical theory ultimately is intended to be part of the general decision-making process, often only the estimation of probabilities is done systematically, and the rest of the decision-making process — for example, the decision whether or not to proceed with further research on medicine AntiAnyVir — is done in informal and unsystematic fashion. This is regrettable, but the fact that this is standard practice is an additional reason why the treatment of statistics and probability in this book is sufficiently complete.
A third reason that this book covers only statistics and not numerical reasoning about decisions is because most college and university statistics courses and books are limited to statistics.
1.5 Limitations of probability and statistics
Statistical testing is not equivalent to research, and research is not the same as statistical testing. Rather, statistical inference is a handmaiden of research, often but not always necessary in the research process.
A working knowledge of the basic ideas of statistics, especially the elements of probability, is unsurpassed in its general value to everyone in a modern society. Statistics and probability help clarify one’s thinking and improve one’s capacity to deal with practical problems and to understand the world. To be efficient, a social scientist or decision-maker is almost certain to need statistics and probability.
On the other hand, important research and top-notch decision-making have been done by people with absolutely no formal knowledge of statistics. And a limited study of statistics sometimes befuddles students into thinking that statistical principles are guides to research design and analysis. This mistaken belief only inhibits the exercise of sound research thinking. Alfred Kinsey long ago put it this way:
… no statistical treatment can put validity into generalizations which are based on data that were not reasonably accurate and complete to begin with. It is unfortunate that academic departments so often offer courses on the statistical manipulation of [data from human behavior] to students who have little understanding of the problems involved in securing the original data. … When training in these things replaces or at least precedes some of the college courses on the mathematical treatment of data, we shall come nearer to having a science of human behavior. (Kinsey, Pomeroy, and Martin 1948, p 35).
In much — even most — research in social and physical sciences, statistical testing is not necessary. Where there are large differences between different sorts of circumstances for example, if a new medicine cures 90 patients out of 100 and the old medicine cures only 10 patients out of 100 — we do not need refined statistical tests to tell us whether or not the new medicine really has an effect. And the best research is that which shows large differences, because it is the large effects that matter. If the researcher finds that s/he must use refined statistical tests to reveal whether there are differences, this sometimes means that the differences do not matter much.
To repeat, then, some or even much research — especially in the physical and biological sciences — does not need the kind of statistical manipulation that will be described in this book. But most decision problems do need the kind of probabilistic and statistical input that is described in this book.
Another matter: If the raw data are of poor quality, probabilistic and statistical manipulation cannot be very useful. In the example of the contractor and her ambulances, if the contractor’s estimate that a given ambulance has a one-in-ten chance of being unfit for service out-of-order on a given day is very inaccurate, then our calculation of the probability that three or more ambulances will be out of order on a given day will not be helpful, and may be misleading. To put it another way, one cannot make bread without flour, yeast, and water. And good raw data are the flour, yeast and water necessary to get an accurate estimate of a probability. The most refined statistical and probabilistic manipulations are useless if the input data are poor — the result of unrepresentative samples, uncontrolled experiments, inaccurate measurement, and the host of other ways that information gathering can go wrong. (See Simon and Burstein (1985) for a catalog of the obstacles to obtaining good data.) Therefore, we should constantly direct our attention to ensuring that the data upon which we base our calculations are the best it is possible to obtain.
1.6 Why is Statistics Such a Difficult Subject?
Why is statistics such a tough subject for so many people?
“Among mathematicians and statisticians who teach introductory statistics, there is a tendency to view students who are not skillful in mathematics as unintelligent,” say two of the authors of a popular introductory text (McCabe and McCabe 1989, p 2). As these authors imply, this view is out-and-out wrong; lack of general intelligence on the part of students is not the root of the problem.
Scan this book and you will find almost no formal mathematics. Yet nearly every student finds the subject very difficult — as difficult as anything taught at universities. The root of the difficulty is that the subject matter is extremely difficult. Let’s find out why.
It is easy to find out with high precision which movie is playing tonight at the local cinema; you can look it up on the web or call the cinema and ask. But consider by contrast how difficult it is to determine with accuracy:
- Whether we will save lives by recommending vitamin D supplements for the whole population as protection against viral infections. Some evidence suggests that low vitamin D levels predispose to more severe lung infections, and that taking supplements can help (Martineau et al. 2017). But, how certain can we be of the evidence? How safe are the supplements? Does the benefit, and the risk, differ by ethnicity?
- What will be the result of more than a hundred million Americans voting for president a month from now; the best attempt usually is a sample of 2000 people, selected in some fashion or another that is far from random, weeks before the election, asked questions that are by no means the same as the actual voting act, and so on;
- How men feel about women and vice versa.
The cleverest and wisest people have pondered for thousands of years how to obtain answers to questions like these, and made little progress. Dealing with uncertainty was completely outside the scope of the ancient philosophers. It was not until two or three hundred years ago that people began to make any progress at all on these sorts of questions, and it was only about one century ago that we began to have reasonably competent procedures — simply because the problems are inherently difficult. So it is no wonder that the body of these methods is difficult.
So: The bad news is that the subject is extremely difficult. The good news is that you — and that means you — can understand it with hard thinking, even if you have no mathematical background beyond arithmetic and you think that you have no mathematical capability. That’s because the difficulty lies in such matters as pin-pointing the right question, but not in any difficulties of mathematical manipulation.