Principles of Statistics

M. G. Bulmer

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486135205
• Topic: Matemáticas
• Subtopic: Probabilidad y estadística

There are many textbooks which describe current methods of statistical analysis, while neglecting related theory. There are equally many advanced textbooks which delve into the far reaches of statistical theory, while bypassing practical applications. But between these two approaches is an unfilled gap, in which theory and practice merge at an intermediate level. Professor M. G. Bulmer's Principles of Statistics, originally published in 1965, was created to fill that need. The new, corrected Dover edition of Principles of Statistics makes this invaluable mid-level text available once again for the classroom or for self-study.

Principles of Statistics was created primarily for the student of natural sciences, the social scientist, the undergraduate mathematics student, or anyone familiar with the basics of mathematical language. It assumes no previous knowledge of statistics or probability; nor is extensive mathematical knowledge necessary beyond a familiarity with the fundamentals of differential and integral calculus. (The calculus is used primarily for ease of notation; skill in the techniques of integration is not necessary in order to understand the text.)
Professor Bulmer devotes the first chapters to a concise, admirably clear description of basic terminology and fundamental statistical theory: abstract concepts of probability and their applications in dice games, Mendelian heredity, etc.; definitions and examples of discrete and continuous random variables; multivariate distributions and the descriptive tools used to delineate them; expected values; etc.
The book then moves quickly to more advanced levels, as Professor Bulmer describes important distributions (binomial, Poisson, exponential, normal, etc.), tests of significance, statistical inference, point estimation, regression, and correlation. Dozens of exercises and problems appear at the end of various chapters, with answers provided at the back of the book. Also included are a number of statistical tables and selected references.

CHAPTER 1

THE TWO CONCEPTS OF PROBABILITY

“But ‘glory’ doesn’t mean ‘a nice knock-down argument ’,” Alice objected.

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

Lewis Carroll: Through the Looking-glass

It is advisable in any subject to begin by defining the terms which are to be used. This is particularly important in probability theory, in which much confusion has been caused by failure to define the sense in which the word probability is being used. For there are two quite distinct concepts of probability, and unless they are carefully distinguished fruitless controversy can arise between people who are talking about different things without knowing it. These two concepts are: (1) the relative frequency with which an event occurs in the long run, and (2) the degree of belief which it is reasonable to place in a proposition on given evidence. The first of these I shall call statistical probability and the second inductive probability, following the terminology of Darrell Huff (1960). We will now discuss them in turn. Much of this discussion, particularly that of the logical distinction between the two concepts of probability, is based on the excellent account in the second chapter of Carnap (1950).

STATISTICAL PROBABILITY

The concept of statistical probability is based on the long run stability of frequency ratios. Before giving a definition I shall illustrate this idea by means of two examples, from a coin-tossing experiment and from data on the numbers of boys and girls born.

Coin-tossing. No one can tell which way a penny will fall; but we expect the proportions of heads and tails after a large number of spins to be nearly equal. An experiment to demonstrate this point was performed by Kerrich while he was interned in Denmark during the last war. He tossed a coin 10,000 times and obtained altogether 5067 heads; thus at the end of the experiment the proportion of heads was •5067 and that of tails •4933. The behaviour of the proportion of heads throughout the experiment is shown in Fig. 1. It will be seen that it fluctuates widely at first but begins to settle down to a more or less stable value as the number of spins increases. It seems reasonable to suppose that the fluctuations would continue to diminish if the experiment were continued indefinitely, and that the proportion of heads would cluster more and more closely about a limiting value which would be very near, if not exactly, one-half. This hypothetical limiting value is the (statistical) probability of heads.

FIG. 1. The proportion of heads in a sequence of spins of a coin (Kerrich, 1946)

The sex ratio. Another familiar example of the stability of frequency ratios in the long run is provided by registers of births. It has been known since the eighteenth century that in reliable birth statistics based on sufficiently large numbers there is always a slight excess of boys; for example Laplace records that among the 215,599 births in thirty districts of France in the years 1800-1802 there were 110,312 boys and 105,287 girls; the proportions of boys and girls were thus •512 and •488 respectively. In a smaller number of births one would, however, expect considerable deviations from these proportions. In order to give some idea of the effect of the size of the sample on the variability of the sex ratio I have calculated in Table 1 the proportions of male births in 1956 (a) in the major regions of England, and (b) in the rural districts of Dorset. The figures for the major regions of England, which are each based on about 100,000 births, range between •512 and •517, while those for the rural districts of Dorset, based on about 200 births each, range between •38 and •59. The larger sample size is clearly the reason for the greater constancy of the former. We can imagine that if the sample were increased indefinitely, the proportion of boys would tend to a limiting value which is unlikely to differ much from •514, the sex ratio for the whole country. This hypothetical limiting value is the (statistical) probability of a male birth.

TABLE1

The sex ratio in England in 1956

(Source: Annual Statistical Review)

Definition of statistical probability

A statistical probability is thus the limiting value of the relative frequency with which some event occurs. In the examples just considered only two events were possible at each trial: a penny must fall heads or tails and a baby must be either a boy or a girl. In general there will be a larger number of possible events at each trial; for example, if we throw a die there are six possible results at each throw, if we play roulette there are thirty-seven possible results (including zero) at each spin of the wheel and if we count the number of micro-organisms in a sample of pond water the answer can be any whole number. Consider then any observation or experiment which can, in principle at least, be repeated indefinitely. Each repetition will result in the occurrence of one out of an arbitrary number of possible outcomes or events, which will be symbolised by the letters A, B, C and so on. If in n repetitions of the experiment the event A has occurred n(A) times, the proportion of times on which it has occurred is clearly n(A)/n, which will be denoted by p(A). In many situations it is found that as the number of repetitions increases p(A) seems to cluster more and more closely about some particular value, as does the proportion of heads in Fig. 1. In these circumstances it seems reasonable to suppose that this behaviour would continue if the experiment could be repeated indefinitely and that p(A) would settle down with ever diminishing fluctuations about some stable limiting value. This hypothetical limiting value is called the (statistical) probability of A and is denoted by P(A) or Prob(A).

Two important features of this concept of probability must be briefly mentioned. Firstly, it is an empirical concept. Statements about statistical probability are statements about what actually happens in the real world and can only be verified by observation or experiment. If we want to know whether the statistical probability of heads is

when a particular coin is thrown in a particular way, we can only find out by throwing the coin a large number of times. Considerations of the physical symmetry of the coin can of course provide good, a priori reason for conjecturing that this probability is about

but confirmation, or disproof, of this conjecture can only come from actual experiment.

Secondly, we can never know with certainty the exact probability of an event. For no experiment can, in practice, be continued indefinitely, since either the apparatus or the experimenter will wear out sooner or later; and even if it were possible to repeat the experiment for ever, we could never reach the end of an endless sequence of relative frequencies to find out what their limiting value is.¹ It follows that the above definition of statistical probability cannot be interpreted in a literal, operational sense. Some authors, such as Jeffreys ( 1961 ), have concluded that the concept of statistical probability is invalid and meaningless; but the philosophical difficulties in defining probability are no greater than those encountered in trying to define precisely other fundamental scientific concepts such as time and should not prevent us from using this concept, whose meaning is intuitively clear. The reader who wishes to pursue this topic further is referred to the books of Braithwaite (1955), von Mises (1957) and Reichenbach (1949).

INDUCTIVE PROBABILITY

The second concept of probability is that of the degree of belief which it is rational to place in a hypothesis or proposition on given evidence. J. S. Mill gives a very clear definition of this concept: “We must remember,” says Mill, “that the probability of an event is not a quality of the event itself, but a mere name for the degree of ground which we, or someone else, have for expecting it.... Every event is in itself certain, not probable: if we knew all, we should either know positively that it will happen, or positively that it will not. But its probability to us means the degree of expectation of its occurrence, which we are warranted in entertaining by our present evidence.” ²

This concept of probability should be familiar to most people; for, as Bishop Butler wrote, “To us, probability is the very guide of life.” It is perhaps most clearly illustrated in the deliberations of juries. The function of the jury in a criminal prosecution is to listen to the evidence, and then to determine the probability that the prisoner committed the crime of which he is accused. If they consider the probability very high they bring in a verdict of guilty; otherwise, a verdict of not guilty. In a civil action, on the other hand, the jury, if there is one, will find for the party which they consider to have the higher probability of being correct in its assertions. The probability which it is the function of the jury to assess is clearly not a statistical probability; for each trial is unique and cannot be considered as one out of a large number of similar trials. What the jury does is to decide, after hearing the evidence, what, as reasonable men, they ought to believe and with what strength they should hold that belief. This concept will be called inductive probability.

The essential difference between the two concepts of probability is that statistical probability is, as we have already seen, an empirical concept while inductive probability is a logical concept. Statements about inductive probability are not statements about anything in the outside world...