Gregersen E. (editor) The Britannica Guide to Statistics and Probability

Подождите немного. Документ загружается.

7 The Britannica Guide to Statistics and Probability 7

science with a rapidity of conquest rivaled only by Attila,

Mohammed, and the Colorado beetle.” The spread of sta-

tistical mathematics through the sciences began, in fact,

at least a century before there were any professional stat-

isticians. Even regardless of the use of probability to

estimate populations and make insurance calculations,

this history dates back at least to 1809. In that year, the

German mathematician Carl Friedrich Gauss published a

derivation of the new method of least squares incorporat-

ing a mathematical function that soon became known as

the astronomer’s curve of error, and later as the Gaussian

or normal distribution.

The problem of combining many astronomical obser-

vations to give the best possible estimate of one or several

parameters was discussed in the 18th century. The ﬁrst

publication of the method of least squares as a solution to

this problem was inspired by a more practical problem,

the analysis of French geodetic measures undertaken to

ﬁx the standard length of the metre. This was the basic

measure of length in the new metric system, decreed by

the French Revolution and deﬁned as 1/40,000,000 of the

longitudinal circumference of the Earth. In 1805 the

French mathematician Adrien-Marie Legendre proposed

to solve this problem by choosing values that minimize

the sums of the squares of deviations of the observations

from a point, line, or curve drawn through them. In the

simplest case, where all observations were measures of a

single point, this method was equivalent to taking an

arithmetic mean.

Gauss soon announced that he had already been using

least squares since 1795, a somewhat doubtful claim. After

Legendre’s publication, Gauss became interested in the

mathematics of least squares, and he showed in 1809 that

the method gave the best possible estimate of a parameter

History of Statistics and Probability 7

if the errors of the measurements were assumed to follow

the normal distribution. This distribution, whose impor-

tance for mathematical probability and statistics was

decisive, was ﬁrst shown by the French mathematician

Abraham de Moivre in the 1730s to be the limit (as the

number of events increases) for the binomial distribution.

In particular, this meant that a continuous function (the

normal distribution) and the power of calculus could be

substituted for a discrete function (the binomial distribu-

tion) and laborious numerical methods. Laplace used the

normal distribution extensively as part of his strategy for

applying probability to very large numbers of events. The

most important problem of this kind in the 18th century

involved estimating populations from smaller samples.

Laplace also had an important role in reformulating the

method of least squares as a problem of probabilities. For

much of the 19th century, least squares was overwhelm-

ingly the most important instance of statistics in its guise

as a tool of estimation and the measurement of uncer-

tainty. It had an important role in astronomy, geodesy, and

related measurement disciplines, including even quantita-

tive psychology. Later, about 1900, it provided a

mathematical basis for a broader ﬁeld of statistics that

came to be used by a wide range of ﬁelds.

sTaTisTical TheoRies

in The sciences

The role of probability and statistics in the sciences was

not limited to estimation and measurement. Equally sig-

niﬁcant, and no less important for the formation of the

mathematical ﬁeld, were statistical theories of collective

phenomena that bypassed the study of individuals. The

social science bearing the name statistics was the prototype

7 The Britannica Guide to Statistics and Probability 7

of this approach. Quetelet advanced its mathematical

level by incorporating the normal distribution into it. He

argued that human traits of every sort, from chest circum-

ference and height to the distribution of propensities to

marry or commit crimes, conformed to the astronomer’s

error law. The kinetic theory of gases of Clausius, Maxwell,

and the Austrian physicist Ludwig Boltzmann was also a

statistical one. Here it was not the imprecision or uncer-

tainty of scientiﬁc measurements but the motions of the

molecules themselves to which statistical understandings

and probabilistic mathematics were applied. Once again,

the error law played a crucial role. The Maxwell-Boltzmann

distribution law of molecular velocities, as it has come to

be known, is a three-dimensional version of this same

function. In importing it into physics, Maxwell drew both

on astronomical error theory and on Quetelet’s social

physics.

Biometry

The English biometric school developed from the work of

the polymath Francis Galton, cousin of Charles Darwin.

Galton admired Quetelet, but he was critical of the statis-

tician’s obsession with mean values rather than variation.

The normal law, as he began to call it, was for him a way to

measure and analyze variability. This was especially impor-

tant for studies of biological evolution, because Darwin’s

theory was about natural selection acting on natural diver-

sity. A ﬁgure from Galton’s 1877 paper on breeding sweet

peas shows a physical model, now known as the Galton

board, that he employed to explain the normal distribu-

tion of inherited characteristics. In particular, he used his

model to explain the tendency of progeny to have the

same variance as their parents, a process he called

History of Statistics and Probability 7

Sir Francis Galton’s Galton board helped explain the normal distribution of

inherited characteristics. SSPL via Getty Images

reversion, subsequently known as regression to the mean.

Galton was also founder of the eugenics movement, which

called for guiding the evolution of human populations the

same way that breeders improve chickens or cows. He

developed measures of the transmission of parental char-

acteristics to their offspring: The children of exceptional

parents were generally somewhat exceptional themselves,

but there was always, on average, some reversion or regres-

sion toward the population mean. He developed the

7 The Britannica Guide to Statistics and Probability 7

elementary mathematics of regression and correlation as a

theory of hereditary transmission and thus as statistical

biological theory rather than as a mathematical tool.

Galton came to recognize that these methods could be

applied to data in many ﬁelds, however, and by 1889, when

he published his Natural Inheritance, he stressed the ﬂexi-

bility and adaptability of his statistical tools.

Still, evolution and eugenics remained central to the

development of statistical mathematics. The most inﬂu-

ential site for the development of statistics was the

biometric laboratory set up at University College London

by Galton’s admirer, the applied mathematician Karl

Pearson. From about 1892 he collaborated with the

English biologist Walter F.R. Weldon on quantitative

studies of evolution, and he soon began to attract an

assortment of students from many countries and disci-

plines who hoped to learn the new statistical methods.

Their journal, Biometrika, was for many years the most

important venue for publishing new statistical tools and

for displaying their uses.

Biometry was not the only source of new develop-

ments in statistics at the turn of the 19th century. German

social statisticians such as Wilhelm Lexis had turned to

more mathematical approaches some decades earlier. In

England, the economist Francis Edgeworth became inter-

ested in statistical mathematics in the early 1880s. One of

Pearson’s earliest students, George Udny Yule, turned

away from biometry and especially from eugenics in favour

of the statistical investigation of social data. Nevertheless,

biometry provided an important model, and many statis-

tical techniques, for other disciplines. The 20th-century

ﬁelds of psychometrics, concerned especially with mental

testing, and econometrics, which focused on economic

time-series, reveal this relationship in their very names.

History of Statistics and Probability 7

Samples and Experiments

Near the beginning of the 20th century, sampling regained

its respectability in social statistics, for reasons that ini-

tially had little to do with mathematics. Early advocates,

such as the ﬁrst director of the Norwegian Central Bureau

of Statistics, A.N. Kiaer, thought of their task primarily in

terms of attaining representativeness in relation to the

most important variables—for example, geographic

region, urban and rural, rich and poor. The London statis-

tician Arthur Bowley was among the ﬁrst to urge that

sampling should involve an element of randomness. Jerzy

Neyman, a statistician from Poland who had worked for a

time in Pearson’s laboratory, wrote a particularly decisive

mathematical paper on the topic in 1934. His method of

stratiﬁed sampling incorporated a concern for representa-

tiveness across the most important variables, but it also

required that the individuals sampled should be chosen

randomly. This was designed to avoid selection biases but

also to create populations to which probability theory

could be applied to calculate expected errors. George

Gallup achieved fame in 1936 when his polls, employing

stratiﬁed sampling, successfully predicted the reelection

of Franklin Delano Roosevelt, in deﬁance of the Literary

Digest’s much larger but uncontrolled survey, which fore-

cast a landslide for the Republican Alfred Landon.

The alliance of statistical tools and experimental

design was also largely an achievement of the 20th cen-

tury. Here, too, randomization came to be seen as central.

The emerging protocol called for the establishment of

experimental and control populations and for the use of

chance where possible to decide which individuals would

receive the experimental treatment. These experimental

repertoires emerged gradually in educational psychology

7 The Britannica Guide to Statistics and Probability 7

during the 1900s and ’10s. They were codiﬁed and given a

full mathematical basis in the next two decades by Ronald

A. Fisher, the most inﬂuential of all the 20th-century stat-

isticians. Through randomized, controlled experiments

and statistical analysis, he argued, scientists could move

beyond mere correlation to causal knowledge even in

ﬁelds whose phenomena are highly complex and variable.

His ideas of experimental design and analysis helped to

reshape many disciplines, including psychology, ecology,

and therapeutic research in medicine, especially during

the triumphant era of quantiﬁcation after 1945.

The Modern Role of Statistics

In some ways, statistics has ﬁnally achieved the

Enlightenment aspiration to create a logic of uncertainty.

Statistical tools are at work in almost every area of life,

including agriculture, business, engineering, medicine,

law, regulation, and social policy, as well as in the physical,

biological, and social sciences and even in parts of the aca-

demic humanities. The replacement of human “computers”

with mechanical and then electronic ones in the 20th cen-

tury greatly lightened the immense burdens of calculation

that statistical analysis once required. Statistical tests are

used to assess whether observed results, such as increased

harvests where fertilizer is applied, or improved earnings

where early childhood education is provided, give reason-

able assurance of causation, rather than merely random

ﬂuctuations. Following World War II, these signiﬁcance

levels virtually came to deﬁne an acceptable result in some

of the sciences and also in policy applications.

From about 1930 there grew up in Britain and

America—and a bit later in other countries—a profession

of statisticians, experts in inference, who deﬁned standards

History of Statistics and Probability 7

of experimentation as well as methods of analysis in

many ﬁelds. To be sure, statistics in the various disci-

plines retained a fair degree of speciﬁcity. There were also

divergent schools of statisticians, who disagreed, often

vehemently, on some issues of fundamental importance.

Fisher was highly critical of Pearson. Neyman and Egon

Pearson, while unsympathetic to father Karl’s meth-

ods, disagreed also with Fisher’s. Under the banner of

Bayesianism appeared yet another school, which, against

its predecessors, emphasized the need for subjective

assessments of prior probabilities. The most immoder-

ate ambitions for statistics as the royal road to scientiﬁc

inference depended on unacknowledged compromises

that ignored or dismissed these disputes. Despite them,

statistics has thrived as a somewhat heterogeneous but

powerful set of tools, methods, and forms of expertise

that continues to regulate the acquisition and interpreta-

tion of quantitative data.

PRoBABILItY tHeoRY

CHAPTER 2

robability theory is the branch of mathematics con-

cerned with the analysis of random phenomena. The

outcome of a random event cannot be determined before

it occurs, but it may be any one of several possible out-

comes. The actual outcome is considered to be determined

by chance.

The word probability has several meanings in ordinary

conversation, two of which are particularly important for

the development and applications of the mathematical

theory of probability. One is the interpretation of proba-

bilities as relative frequencies, for which simple games

involving coins, cards, dice, and roulette wheels provide

examples. The distinctive feature of games of chance is

that the outcome of a given trial cannot be predicted with

certainty, but the collective results of a large number of tri-

als display some regularity. For example, the statement that

the probability of “heads” in tossing a coin equals one-half,

according to the relative frequency interpretation, implies

that in a large number of tosses the relative frequency with

which “heads” actually occurs will be approximately one-

half, but it contains no implication concerning the outcome

of any given toss. There are many similar examples involv-

ing groups of people, molecules of a gas, genes, and so on.

Actuarial statements about the life expectancy for persons

of a certain age describe the collective experience of a large

number of individuals but do not purport to say what will

happen to any particular person. Similarly, predictions

about the chance of a genetic disease occurring in a child of

parents having a known genetic makeup are statements

Probability Theory 7

Probability theory is exempliﬁed by roulette: players bet on within which red

or black numbered compartment of a revolving wheel a small ball will come

to rest. Jeff T. Green/Getty Images