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

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Combinatorial Geometry 232

Some Historically Important

Topics of Combinatorial

Geometry 234

Packing and Covering 234

Polytopes 236

Incidence Problems 238

Helly’s Theorem 238

Methods of Combinatorial

Geometry 240

Exhausting the Possibilities 240

Use of Extremal Properties 241

Use of Transformations

Between Different Spaces

and Applications of

Helly’s Theorem 243

Chapter 6: Biographies 244

Jean Le Rond d’Alembert 244

Thomas Bayes 248

Daniel Bernoulli 249

Jakob Bernoulli 251

Bha¯skara II 253

Ludwig Eduard Boltzmann 254

George Boole 255

Girolamo Cardano 257

Arthur Cayley 259

Francis Ysidro Edgeworth 261

Pierre de Fermat 262

Sir Ronald Aylmer Fisher 265

John Graunt 267

Pierre-Simon, Marquis de Laplace 268

Adrien-Marie Legendre 271

Abraham de Moivre 274

John F. Nash, Jr. 275

Jerzy Neyman 276

Karl Pearson 277

270

236

252

Sir William Petty 279

Siméon-Denis Poisson 280

Adolphe Quetelet 282

Jakob Steiner 283

James Joseph Sylvester 284

John von Neumann 286

Chapter 7: Special Topics 291

Bayes’s Theorem 291

Binomial Distribution 293

Central Limit Theorem 295

Chebyshev’s Inequality 296

Decision Theory 297

Distribution Function 298

Error 298

Estimation 299

Indifference 300

Inference 301

Interval Estimation 301

Law of Large Numbers 302

Least Squares Approximation 303

Markov Process 305

Mean 305

Normal Distribution 308

Permutations and Combinations 310

Point Estimation 313

Poisson Distribution 314

Queuing Theory 315

Random Walk 316

Sampling 316

Standard Deviation 318

Stochastic Process 318

Student’s T-Test 319

Glossary 321

Bibliography 324

Index 326

309

289

294

Introduction 7

his volume presents a multifaceted view of statistics

and probability. Through the eyes of the discoverers

we ﬁnd the thrilling aspects of mathematical applications

that changed the lives of the innovators themselves, as

well as the world at large. The technology that speeds us

through our modern age of discovery has depended upon

statistical knowledge and probability theory for guidance.

Within these pages readers will ﬁnd the history of these

important disciplines of mathematics: the geniuses of

invention and theory, many practical applications of the

math, as well as explanations of the major topics. Statistics

and probability may seem forbidding terrain to some, but

this collective branch of study has proven its practical use-

fulness everywhere from how to play a hand at a card table

to evaluating SAT scores to ensuring the safety of rockets

in outer space.

First, to space.

In 1960 an invitation was extended to select incom-

ing engineering freshmen at a Midwestern university.

These students could apply to participate in a scientiﬁc

study that would provide necessary information for space

travel. At the time, no one really knew how people locked

in a space capsule would behave. Would crew members

who were isolated and sequestered for a number of days

at a stretch sleep well? Would they argue and get on each

other’s nerves? Would their dietary patterns be affected?

Would they suffer anxiety attacks?

NASA was developing a program to send people into

outer space. As there was no data on what happened to

human beings once they left the conﬁnes of the planet,

statistical data under simulated conditions was crucial. If

several people were sequestered in a capsule under pres-

sures of risk, the denial of home comforts, and with the

added factor of personality differences, might they tend

to push the wrong buttons on the control panel?

7 The Britannica Guide to Statistics and Probability 7

Not only was statistical data necessary, probability

theory was crucial. These days, the common high school

student who has watched the World Series of Poker tourna-

ments on television knows that knowledge of the odds can

and often does determine a player’s stake. But poker, ruth-

less as it might be at times, is merely a game. Sending

people off in a rocket for the ﬁrst time ever is not.

Scientists and mathematicians, of course, were fairly

sure of certain forces and events, such as gravitational pull,

centrifugal force, friction, mathematical relationships

governing ellipses and parabolas and such, to name a few.

But add people—a rocket full of NASA crew members

blasting off from the face of the earth—and could those

scientists tell the pilots for sure—for certain—exactly what

would happen? The answer was no. Everyone knew that

risks existed. Mathematicians were called in to determine

to the best of their abilities what those risks might be, and

how conﬁdent one might be that the anticipated scientiﬁc

responses and behaviours would indeed occur.

For example, during the all-important re-entry phase

of the space journey, if the curvature of the ﬂight path of a

speeding spacecraft from one destination in space to a

moving, spinning earth thousands of miles away was

undertaken, what were the chances of a meteorite inter-

fering? What were the odds of engine failure or abnormal

frictional forces? What were the probabilities of the

spacecraft and its occupants hitting the ocean instead of

the Himalayas?

One must stand in awe of the mathematics that these

theoreticians were asked to deliver. The results certainly

eclipsed whether or not a straight ﬂush would appear

to assure a winning poker hand. To their best knowl-

edge, these mathematicians were assessing the chances

of life or death. Unlike the college classroom, partial

credit on this exam would not be acceptable. And yet the

mathematicians were not dealing with an exact science.

They were hoping for probabilities that covered all related

factors as far as they knew. What would probably happen?

(And if the theorists had trouble sleeping at night, imag-

ine the training space crew.)

Mathematical tension was rampant. In fact, news foot-

age of NASA scientists in front of computers monitoring

space ﬂights showed them chain smoking, frequently rub-

bing their faces with their open palms, shifting with the

jitters, and ﬁnally, ecstatic as football fans when a satisfac-

tory mission ended and the words came: “Houston, we

have recovery.”

While probability and statistics look innocent, appar-

ently composed of peaceful numbers and placid formulas

about what might happen over the course of a certain

event, we understand the inner turmoil beneath a calm

exterior. And one isn’t required to be a NASA math-

ematician to suffer from these statistical tensions. Take

the average high school student trying to enter college,

whose selection and application process might very well

involve at least one fall Saturday morning spent taking

the SAT test. One can feel one’s blood pressure rising at

the thought. It seems to the students that the culprits in

student discomfort are the test questions. But the hidden

instigators are actually statistical measures, standard devi-

ations. After all, a student might miss many questions on

the test and reach an acceptable score. The real concern

is how far from the average student is the test taker? That

is the measure college admissions ofﬁcers would like to

know. And the statistical standard deviation, converted to

a score that is more understandable and easier to read and

compare, is the cause of all that student agony. In any given

SAT test, students are competing with the other students

who are taking that same test. If every test taker were sta-

tistically average, no measurable standard deviation would

7 Introduction 7

7 The Britannica Guide to Statistics and Probability 7

exist, and nobody would score higher than anyone else.

The college admissions people would have to ﬁnd another

way to make their decisions.

Making use of terms such as agony to discuss a mathe-

matical tool seems melodramatic. Yet that term and

others, including downright pejoratives, have been used

to describe the applications of statistics. Recall author

Darrell Huff’s bestselling book, How to Lie with Statistics.

If statistics can convince one to follow a certain path—a

wrong path—then perhaps statistics alone are not enough

for making a wise decision. Morality must be applied, as

well. To use the term sinister when considering possible

statistic applications might be reasonable, as will be

explained shortly.

The math discipline often fondly referred to as “stats”

by its students comes with an ingenious side, and also

caveats. One wonders if Carl Friedrich Gauss (1777–1855)

foresaw such developments when his probability distribu-

tion equations led to the still-popular bell curve, at the

foundation of statistical measures.

The plotted curve demonstrates visually the distribu-

tion of a population, mean (or average), and standard

deviation. The area under the curve can be made to illus-

trate the percents of the total population falling in certain

standard deviation intervals. As the previous sentence

shows, just the verbiage in describing this mathematical

graph and its statistical measuring requires enormous

amounts of detail held in the brain. By contrast, the rather

beautiful curve itself gently relates its properties pictori-

ally, aesthetically, and perhaps more effectively, especially

for the novice.

The bell curve is also called the normal curve, or the

curve showing normal distribution of the population

members under study. This choice of expression, “nor-

mal,” returns us to the caution required when entering the

world of statistics. To study a population with the normal

curve, one must be careful about assuming what is normal

and what is not. The statistics being reached might just

bleed off unintended inference: the bias, bigotry, political

leanings, and even those sinister intentions mentioned

earlier. On the positive side, statistics have helped pave

the way for space travel, inoculations to wipe out polio,

and even supplied sports information that helped the

Boston Red Sox win a World Series title. This last advance

(an advance depending on whom you root for, that is)

came thanks to Red Sox statistician Bill James and his

innovative view on what is important in baseball as

opposed to what people had thought was important in

baseball. On the negative side of statistics, consider a little

Nazi statistical undertaking that involved a key Polish

mathematician victim during the early 1940s.

Stefan Banach (1892–1945) founded functional analysis

and helped develop the theory of topology, vector space,

and normed linear spaces (which are now known as Banach

spaces). These ingenious discoveries were all good things

intended to help mankind and further human knowledge,

our understanding of ourselves, and make life easier for

succeeding generations. The 1920s and ’30s were good

years for Banach, but his life was destined to change quite

abruptly. From 1941 to 1944, under the Nazi occupation,

Banach was compelled to take work as a lice feeder,

thereby becoming infested. For three years he was forced

to become a virtual lice farm as the Nazis studied him,

gathering statistics on infectious diseases. This brilliant

mathematician died of lung cancer in 1945, the last years

of his life spent not as a statistical analyst but rather as

a subject. As previously mentioned, statistics can have a

seamy side or a wonderfully illuminating side. How the

stats are arrived at and how they are presented may make

all the difference. Inferences are often crucial.

7 Introduction 7

7 The Britannica Guide to Statistics and Probability 7

While the Nazis were taking statistics to a barbaric

level, during another time in history in one of those com-

plete twists of human nature that demonstrates caring and

fair play, earlier statistical work from a brilliant German

physicist helped unite previous rivals. The brilliancies in

both discovery and collegiality are found in the work of

Ludwig Eduard Boltzmann (1844–1906). Boltzmann’s sta-

tistical mechanics helped explain and make available

predictions of how the properties of atoms (their mass,

charge, and structure) determine the properties of matter

that become observable to scientists (for instance, viscos-

ity, thermal conductivity, and diffusion). Boltzmann

applied the theory of probability of the motions of atoms

to the second law of thermodynamics. The second law was

shown to be statistical. Its investigations led to the theo-

rem of equipartition of energy (the Maxwell-Boltzmann

distribution law). And perhaps the dual names in sponsor-

ship of that equipartition law suggest traits of Boltzmann’s

character and ingenuity as well as the importance and ben-

eﬁts of a cooperative approach toward discovery. First a

brief step back in time is required.

In the 1680s Isaac Newton (England) and Gottfried

Wilhelm Leibniz (Germany) had simultaneously and

independently discovered calculus. While both discov-

eries were accomplished in different ways, both were

legitimate and provided a long-sought-after mathematical

tool for future math discovery and scientiﬁc achievement.

Unfortunately, a rivalry developed between the followers

of Newton and Leibniz. The reticent Newton was content

to achieve with rigor and with silence. Leibniz was a mas-

ter of getting the word out about his work. Instant fame

went to Leibniz. Leadership in mathematics discovery

therefore shifted from England across the Channel to the

Leibniz camp and the continent, remaining on the conti-

nent for quite some time.

Enter the aforementioned Ludwig Boltzmann in the

late 1800s. He was one of the ﬁrst continental scientists

to recognize the importance of the electromagnetic the-

ory proposed by James Clerk Maxwell of England.

Maxwell’s work had long been under attack. The support

and recognition of Ludwig Boltzmann gave substance to

belief in Maxwell’s work. Discoveries in atomic physics

now proved Maxwell correct. His Brownian motion

investigations could be explained only by the statistical

mechanics furthered by Boltzmann. (Brownian motion is

the random movement of microscopic particles sus-

pended in a ﬂuid and is named for Scottish botanist

Robert Brown, the ﬁrst to study such ﬂuctuations.) In

reaching across the Channel, as Boltzmann did with

Maxwell, we observe the growth of knowledge, discovery,

innovation, and the achievements of modernity. One is

left to wonder how much greater the discoveries might

have been had Leibniz been able to reach out to Newton,

if indeed that was even possible at the time, or if the Nazi

regime had nurtured a Polish mathematician and encour-

aged discovery rather than generate statistics based upon

the bite marks on his trunk and scalp. It seems we humans

do best when we observe the achievements of past

geniuses and grow from that. But we must be cautious in

the process, such as statistically omitting from college

ranks what a single test might point out as below normal,

and from applying too strictly the numbers that arise

from numbers.

We must admit that statistics can tell lies. We must

make sure that they do not.

7 Introduction 7