Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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shared the same memory for code and data, a design that
greatly facilitated the “conditional loops” at the heart of
all subsequent coding.
Another important consultancy was at the RAND
Corporation, a think tank charged with planning nuclear
strategy for the U.S. Air Force. Von Neumann insisted on
the value of game-theoretic thinking in defense policy.
He supported development of the hydrogen bomb and
was reported to have advocated a preventive nuclear
strike to destroy the Soviet Union’s nascent nuclear capa-
bility circa 1950.
Von Neumann’s shift to applied mathematics after the
midpoint of his career mystified colleagues, who felt that
a genius of his calibre should concern himself with “pure”
mathematics. In an essay written in 1956, von Neumann
made an eloquent defense of applied mathematics. He
praised the invigorating influence of “some underlying
empirical, worldly motif” in mathematics, warning that
“at a great distance from its empirical source, or after
much abstract inbreeding, a mathematical subject is in
danger of degeneration.” With his pivotal work on game
theory, quantum theory, the atomic bomb, and the com-
puter, von Neumann likely exerted a greater influence on
the modern world than any other mathematician of the
20th century.
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sPeCIAL toPICs
S
ome topics discussed earlier are treated here in greater
detail. For example, you will learn how Sir Ronald
Fisher used Bayes’s theorem to throw suspicion on a
famous scientist. You may also be amazed at how statistics
was used to settle a controversy about Earth’s shape.
bayes’s TheoReM
Bayes’s theorem is a means for revising predictions in light
of relevant evidence, also known as conditional probabil-
ity or inverse probability. The theorem was discovered
among the papers of the English Presbyterian minister
and mathematician Thomas Bayes and posthumously
published in 1763. Related to the theorem is Bayesian
inference, or Bayesianism, based on the assignment of
some a priori distribution of a parameter under investiga-
tion. In 1854 the English logician George Boole criticized
the subjective character of such assignments, and
Bayesianism declined in favour of “confi dence intervals”
and “hypothesis tests”—now basic research methods.
As a simple application of Bayes’s theorem, consider
the results of a screening test for infection with the human
immunodefi ciency virus (HIV). Suppose an intravenous
drug user undergoes testing where experience has indi-
cated a 25 percent chance that the person has HIV. A
quick test for HIV can be conducted, but it is not infal-
lible: Almost all individuals who have been infected long
enough to produce an immune system response can be
detected, but very recent infections may go undetected.
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In addition, “false positive” test results (i.e., a false indica-
tion of infection) occur in 0.4 percent of people who are
not infected. Hence, positive test results do not prove that
the person is infected. Nevertheless, infection seems more
likely for those who test positive, and Bayes’s theorem
provides a formula for evaluating the probability.
The logic of this formula is explained as follows:
Suppose that there are 10,000 intravenous drug users in
the population, of which 2,500 are infected with HIV.
Suppose further that if all 2,500 people are tested, 95
percent (2,375 people) will produce a positive test result.
The other 5 percent are known as “false negatives.” In
addition, of the remaining 7,500 people who are not
infected, about 0.4 percent, or 30 people, will test positive
(“false positives”). Because there are 2,405 positive tests
in all, the probability that a person testing positive is
actually infected can be calculated as 2,375/2,405, or about
98.8 percent.
Applications of Bayes’s theorem used to be limited
mostly to such straightforward problems, even though
the original version was more complex. There are two key
difficulties in extending these sorts of calculations, how-
ever. First, the starting probabilities are rarely so easily
quantified. They are often highly subjective. To return to
the HIV screening previously described, a patient might
appear to be an intravenous drug user but might be unwill-
ing to admit it. Subjective judgment would then enter
into the probability that the person indeed fell into this
high-risk category. Hence, the initial probability of HIV
infection would in turn depend on subjective judgment.
Second, the evidence is not often so simple as a positive
or negative test result. If the evidence takes the form of a
numerical score, the sum used in the denominator of the
above calculation must be replaced by an integral. More
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complex evidence can easily lead to multiple integrals
that, until recently, could not be readily evaluated.
Nevertheless, advanced computing power, along with
improved integration algorithms, has overcome most cal-
culation obstacles. In addition, theoreticians have
developed rules for delineating starting probabilities that
correspond roughly to the beliefs of a “sensible person”
with no background knowledge. These rules can often be
used to reduce undesirable subjectivity. These advances
have led to a recent surge of applications of Bayes’s theo-
rem, more than two centuries since it was first put forth.
It is now applied to such diverse areas as the productivity
assessment for a fish population and the study of racial
discrimination.
binoMial disTRibuTion
The binomial distribution is a common distribution func-
tion for discrete processes in which a fixed probability
prevails for each independently generated value. First
studied in connection with games of pure chance, the
binomial distribution is now widely used to analyze data in
virtually every field of human inquiry. It applies to any
fixed number (n) of repetitions of an independent process
that produces a certain outcome with the same probabil-
ity (p) on each repetition. For example, it provides a
formula for the probability of obtaining 10 sixes in 50 rolls
of a die. Swiss mathematician Jakob Bernoulli, in a proof
posthumously published in 1713, determined that the
probability of k such outcomes in n repetitions is equal to
the kth term (where k starts with 0) in the expansion of
the binomial expression (p + q)n, where q = 1 − p. (Hence the
name binomial distribution.) In the example of the die, the
probability of turning up any number on each roll is 1 out
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of 6 (the number of faces on the die). The probability of
turning up 10 sixes in 50 rolls, then, is equal to the 10th
term (starting with the 0th term) in the expansion of
(5/6 + 1/6)
50
, or 0.115586.
In 1936 the British statistician Ronald Fisher used
the binomial distribution to publish evidence of possible
scientific chicanery in the famous experiments on pea
genetics reported by the Austrian botanist Gregor Mendel
in 1866. Fisher observed that Mendel’s laws of inheritance
would dictate that the number of yellow peas in one of
Mendel’s experiments would have a binomial distribution
with n = 8,023 and p =
3
⁄
4
, for an average of np ≅ 6,017 yel-
low peas. Fisher found remarkable agreement between
this number and Mendel’s data, which showed 6,022 yel-
low peas out of 8,023. One would expect the number to
be close, but a figure that close should occur only 1 in 10
times. Fisher found, moreover, that all seven results in
Binomial distribution provides a formula for the probability of obtaining 10
sixes in 50 rolls of a die. Shutterstock.com
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Mendel’s pea experiments were extremely close to the
expected values—even in one instance where Mendel’s
calculations contained a minor error. Fisher’s analysis
sparked a lengthy controversy that remains unresolved to
this day.
cenTRal liMiT TheoReM
The central limit theorem establishes the normal distribu-
tion as the distribution to which the mean (average) of
almost any set of independent and randomly generated
variables rapidly converges. The central limit theorem
explains why the normal distribution arises so commonly
and why it is generally an excellent approximation for the
mean of a collection of data (often with as few as 10
variables).
The standard version of the central limit theorem,
first proved by the French mathematician Pierre-Simon
Laplace in 1810, states that the sum or average of an infi-
nite sequence of independent and identically distributed
random variables, when suitably rescaled, tends to a nor-
mal distribution. Fourteen years later the French
mathematician Siméon-Denis Poisson began a continuing
process of improvement and generalization. Laplace and
his contemporaries were interested in the theorem pri-
marily because of its importance in repeated measurements
of the same quantity. If the individual measurements could
be viewed as approximately independent and identically
distributed, their mean could be approximated by a nor-
mal distribution.
The Belgian mathematician Adolphe Quetelet (1796–
1874), famous today as the originator of the concept of the
homme moyen (“average man”), was the first to use the nor-
mal distribution for something other than analyzing error.
For example, he collected data on soldiers’ chest girths
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and showed that the distribution of recorded values cor-
responded approximately to the normal distribution. Such
examples are now viewed as consequences of the central
limit theorem.
The central limit theorem also plays an important role
in modern industrial quality control. The first step in
improving the quality of a product is often to identify the
major factors that contribute to unwanted variations.
Efforts are then made to control these factors. If these
efforts succeed, any residual variation will typically be
caused by a large number of factors, acting roughly inde-
pendently. In other words, the remaining small amounts
of variation can be described by the central limit theorem,
and the remaining variation will typically approximate a
normal distribution. For this reason, the normal distribu-
tion is the basis for many key procedures in statistical
quality control.
chebyshev’s inequaliTy
Chebyshev’s inequality (also called the Bienaymé-Chebyshev
inequality) characterizes the dispersion of data away from
its mean (average). Although the general theorem is attrib-
uted to the 19th-century Russian mathematician Pafnuty
Chebyshev, credit for it should be shared with the French
mathematician Irénée-Jules Bienaymé, whose (less gen-
eral) 1853 proof predated Chebyshev’s by 14 years.
Chebyshev’s inequality puts an upper bound on the
probability that an observation should be far from its mean.
It requires only two minimal conditions: (1) that the under-
lying distribution have a mean and (2) that the average size
of the deviations away from this mean (as gauged by the
standard deviation) not be infinite. Chebyshev’s inequality
then states that the probability that an observation will be
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more than k standard deviations from the mean is at most
1/k
2
. Chebyshev used the inequality to prove his version of
the law of large numbers.
Unfortunately, with virtually no restriction on the
shape of an underlying distribution, the inequality is so
weak as to be virtually useless to anyone looking for a pre-
cise statement on the probability of a large deviation. To
achieve this goal, people usually try to justify a specific
error distribution, such as the normal distribution as pro-
posed by the German mathematician Carl Friedrich
Gauss. Gauss also developed a tighter bound, 4/9k
2
(for
k > 2/√
-
3), on the probability of a large deviation by impos-
ing the natural restriction that the error distribution
decline symmetrically from a maximum at 0.
The difference between these values is substantial.
According to Chebyshev’s inequality, the probability that
a value will be more than two standard deviations from the
mean (k = 2) cannot exceed 25 percent. Gauss’s bound is 11
percent, and the value for the normal distribution is just
less than 5 percent. Thus, it is apparent that Chebyshev’s
inequality is useful only as a theoretical tool for proving
generally applicable theorems, not for generating tight
probability bounds.
decision TheoRy
In statistics, decision theory is a set of quantitative meth-
ods for reaching optimal decisions. A solvable decision
problem must be capable of being tightly formulated in
terms of initial conditions and choices or courses of action,
with their consequences. In general, such consequences
are not known with certainty but are expressed as a set of
probabilistic outcomes. Each outcome is assigned a “util-
ity” value based on the preferences of the decision maker.
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An optimal decision, following the logic of the theory, is
one that maximizes the expected utility. Thus, the ideal of
decision theory is to make choices rational by reducing
them to a kind of routine calculation.
disTRibuTion funcTion
The mathematical expression that describes the probabil-
ity that a system will take on a specific value or set of
values is called a distribution function. The classic exam-
ples are associated with games of chance. The binomial
distribution gives the probabilities that heads will come
up a times and tails n − a times (for 0 ≤ a ≤ n), when a fair
coin is tossed n times. Many phenomena, such as the dis-
tribution of IQs, approximate the classic bell-shaped, or
normal, curve. The highest point on the curve indicates
the most common or modal value, which in most cases
will be close to the average (mean) for the population. A
well-known example from physics is the Maxwell-
Boltzmann distribution law, which specifies the
probability that a molecule of gas will be found with veloc-
ity components u, v, and w in the x, y, and z directions. A
distribution function may take into account as many vari-
ables as one chooses to include.
eRRoR
In applied mathematics, the error is the difference
between a true value and an estimate, or approximation,
of that value. In statistics, a common example is the dif-
ference between the mean of an entire population and the
mean of a sample drawn from that population. In numeri-
cal analysis, round-off error is exemplified by the difference
between the true value of the irrational number π and the
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value of rational expressions such as 22/7, 355/113, 3.14, or
3.14159. Truncation error results from ignoring all but a
finite number of terms of an infinite series. For example,
the exponential function e
x
may be expressed as the sum of
the infinite series
1 + x + x
2
/2 + x
3
/6 + ⋯ + x
n
/n! + ⋯
Stopping the calculation after any finite value of n will
give an approximation to the value of ex that will be in
error, but this error can be made as small as desired by
making n large enough.
The relative error is the numerical difference divided
by the true value, and the percentage error is this ratio
expressed as a percent. The term random error is sometimes
used to distinguish the effects of inherent imprecision
from so-called systematic error, which may originate in
faulty assumptions or procedures. The methods of math-
ematical statistics are particularly suited to the estimation
and management of random errors.
esTiMaTion
In statistics, any procedure used to calculate the value of
some property of a population from observations of a
sample drawn from the population is called an estimation.
A point estimate, for example, is the single number most
likely to express the value of the property. An interval esti-
mate defines a range within which the value of the property
can be expected (with a specified degree of confidence) to
fall. The 18th-century English theologian and mathemati-
cian Thomas Bayes was instrumental in the development
of Bayesian estimation to facilitate revision of estimates
on the basis of further information.