Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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7 The Britannica Guide to Statistics and Probability 7
320
appropriate reference distribution, the null hypothesis is
rejected. The appropriate reference distribution for the
t -statistic is the t distribution. The critical value depends
on the signifi cance level of the test (the probability of
erroneously rejecting the null hypothesis).
For example, suppose a researcher wishes to test the
hypothesis that a sample of size n = 25 with mean x = 79
and standard deviation s = 10 was drawn at random from
a population with mean μ = 75 and unknown standard
deviation. Using the formula for the t -statistic,
the calculated t equals 2. For a two-sided test at a common
level of signifi cance α = 0.05, the critical values from the t
distribution on 24 degrees of freedom are −2.064 and
2.064. The calculated t does not exceed these values, hence
the null hypothesis cannot be rejected with 95 percent
confi dence. (The confi dence level is 1 − α.)
A second application of the t distribution tests the
hypothesis that two independent random samples have
the same mean. The t distribution can also be used to
construct confi dence intervals for the true mean of a pop-
ulation (the fi rst application) or for the difference between
two sample means (the second application).
321
GLossARY
Bayesian estimation Technique for calculating the
probability of the validity of a proposition based
on a prior estimate of its probability and new rele-
vant evidence.
binomial distribution A common distribution
function for discrete processes in which a fi xed
probability prevails for each independently gener-
ated value.
biometry English biometric school developed from the
work of the polymath Francis Galton.
catenary Curve formed by a perfectly fl exible chain sus-
pended between its two fi xed extremities.
combinatorics Concerned with problems of selection,
arrangement, and operation within a fi nite or discrete
system (also called combinatorial mathematics).
confi dence intervals Interval estimates of population
parameters.
distribution function A mathematical expression that
describes the probability that a system will take on a
specifi c value or set of values.
eugenics Selection of desired heritable characteristics to
improve future generations, typically referring
to humans.
hypothesis testing Draws on data from a sample to
make conclusions about a population parameter or a
population probability distribution.
interval estimates Estimate that includes a statement
concerning the degree of confi dence that the interval
contains the estimated population parameter.
7 The Britannica Guide to Statistics and Probability 7
322
isomorphic Being of identical or similar form, shape, or
structure.
nonparametric method A statistical method requiring
fewer assumptions about a population or probability
distribution.
p-value Measure of how possible the sample results are,
assuming a true null hypothesis. A smaller p-value
indicates less likely sample results.
partition Division of a set of objects into a family of
subsets that are mutually exclusive and jointly
exhaustive.
Philosophe Any 18th-century French writer, scientist,
and thinker convinced of the supremacy and
efficacy of human reason and nature.
point estimate Value of a sample statistic used as
a single estimate of a population parameter,
without statements regarding the quality or
precision.
probability density function A function whose
integral is calculated to find probabilities associated
with a continuous random variable.
qualitative data Provide labels or names for groups of
comparable items.
quantitative data Measure either how much or how
many of something.
regression A process to determine a line or curve best
representing the general trend of a data set.
regression to the mean A model in which progeny tend
to have the same variance as their parents.
residual The difference between the observed value of y
and the value of y predicted by the estimated regres-
sion equation.
sample space The set of all possible outcomes of an
experiment.
323
7
Glossary 7
statistical test Assesses whether observed results give
reasonable assurance of causation, rather than merely
random fluctuations.
stochastic process A process in probability theory
involving the operation of chance.
t distribution A family of curves in which the number of
degrees of freedom specifies a particular curve.
z-score Represents the relative position of the data
value by indicating the number of standard deviations
it is from the mean.
324
BIBLIoGRAPHY
General Works
Ian Hacking , The Emergence of Probability: A Philosophical
Study of Early Ideas About Probability, Induction, and Statistical
Inference (1975, reissued 1991), discusses the history of prob-
ability and its interpretations in relation to a broad intellectual
background, up to about 1750. Lorraine Daston , Classical
Probability in the Enlightenment (1988, reprinted 1995), consid-
ers probability theory in the 17th and 18th centuries and how
it was understood as the mathematics of good sense. Ian
Hacking , The Taming of Chance (1990), covers statistical ideas
of regularity and order, set against a background of scientifi c
activity and bureaucratic intervention, in the 19th century.
Theodore M. Porter , The Rise of Statistical Thinking, 1820–1900
(1986), examines statistics as a strategy for dealing with large
numbers, its emergence in bureaucratic social science, and
its extension to the natural sciences, and his Trust in Numbers:
The Pursuit of Objectivity in Science and Public Life (1995) scru-
tinizes numbers, calculation, and objectivity, understood
as administrative tools. Gerd Gigerenzer et al., The Empire
of Chance: How Probability Changed Science and Everyday
Life (1989, reprinted 1991), contains essays on the develop-
ment of probability and statistics, from its roots in gambling
and insurance to its relations to science and philosophy, and
on to more recent applications in polling and baseball.
Probability Theory
F.N. David, Games, Gods, and Gambling: The Origins and
History of Probability and Statistical Ideas from the Earliest Times
325
7
Bibliography 7
to the Newtonian Era (1962), covers the early history of prob-
ability theory. Stephen M. Stigler, The History of Statistics:
The Measurement of Uncertainty Before 1900 (1986), describes
the attempts of early statisticians to use probability theory
and to understand its significance in scientific problems.
Statistics
Overviews are provided in David R. Anderson, Dennis J.
Sweeney, and Thomas A. Williams, Introduction to Statistics:
Concepts and Applications, 3rd ed. (1994), an introductory
treatment with modest mathematical prerequisites; Judith
M. Tanur et al., Statistics: A Guide to the Unknown, 3rd ed.
(1989), containing a variety of statistical applications on
topics of interest to the general reader; David Freedman
et al., Statistics, 2nd ed. (1991), an innovative treatment of a
variety of topics at the introductory level; and David S.
Moore and George P. McCabe, Introduction to the Practice of
Statistics, 2nd ed. (1993).
Game Theory
The seminal work in game theory is John von Neumann
and Oskar Morgenstern, Theory of Games and Economic
Behavior, 3rd ed. (1953, reprinted 1980). Two introductions
that require only high school algebra are Avinash K. Dixit
and Susan Skeath, Games of Strategy (1999); and Philip D.
Straffin, Game Theory (1993). Applications of game theory
are presented in Nesmith C. Ankeny, Poker Strategy: Winning
with Game Theory (1981, reprinted 1982); and Karl Sigmund,
Games of Life: Explorations in Ecology, Evolution, and Behavior
(1993). Histories of game theory can be found in William
Poundstone, Prisoner’s Dilemma: John von Neumann, Game
Theory, and the Puzzle of the Bomb (1992); and E. Roy
Weintraub (ed.), Toward a History of Game Theory (1992)
326
InDeX
A
Abel, Niels Henrik, 273
absorbing processes, 105
acceptance sampling, 148–149
additivity, principle of, 53–54, 82
Alembert, Jean Le Rond d’, 30,
244–248, 269
Analogy of Religion (Butler), 26
analysis of variance, 138,
141–142, 267
Appel, Kenneth, 230
a posteriori distribution, 75
Arbuthnot, John, 27, 30
Arnauld, Antoine, 26
Ars conxctandi (J. Bernoulli),
30–31, 251
Axelrod, Robert, 174
B
balanced incomplete block (BIB)
designs, 215–218
Banzhaf, John F., III, 192, 195
Banzhaf index, 193, 195
Banzhaf value, 192–196
Barrow, Isaac, 251
Bayes, Thomas, 31, 64, 135,
248–249, 291, 299, 314
Bayesian methods, 135–136, 314
Bayes’s theorem, 31, 64–65, 74,
135, 291–293
bell curve, 308
Berkeley, Bishop George, 249
Bernoulli, Daniel, 30, 74, 249–251
Bernoulli, Jakob, 30–31, 38, 54,
74, 76, 78, 251–252, 275, 300,
302, 314
Bernoulli, Nicolas, 28–30
Bhāskara II, 199, 253–254
Bhattacharya, K. N., 218
Bienaymé, Irénée-Jules, 296
Bienaymé-Chebyshev
inequality, 296
binomial distribution, 41, 94,
125–126, 293–295
biometry, 44
Birkhoff, George David, 103
birthday problem, 57–59
Boltzmann, Ludwig Eduard, 42,
254–255
Boole, George, 199, 255–257, 291
Boolean algebra, 255, 257
Borel, Émile, 84
Bose, Raj Chandra, 216, 219, 223
Bowley, Arthur, 45
Box-Jenkins autoregressive
integrated moving average,
145–146
Brahmagupta, 253
Brams, Steven J., 174
British Association for the
Advancement of Science, 35
Brown, Robert, 95
Brownian motion, 64, 86–87,
104, 110, 255
327
7
Index 7
Brownian motion process,
95–101
Bruck, Richard H., 228
Buckle, Henry Thomas, 38, 39
Butler, Joseph, 26
C
calculus, 31, 41, 101, 249, 251, 313
Cardano, Girolamo, 24, 257–259
Cassini, Jacques-Dominique,
272, 304
Cauchy distribution, 92
Cayley, Arthur, 199, 225, 237,
259–261, 284, 285
census, 21, 34–35, 39, 115
central limit theorem, 64, 78–80,
295–296
chance, games of, 21, 23, 24, 48
Chebyshev, Pafnuty, 296, 297, 303
Chebyshev’s inequality, 77, 78,
296–297, 303
chess, 157, 159–160, 287
Clarke, R. D., 314, 315
Clatworthy, Willard H., 220
Clausius, Rudolf, 38, 42
coding theory, 200
combinatorial geometry, 197,
232–234
methods of, 240–243
packing and covering and,
234–236
polytopes and, 236–238
combinatorial theory, 201
combinatorics, 197
history of, 198–202
completely randomized design,
136–137
conditional expectation, 92–94
conditional probability,
59–60, 64
applications of, 60–63
Condorcet, Marie-Jean-Antoine-
Nicolas de Caritat, marquis
de, 31, 32, 184
Condorcet voting paradox, 188
Connor, W. S., 218, 220
correlation, 144–145
countable additivity, axiom of,
87, 88
Craig, John, 32
Cramér, Harald, 112
crime rates, 37
criminal justice, 32, 33
D
Darwin, Charles, 42, 266
data, defi nition of, 115
decision analysis, 152–153
decision theory, 297–298
De Morgan, Augustus, 229, 256
derangement, 209, 210
Descartes, René, 251, 263, 264
design, theory of, 200, 215–220
Diderot, Denis, 246
diffusion processes, 99
Diophantus, 264, 265
distribution function, 298
Divine Providence, 27, 34
Doob, Joseph Leo, 114
E
econometrics, 44
and forecasting, 145, 146
Edgeworth, Francis Ysidro, 44,
261–262
7 The Britannica Guide to Statistics and Probability 7
328
Ehrenfest, Paul, 105
Ehrenfest model of diffusion,
105–108
Einstein, Albert, 95, 97, 98,
101, 288
electoral probabilities, 31
Electronic Numerical Integrator
and Computer (ENIAC)
computer, 289–290
Enlightenment, 28, 31, 46, 184
enumeration, problems of,
202–213
Equitable (insurance society), 26
Erdös, Paul, 215
ergodic theorem, 103
estimation, 299–300
interval, 301–302
point, 313–314
Euclid, 272
eugenics, 43, 44, 94, 266, 278
Euler, Leonhard, 199, 209, 210,
211, 221, 231, 233, 237–238
Eulerian cycle, 231
European Economic
Community (EEC), 194
evolution, 42, 44
expectation, 23
experimental design, 136–138
exploratory data analysis,
121–122
F
factorial design, 136, 137–138, 223
fair contract, 25
Federal Communications
Commission, 196
Federico, P. J., 237
Feller, William, 99
Fermat, Pierre de, 21, 23, 24,
199, 258, 262–265, 273,
274, 311
Ferrers diagram, 206–209
Finetti, Bruno de, 74
finite sample spaces, 82
Fisher, Sir Ronald Aylmer, 46, 47,
200, 265–267, 279, 294
Fisher’s inequality, 216
forecasting, 145
four-color theorem, 229–231
Fourier, Joseph, 282
French Revolution, 35, 40, 184
frequency distribution, 117
G
Galbraith, John Kenneth, 262
Gallup, George, 45
Galton, Francis, 42–44, 94,
278, 279
gambling, 21–24, 25, 105, 111,
112–113, 165
games
classification of, 156–158
cooperative, 168–169
n-person, 178–179, 189
one-person, 158
two-person constant-sum,
159–164, 165, 166, 167,
168, 169
two-person variable-sum,
166–167, 168, 171
game theory, 154–156, 163–164,
165, 166, 170, 195–196, 287,
288–289, 290
biological applications of,
176–178
n-person, 183
329
7
Index 7
Gauss, Carl Friedrich, 40, 265,
273, 297, 304, 309–310
Gaussian distribution, 40, 271,
308, 309
generating functions, 204–205, 208
Gibbs, J. Willard, 103
Gödel, Kurt, 287
Gosset, William Sealy, 319
graph theory, 199, 200–201,
224–228
applications of, 228–232
Graunt, John, 33, 267–268
Gregory, David, 234
group theory, 201
Guthrie, Francis, 229
H
Hadwiger, Hugo, 232
Haken, Wolfgang, 230
Hales, Thomas, 235
Hall, M., Jr., 214, 218
Halley, Edmond, 267, 274
Hall’s theorem, 214
Hamming, Richard W., 223
Hartley, David, 31
Heawood, P. J., 230
Heisenberg, Werner, 287
Helly, Eduard, 238
Helly’s theorem, 238–240, 243
Herschel, John, 39
Hilbert, David, 287
History of Civilization in England
(Buckle), 38, 39
Hocquenghem, Alexis, 224
Hume, David, 32
Huygens, Christiaan, 24, 25,
264, 274
hypothesis testing, 132–134
I
I Ching, 198
inclusion and exclusion,
principle of, 209
indifference, 300–301
infectious diseases, 30
inference, 301
infinite probability spaces, 80
infinite sample spaces,
82–84
information theory, 200
insurance, 21, 25–26, 165
insurance risk theory, 111–112
inverse probability, law of, 64
Ising problem, 212, 213
Itō Kyosi, 101
Ito stochastic calculus, 101
J
Jacobi, Karl, 261, 273
Jenner, Edward, 30
judicial probabilities, 31
K
Kempe, A. B., 230
Kendall, Maurice, 39–40
Kepler, Johannes, 235
Keynes, John Maynard, 300
Kiaer, A. N., 45
Kirkman, Thomas P., 200,
218, 237
Klee, Victor, L., 238
Kolmogorov, Andrey
Nikolayevich, 86
Königsberg bridge problem,
231–232