INTRODUCTION TO STATISTICS
IN PATHOLOGY.
(Procedure 152).
G. William Moore, MD, PhD.
Chief, Quality Assurance Section.
Chief, Autopsy Section.
http://www.netautopsy.org/axsop/axsop152.htm
NEXT PAGE
PREVIOUS PAGE
RETURN TO TABLE OF CONTENTS
United States Government Work, uncopyrighted, public-domain,
DRAFT COPY ONLY. This document does not necessarily represent the views
or policies of any United States Government agency. This document
is provided "as is", without warranty of any kind, express or implied,
including but not limited to the warranties of merchantability,
fitness for a particular purpose, and noninfringement. In no event
shall the authors be liable for any claim, damages or other liability,
whether in an action of contract, tort or otherwise, arising from,
out of, or in connection with the document or the use or other dealings
made with the document..
PRINCIPLE OF THE TEST.
Statistics is an important discipline in evaluating
quality assurance data in pathology.
SPECIMEN REQUIRED.
Not applicable.
REAGENTS, INSTRUMENTATION.
Not applicable.
STEP-BY-STEP DESCRIPTION.
1. TABLE OF CONTENTS.
1. Table of Contents.
2. Introduction.
Probability.
The mathematical study of collections of repeated, chance events.
Statistics.
The mathematical study of repeated sampling from a larger population.
3. Descriptive Statistics. Simple Statistics.
Histogram.
Summation, ∑.
Central Tendency.
(Arithmetic) Mean. Median. Mode. Harmonic Mean. Geometric Mean.
Sample mean, x.
Population mean, μ.
Scatter.
Range.
Percentile, Decile, Quintile, Quartile.
Sample variance, s2.
Sample standard deviation, s.
Sample standard error, sx.
4. Sample versus Population.
Sample.
Sample mean, x.
Sample variance, s2.
Sample standard deviation, s.
Sample standard error, sx.
Population.
Population mean, μ.
Population variance, σ2.
Population standard deviation, σ.
Population standard error, σx.
5. Design of Experiments.
Random sample.
Random numbers.
(Pseudo)-Random number generators.
Stratification.
Sampling Plans.
Control all relevant variables.
Random assignment to treatment and no-treatment groups.
Randomized double-blind study.
6. Estimation. Hypothesis Testing.
Estimation.
Expected Value.
Method of Least Squares (Gauss).
Hypothesis Test.
Type I Error and Type II Error.
Null Hypothesis.
Alternative Hypothesis.
Significance Level.
Confidence Limits.
7. Theory of Probability.
Set Theory.
Axioms of Probability.
Conditional Probability. P(B|A) = P(A∩B)/P(A).
Binary Tree.
Bayes' Theorem.
Normal=Gaussian Distribution. Invented by de Moivre (1720).
Binomial Distribution.
Poisson Distribution.
8.
Correlation, Linear Regression.
Linear Regression.
Correlation.
9. Contingency Table Analysis.
Contingency Table.
Chisquare Test. χ2.
Fisher Exact Test.
Token Swap Test.
10. History of Statistics.
10. History of Statistics.
Aristotle (384 BCE-322 BCE).
Archimedes (287? BCE-212 BCE).
Euclid (?-?300 BCE).
Leonardo Pisano Fibonacci (1170?-1240?).
Occam. William of Ockham (1300?-1349).
John Napier, Laird of Merchiston (1550-1617).
Galileo Galilei (1564-1642).
René Descartes (1596-1650).
Sir Isaac Newton (1642-1727).
Abraham DeMoivre (1667-1754).
Daniel Bernoulli (1700-1782).
Leonhard Euler (1707-1783).
Pierre de Fermat (1601-1665).
John Graunt. Eighteenth century British gentleman.
Rev. Theodore Bayes.
Karl F. Gauss (1777-1855).
Gregor Mendel (1822-1884).
Sir Francis Galton (1822-1911).
Samuel Clemens (Mark Twain) (1835-1910).
Benjamin Disraeli, Earl of Beaconsfield (1804-1881).
John Maynard Keynes (1883-1946).
Karl Pearson. Early twentieth century British statistician.
Aleksander N. Kolmogorov. Great 20th c. Russian statistician.
Jerzy Neyman (1894-1981).
Joseph Berkson.ÿ2020th century British statistician.
Albert Einstein (1879-1955).
Kurt Gödel. 20th c. Austrian-American mathematician.
Alan B. Turing. 20th c. British mathematician.
William S. Gossett (Student). An employee of the Guinness Brewery.
Sir Ronald A. Fisher. Greatest British statistician of the twentieth century.
11. References.
2. INTRODUCTION.
NEXT PAGE.
RETURN TO TABLE OF CONTENTS.
PROBABILITY is the mathematical discipline that studies
the behavior of repetitive, chance events. Probability has been an area
of interest since games of chance (variations of dice, played with
animal bones) were played in ancient Egypt and in ancient China.
Probability was developed as a systematic mathematical discipline
in the eighteenth century in Western Europe, as the idle-rich noblemen
wished to better their performance at the gaming tables, and called upon
their court wizards/philosophers in mathematics to improve their winnings.
STATISTICS is the mathematical discipline devoted to
drawing inferences from large quantities of data, collected repetitively.
The name derives from STATUS (Latin: the State), because
the first statistics were the BILLS OF MORTALITY, collected by
the British government in 18th century London. In European countries with
inheritance taxes, dead citizens and the causes of their demise
have always been a serious matter to the government. The CORONER
(Latin: CORONA=crown), was originally a British tax official who determined
the cause-of-death, and thus the disposition of the decedent's estate.
The 18th century British gentleman, John Graunt
was the first person to organize these death statistics
(i.e., summary data on many persons),
and use them for descriptive and predictive purposes.
Graunt documented the plague epidemic in London,
and was able to conclude that the outbreak originated from
unsanitary conditions of the poorer districts of London,
leading to social reforms in public health.
Since probability and statistics both involve the study of
repetitive chance events (Roll of dice; death in London),
they are intellectually cognate disciplines.
3. DESCRIPTIVE STATISTICS. SIMPLE STATISTICS.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
HISTOGRAM.
A HISTOGRAM is a block diagram, as illustrated below,
in which each block corresponds to a case or a subject or a patient:
y= | _____
| | |
8 | |___|____
| | | |
7 | |___|___|
| | | |
6 | ____|___|___|
| | | | |
5 | |___|___|___|____
| | | | | |
4 | ____|___|___|___|___|____
| | | | | | | |
3 | |___|___|___|___|___|___|
| | | | | | | |
2 |_______|___|___|___|___|___|___|____
| | | | | | | | | |
1__|___|___|___|___|___|___|___|___|___|____
x= 0 1 2 3 4 5 6 7 8
The horizontal axis, or abscissa, or x-axis, corresponds to the
value observed for each case. The vertical axis, or ordinate,
or y-axis, corresponds to the number of observations with
that value. In the above histogram, there is a total of 33 cases,
A SAMPLE SIZE OF n=33, including:
One case with value x=0.
One case with value x=1.
Three cases with value x=2.
Five cases with value x=3.
Eight cases with value x=4.
Seven cases with value x=5.
Four cases with value x=6.
Three cases with value x=7.
One case with value x=8.
Total of 33 cases.
A STATISTIC is a calculation performed on the observations
in the histogram. The calculation of a statistic has three purposes:
estimation of the CENTRAL TENDENCY of the histogram;
estimation of the SCATTER of the histogram;
and a HYPOTHESIS TEST performed upon the histogram.
The most popular statistic
for estimating the central tendency of a histogram is the
SAMPLE ARITHMETIC MEAN for this histogram is obtained by adding up
all the observation-values, and dividing by the sample size, n.
Summation,
∑i=1 n xi
= (x1 + x2 ... + xn)
read: the sum of xi from i equals 1 to n,
For the example histogram, the mean is:
(0 + 1 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4
+ 5 + 5 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 8)/33 = 4.2.
Central Tendency is a class of statistics
that measures the central region of the histogram.
Measures of Central Tendency include:
Sample (Arithmetic) Mean. Median. Mode. Harmonic Mean. Geometric Mean.
Sample arithmetic mean,
x.
x = (∑i=1 n xi)/n.
Median.
The x-value in the histogram where half of the observations
are below the median and half of the observations are above the median.
Mode.
Highest y-value on the histogram.
There may be several modes on the same histogram.
Harmonic Mean. Geometric Mean.
For specialized scientific problems.
Scatter/Dispersion
is a statistic that measures the scatter or width of the histogram.
Measures of Scatter/Dispersion include:
Range.
Percentile.
Decile.
Quintile.
Quartile.
Binile=Median.
Range.
The lowest and highest observed value.
For the example histogram, the range is 0 to 8.
Percentile.
Division of the histogram into 100 parts.
The first percentile is the value below which 1% of the observations fall.
The second percentile is the value below which 2% of the observations fall.
The third percentile is the value below which 3% of the observations fall.
....
The 99th percentile is the value below which 99% of the observations fall.
Decile.
Division of the histogram into 10 parts.
The first decile is the value below which 10% of the observations fall.
The second decile is the value below which 20% of the observations fall.
....
The ninth decile is the value below which 90% of the observations fall.
Quintile.
Division of the histogram into 5 parts.
The first quintile is the value below which 20% of the observations fall.
The second quintile is the value below which 40% of the observations fall.
....
The fourth quintile is the value below which 80% of the observations fall.
Quartile.
Division of the histogram into 4 parts.
The first quartile is the value below which 22% of the observations fall.
The second quartile is the value below which 50% of the observations fall.
The third quartile is the value below which 75% of the observations fall.
Binile=Median.
Division of the histogram into 2 parts.
The median is the value below which 50% of the observations fall.
Sample variance,
s2 = (∑ni=1
(xi) - x)2)/(n-1).
Sample standard deviation, s = √(s2).
Sample standard error, sx =
√(s2/n).
4. SAMPLE VERSUS POPULATION.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
Sample.
The set of ACTUAL OBSERVATIONS THAT YOU COLLECTED.
The SAMPLE SIZE is usually denoted n.
Sample mean, x.
Sample variance, s2.
Sample standard deviation, s.
Sample standard error, sx.
Population.
The set if ALL POSSIBLE OBSERVATIONS.
Population mean, μ.
Population variance, σ2.
Population standard deviation, σ.
Population standard error, σx.
STATISTICS is the mathematical discipline
that determines properties of the population from measurements
on the individual sample.
5. DESIGN OF EXPERIMENTS.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
Random sample.
A random sample is a sample drawn from a population
which, after repeated samples, has the same statistical behavior
as the population itself.
One of the great missteps of Baltimore journalistic/statistical history
was the prediction in 1932 by the American Mercury magazine,
edited by the late Baltimore pundit,
H. L. Mencken, that Herbert Hoover would win re-election to the
U.S. presidency by a large margin.
A telephone survey was conducted among a sample of U. S. citizens,
and a majority of the respondents preferred Hoover over his Democratic
opponent, Franklin D. Roosevelt. In actual fact, Roosevelt won
by a landslide, and soon thereafter, the American Mercury
went out of business. The U.S.A. was in the middle of
a deep economic depression (as was the rest of the western world),
and the American people
held Mr. Hoover, the sitting president, responsible for this disaster.
What was wrong with the survey by the American Mercury?
Answer: The survey was confined exclusively to telephone owners,
who were not a random sample of the U. S. voting population,
and who selectively represented richer citizens who were not suffering
from the economic depression. In this example, the telephone sample
was not drawn from the population of interest, namely U. S. voters,
and did not have the same statistical behavior as the population itself.
Random number sequence.
is a sequence of numbers, usually integers between a given range,
say, 0 through 99, which have no repeating sequence.
For example, 1/7 is NOT a random sequence, because it repeats:
1/7=0.142857142857142857142857142857142857142857142857142857....
CYCLES after 6 digits.
However, pi = 3.141592.... IS a random sequence.
(Pseudo)-Random number generator.
A computer program that generates a pseudorandom number sequence.
Theoretically, there can never be a computer random number generator,
because a computer is a finite machine, and every finite machine
must have a cycle time (Turing's Theorem). However, available
pseudorandom number generators have cycle sizes so huge (i.e., larger cycle
number than seconds left in the life of the known universe), that they are
effectively true random generators for ordinary statistical studies.
Stratification.
Sampling Plans.
Control all relevant variables.
Random assignment to treatment and no-treatment groups.
Randomized double-blind study.
Patients selected randomly by computer.
Neither
the subject nor the investigator
(e.g., neither the patient nor the doctor)
knows who gets the TREATMENT and who gets the PLACEBO.
Look for:
1. sample drawn randomly from desired population.
2. confounding factors.
3. event space.
6. ESTIMATION. HYPOTHESIS TESTING.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
ESTIMATION.
Expected Value.
The population value for a statistic.
Examples: population mean, population standard deviation.
Method of Least Squares (Gauss).
Method for determining the best-fit for a sample-statistic,
by minimizing the sum of squared deviations from the expected value.
HYPOTHESIS TEST.
Test for determining the validity of a particular statistical hypothesis,
such as whether the mean of the example histogram is greater than zero.
TYPE I ERROR AND TYPE II ERROR.
TYPE I ERROR, or false positive, or error of commission, is where
you claim that the null hypothesis is false, when it is actually true.
TYPE II ERROR, or false negative, or error of omission, is where
you claim that the null hypothesis is true, when it is actually true.
NULL HYPOTHESIS.
Statement of the CONSERVATIVE STATISTICAL HYPOTHESIS,
such as the assertion that the mean of the example histogram
is equal to zero.
Most statistical tests
calibrate the proportion of Type I Error, e.g.,
p<0.05 means that the proportion of Type I Errors
is less than 5%.
ALTERNATIVE HYPOTHESIS.
Any NEGATION OF CONSERVATIVE STATISTICAL HYPOTHESIS,
such as the assertion that the mean of the example histogram
is not equal to zero.
SIGNIFICANCE LEVEL.
Percentage below which one
REJECTS THE NULL HYPOTHESIS.
Typically, 5%, 1%, or 0.1%.
Typically denoted:
p<0.05,
p<0.01, or
p<0.001.
CONFIDENCE LIMITS.
Upper and lower limits around the sample mean before one
REJECTS THE NULL HYPOTHESIS
y= | |
| |
8 | |
| |
7 | _____ | _____
| | | | | |
6 | |___| | ____|___|
| | | | | | |
5 | |___| | |___|___|____
| | | | | | | |
4 | ____|___| | |___|___|___|
| | | | | | | | |
3 | |___|___| | |___|___|___|
| | | | | | | | |
2 |_______|___|___| | |___|___|___|____
| | | | | | | | | | |
1__|___|___|___|___|_|_|___|___|___|___|____
x= 0 1 2 3 4 5 6 7 8
|
|
7. THEORY OF PROBABILITY.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
Set Theory
is the theory of collections, or sets, of abstract mathematical objects.
Simple set theory can be used to describe the basic axioms and properties
of probability theory.
There are two basic concepts of set theory: the null set, or empty set,
denoted Ø or {}; and set membership, denoted ∈.
The null set is the set containing no members,
i.e., there exists no x ∈Ø.
Subset, ⊆.
A ⊆ B means that every member of A is also a member of B.
Set subtraction, ~.
(A~B) is the set of all elements belonging to A but not B.
Set union, ∪.
(A ∪ B) is the set of all elements belonging to either A or B or both.
Set intersection, ∩.
(A ∩ B) is the set of all elements belonging to both A and B.
The event space, E,
is the set of
all possible events.
For example, the event space for a single coin toss is: {H,T}
(two members).
The event space for a double coin toss is: {HH, HT, TH,TT}
(four members).
The event space for a triple coin toss is:
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (eight members).
For fair coin tosses, all events in the event space are
EQUIPROBABLE.
Thus, for a single fair coin toss, each event has probability 1/2.
For a double fair coin toss, each event has probability 1/4.
For a triple fair coin toss, each event has probability 1/8.
Axioms of Probability.
1. P(Ø)=0. The probability of no events is zero.
2. P(E)=1. The probability of all events is one.
3. If A, B ⊆ E, and (A∩B) = Ø,
then P(A∪B) = P(A)+P(B).
If two events, A and B, are mutually exclusive
then the probability of one or the other event equals
the probability of one event plus the probability of the other event.
Conditional Probability. P(B|A) = P(A∩B)/P(A).
read: The probability of B given A equals the probability of A and B,
divided by the probability of A.
Binary Tree.
Many probablistic chains-of-events can be diagrammed with a
BINARY TREE.
___________ P(A&B)
|
P(A) |
___________| P(B)
| |
| |__________ P(A-B)
P(E) |
________|
|
| ___________ P(B-A)
| |
|__________| P(E-B)
P(E-A) |
|
|_________ P(E-A-B)
Bayes' Theorem.
Normal=Gaussian Distribution. Invented by de Moivre (1720).
Binomial Distribution.
Poisson Distribution.
8. LINEAR REGRESSION, LINEAR CORRELATION.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
LINEAR REGRESSION is a method for estimating
the best-possible line of the form
y = ax + b
on an x-y plane, to pass through a collection of (x,y)-datapoints.
The least-squares estimates for the slope (a) and the y-intercept (b) are:
a = (∑xy-((∑x∑y)/n))/((∑x2)-((∑x)2)/n));
b= ((∑y-a∑y)/n)
LINEAR CORRELATION
is the measure of how well the best-fit line
fits the observed data.
The usual measure of correlation is
LINEAR CORRELATION COEFFICIENT (PEARSON'S r),
where r2
ranges in value from 0 (no correlation) to 1 (perfect correlation).
The formula for Pearson's r is:
r = (∑xy-((∑x∑y)/n))/√(((∑x2)-((∑x)2)/n)×((∑y2)-((∑y)2)/n))
9. CONTINGENCY TABLE ANALYSIS.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
Contingency Table.
A contingency table is a rectangular table with rows
and columns. The simplest contingency table is the
2×2 Contingency Table
(2×2CT),
with two rows and two columns.
Typically a
(2×2CT) is a contest between a
new test
for a particular disease or condition, versus an established test
or
gold standard, as follows:
Gold Std: No Yes
________________________________
| | |
Test: No | a | b | v
| | |
________________________________
| | |
Test: Yes | c | d | w
| | |
________________________________
| | |
| x | y | z
| | |
In this
2×2 Contingency Table,
the
CELL TOTALS
are
a, b, c, d.
That is, the number of patients with Gold Standard=No, Test=No is
cell total a.
The number of patients with Gold Standard=Yes, Test=No is
cell total b.
The number of patients with Gold Standard=No, Test=Yes is
cell total c.
The number of patients with Gold Standard=Yes, Test=Yes is
cell total d.
Cell totals
a, d
represent patients with a correct outcome, that is the new test matches
matches the gold standard.
Cell total
b
represents a
false positive, or false alarm
where the gold standard is no but the new test is yes.
Cell total
c
represents a
false negative, or unintended miss
where the gold standard is yes but the new test is no.
Row marginal totals
u, v,
represent the sum of both cells for a particular row.
That is,
u = a + b,
v = c + d.
Column marginal totals
x, y,
represent the sum of both cells for a particular column.
That is,
x = a + c,
y = b + d.
Grand total, z,
is the sum of all four cell totals.
That is, z = a + b + c + d = u + v = x + y.
The CHISQUARE TEST, χ2.
χ2 = ∑((o-e)2)/e, where
o=observed, e=expected.
FISHER EXACT TEST. F = [n!/(k!×(n-k)!)] ×
[pk×(1-p)(n-k)],
where n is the grand total, and k is a given cell total.
10. FIGURES.
Figure 1. Histogram.
Figure 2. Bell-shaped curve.
Figure 3. Histogram and bell-shaped curve.
Figure 4. Linear correlation (xy-plot).
11. HISTORY OF STATISTICS.
NEXT PAGE.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
The following list of the greats of statistics includes both its founders
as well as some of its famous detractors (including Benjamin Disraeli,
Mark Twain, and Albert Einstein).
Aristotle (384 BCE-322 BCE).
Greek philosopher, who compiled an encyclopedia of all scientific
and other human knowledge available at that time. Aristotle's Rule:
for every positive y such that x > y,
there exists an n > 0 such that y > x.
Loosely speaking, Aristotle was the first biostatistician,
since his works contain discussions of biology and chance phenomena.
See:
http://en.wikipedia.org/wiki/Aristotle
Archimedes (287? BCE-212 BCE).
Ancient Greek mathematician, one of the
three greatest mathematicians of all time (Archimedes, Newton, Gauss).
Archimedes' The Sand Reckoner is the first serious effort
to deal with large-number problems, in this case, the number of grains
of sand on the entire Earth. The elements of calculus are present
in this document. Believe it or not, Archimedes was pretty close
to the right number of grains! Statistics is the mathematical study
of repeated sampling from a larger, possibly infinite, population.
Archimedes laid the foundation for such large-number studies.
See:
http://en.wikipedia.org/wiki/Archimedes
Euclid (?-?300 BCE)
Ancient Greek mathematician, who summarized
the rules of geometry, or Elements, which had been known as
an empirical science by the ancient Egyptians for at least
a millennium previously. Euclid's main contribution is that he collected
the known truths of geometry and derived all geometric theorems from
two undefined concepts (point, line) and five postulates, using the idea of
proof by deduction. The Pythagorean theorem, namely that
in a right triangle with legs a, b, and hypoteneuse c:
a2 + b2 = c2
is the basis for the Theory of Least Squares (see below),
one of the bedrock methods of statistics.
See:
http://en.wikipedia.org/wiki/Euclid
Leonardo Bigollo Pisano Fibonacci (1170?-1240?).
Leonard of Pisa, pre-renaissance Italian mathematician. Fibonacci's
Liber Abaci (1202) (Latin: Book of the Abacus)
introduces Arabic numerals to European mathematics, namely:
٠١٢٣٤٥٦٧٨٩
Yes, the real Arab numerals are: 0=٠, 1=١, 2=٢,
3=٣, 4=٤, 5=٥, 6=٦, 7=٧, 8=٨, 9=٩.
Do you see the resemblance? These numerals are used today in the Middle-East,
and are used to enumerate verses in the Holy Quran (written in 7th century,
Classical Arabic). Europe had been in an intellectual dark ages between
the sacking of Rome in 476 C.E. until the Renaissance (13th-15th centuries).
Fibonacci imported Arabic numerals to 13th century Italy.
Arabic numerals, including zero, are far superior to Roman numerals
(or the even more primitive numerals of Ancient Greek, Hebrew, and Chinese
cultures) for both accounting and mathematics.
Fibonacci was the son of a merchant from Pisa, Italy, living in Algeria,
where he was educated by Arab teachers, who were familiar
with the Arab numeral system, and the great Arab masters of mathematics,
such as Al-Khwárizmí and Abu-Kamil.
The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...,
where each number in the sequence is the sum of the two previous numbers.
The ratio of a given Fibonacci number to its immediate predecessor
approaches the so-called Golden Ratio or Divine Proportion,
(1+√5)/2 = 1.6181....,
for large Fibonacci numbers (in the language of calculus, Fibonacci numbers
approaching infinity). The Golden Ratio is observed in many areas of nature
and esthetics. For example, the petals of a rose and a sunflower
are arranged in this ratio; allegedly the ratio of body-height to
the height of the umbilicus (belly-button) of a beautiful woman
satisfy this ratio; and the dimensions of nearly all national flags
(including the U.S. flag, of course) are in the Golden Ratio.
See:
http://en.wikipedia.org/wiki/Fibonacci
Occam. William of Ockham (1300?-1349).
English philosopher.
Occam's Razor: Entia praeter necessitatem non sunt multiplicanda.
Latin: Entities should not be multiplied beyond necessity.
Paraphrased: The simplest explanation is best.
See:
http://en.wikipedia.org/wiki/William_of_Ockham
John Napier, Laird of Merchiston (1550-1617).
Sixteenth century
Scottish nobleman, gentleman of leisure, and the inventor of logarithms,
which transformed multiplication problems into addition problems,
and division problems into subtraction problems. This advance
in calculation methods led the way to increased accuracy of navigation
and astronomy. Essentially there would never have been a Galileo,
a Kepler, or a Newton, without Napier's incredibly powerful advance
in calculation, the 16th century equivalent of the digital computer.
Napier's discovery is a simple extension of the idea that:
bi×bj = bi+j
which had been known since Euclid and Archimedes.
For example:
8×16 = 23×24 = 23+4
= 27 = 128.
That is, if you want to multiply the two numbers, x and y, where
x = bi and y = bj, all you have to do is look up
the value of i for which x = bi
and the value of j for which y = bj.
Then add i+j (a lot easier than multiplication), and look up
the value z = bi+j. Napier spent twenty years developing
a table of LOGARITHMS, where the value of i for which
x = bi is called the logarithm of x to the base b,
or i = logbx. A reversal of this same process converts
long division into subtraction, which is worlds of difficulty easier.
Napier used 10,000,000 as the base for his logarithm table,
and a contemporary mathematician, Wolfgang Burgi, converted
Napier's tables into base-10, which is to be found in most
high-school algebra texts. The so-called
BASE OF NAPIERIAN LOGARITHMS, e, is the value
2.71......., has a deep significance in calculus,
is named in honor of Napier, but was not known during Napier's lifetime.
See:
http://en.wikipedia.org/wiki/John_Napier
Galileo Galilei (1564-1642).
Seventeenth century Italian physicist and astronomer, who discovered
four of the moons of Jupiter and provided a mathematical basis for
Polish physicist Copernicus' hypothesis that the Sun, not the Earth,
lay at the center of the universe.
For making the latter assertion, Galileo was placed under house arrest
by the Papal Court in 1630 (a better fate than that of
Italian physicist Giordano Bruno,
who was burnt at the stake in 1630 for a similar crime).
In the late 17th century, Catholic university physics departments
began to teach Galileo's doctrines.
In 1993, more than three-and-a-half centuries after Galileo's arrest,
Pope John Paul II exonerated him [Beckmann, 1960].
See:
http://en.wikipedia.org/wiki/Galileo
René Descartes (1596-1650).
Inventor of Analytic Geometry, an algebraic mirror of Euclidean geometry.
In an instant, all the true statements of geometry were true for algebra,
and all the true statements of algebra were true for geometry,
thus doubling the body of provable statements (theorems) in mathematics.
This exact relationship between the statements of algebra and
the statements of geometry is called a ONE-TO-ONE MAPPING.
Dr. Lawrence A. Brown has suggested a Biblical prophesy foreshadowing
Descartes's Theory is Matt 16:18 (known to all educated Roman Catholics):
"Thou art Peter, and upon this rock shall I build my church,
and I shall give thee the Keys to the Kingdom of Heaven,
and whatsoever is bound on earth will be bound in heaven;
and whatsoever is loosed on earth will be loosed in heaven."
St. Peter, the first Roman Catholic pope, serves as a one-to-one mapping
between heaven and earth.
See:
http://en.wikipedia.org/wiki/Descartes
Sir Isaac Newton (1642-1727).
Seventeenth century British physicist
and mathematician, one of the three greatest mathematicians of all time
(Archimedes, Newton, Gauss). Inventor of Classical Physics,
in his classic book: Principia Mathematica Philosophiae Naturalis (Latin:
Mathematical Principles of Natural Philosophy. Co-inventor, with
Blaise Pascal, of the Binomial Formula, which is used to build
the Method of Least Squares. Co-inventor, with Gottlob Leibniz,
of differential and integral calculus, which are used in proving
the Methods of Least Squares.
See:
http://en.wikipedia.org/wiki/Isaac_Newton
Abraham DeMoivre (1667-1754).
Inventor of the Normal Distribution.
See:
http://en.wikipedia.org/wiki/de_Moivre
Blaise Pascal (1623-1662).
Pascal's Wager. Pascal's Triangle.
Pascal's Wager is a bet with the Lord God that He exists,
using risk-benefit analysis.
That is, the risk of unbelief (infinite) is so great, compared
to the cost of belief (i.e., regular religious observance),
that one is better off being religiously observant. This analysis
is considered to be one of the first formal, mathematical treatments
of risk-benefit analysis.
See:
http://en.wikipedia.org/wiki/Blaise_Pascal
Daniel Bernoulli (1700-1782).
Swiss mathematician. Bernoulli Trial.
See:
http://en.wikipedia.org/wiki/Daniel_Bernoulli
Leonhard Euler (1707-1783).
Swiss mathematician. Inventor of topology
(the famous Königsberg bridge problem).
It is said that every calculus textbook is either Euler,
a copy of Euler, or a copy of a copy of Euler [Agnew, 1960].
Worked extensively with Pascal's triangle and the binomial distribution.
The Swiss government office has honored Dr. Euler on their
ten-franc note.
Pierre de Fermat (1601-1665).
Seventeenth century French civil servant and amateur mathematician.
In Fermat's literary estate, there is an assertion written in one
of the books in Fermat's library (the works of Diophantus, the ancient
Greek mathematician, famed for his work with integer arithmetic), that
for integers x, y, z, and k, the equation:
xk = yk + zk
is only true for k=2, so-called Fermat's Last Theorem.
When k=2, there are numerous examples of this equation,
the most being the famous 3-4-5 triangle, for which
32 = 42 + 52,
where 3 and 4 are legs of a right triangle with hypoteneuse equal 5.
Fermat wrote in the margin that he knew a proof, but didn't have
room to write it into the book margins.
No proof and no counterexample were found for the next 350 years.
Sir Andrew Wiles, mathematician from Cambridge and Princeton,
and his student, Dr. Richard Taylor, finally
proved this theorem in the early 1990s, three-and-a-half centuries,
after it had tormented the minds of virtually every great mathematician
living during those years.
See:
http://en.wikipedia.org/wiki/Fermat
Rev. Thomas Bayes.
British Anglican priest who developed the theory of conditional probability.
See:
http://en.wikipedia.org/wiki/Bayes
John Graunt.
Eighteenth century British gentleman,
who first organized death statistics, London Bills of Mortality.
and used them for descriptive and predictive purposes.
Graunt documented the plague epidemic in London,
and was able to conclude that the outbreak originated from
unsanitary conditions of the poorer districts of London,
leading to social reforms in public health.
See:
http://en.wikipedia.org/wiki/John_Graunt
Karl F. Gauss (1777-1855).
Nineteenth century German mathematician, one of the three greatest
mathematicians of all time (Archimedes, Newton, Gauss). Gauss invented
the method of least squares, which forms the backbone of
modern statistical theory. (Actually, Gauss always claimed that he copied
the method from a mathematical colleague [Bühler, 1980].)
Gauss also created a mathematical framework for arithmetic
and classical physics.
Gauss attempted to verify from actual physical measurements whether
Euclid's controversial theorem, that the angles of a triangle
always add up to 180o, was actually true.
See:
http://en.wikipedia.org/wiki/Carl_Gauss
Gregor Mendel (1822-1884).
Czech-German geneticist who first discerned the principles of inheritance,
from experiments on pea-plants: Law of Recessive Inheritance.
Law of Segregation.
Sir Ronald A. Fisher later demonstrated statistically that Mendel
had probably fudged his data a little bit. Copies of the paper
(English and the original German) are available on the Internet.
See:
http://en.wikipedia.org/wiki/Gregor_Mendel
Sir Francis Galton (1822-1911).
Nineteenth century British statistician and biologist,
who studied the statistical behavior of inherited traits
in human populations. Galton's name is (falsely) associated
with racist doctrines common in 19th century Britain,
regarding the supposed genetic superiority of the British people.
See:
http://en.wikipedia.org/wiki/Francis_Galton
Samuel Clemens (Mark Twain) (1835-1910).
American humorist. Very few professions were spared
from Mark Twain's acerbic wit, including statisticians.
In his book, Life on the Mississippi,
Twain speculates on the fact that the Mississippi River is continually
becoming shorter (because stagnant ox-bow lakes in the Mississippi delta
become truncated). According to statisticians (using linear regression
methods, see below), Twain says, during the Roman Empire, New Orleans
must have been as distant from St. Louis as it was from the moon;
whereas in the 21st century, we can expect that New Orleans will become
a suburb of St. Louis.
The Mississippi River begins in St. Louis at the confluence
of the .... rivers, and ends south of New Orleans, where
the Mississippi River empties into the Gulf of Mexico.
The flow of the Mississippi River slows down near its termination,
so that the river becomes very tortuous (twisted) near its mouth,
the Mississippi Delta.
The twist in the river forms a so-called oxbow lake.
Everytime this happens, the length of the entire Mississippi river
shortens by a few miles, so that during Mark Twain's lifetime,
The Mississippi River lost several hundred miles of its entire length.
Using linear regression analysis, you would predict that the
Mississippi River
was tens of thousands of miles long during the Roman Empire,
but would become only several miles south of New Orleans
in the next millennium.
See:
http://en.wikipedia.org/wiki/Mark_Twain
Benjamin Disraeli, Earl of Beaconsfield (1804-1881).
Conservative British Prime Minister during the Victorian Era.
"There are lies, damn lies, and statistics." It is not an accident
that statistics developed in Great Britain, and that the world's best
statisticians still live and work there. Great Britain is an island nation,
and has always made its national livelihood from maritime trade.
Ships at sea, like dice at a gaming table, are subject to chance
occurrences. In his career, Disraeli must have seen more than his share
of deceptive statistics.
See:
http://en.wikipedia.org/wiki/Benjamin_Disraeli
John Maynard Keynes (1883-1946).
"In the long run, we're all dead."
British economist, who developed concepts of national fiscal
and monetary policy. Many economic theories distinguish between
short-run and long-run processes, without really specifying how long
is long-run. This quote is Keynes's ridicule of this particular paradox
of academic economics.
See:
http://en.wikipedia.org/wiki/John_Maynard_Keynes
Karl Pearson.
Early twentieth century British statistician, who introduced the
correlation coefficient, or Pearson's r.
Father of E. S. Pearson, another twentieth century statistical giant.
See:
http://en.wikipedia.org/wiki/Karl_Pearson
Aleksander N. Kolmogorov.
Great 20th c. Russian statistician and mathematician, who introduced many
non-parametric methods in statistics, including the Kolmogorov-Smirnov test.
See:
http://en.wikipedia.org/wiki/Kolmogorov
Jerzy Neyman (1894-1981).
Polish-American statistician, who developed a method for assessing
the robustness of a statistical formula, the Neyman-Pearson Theorem.
Neyman was a playful genius, who called the null hypothesis, "the devil".
I once attended a lecture that he gave at North Carolina State University
Department of Statistics, where I received my PhD.
Joseph Berkson.
ÿ2020th century British statistician. Berkson's Paradox.
In short, if you don't die of one thing, you'll die of another. For example,
if you collect a sample of autopsied patients who died of cancer, you will
discover that a lower proportion of these patients have significant
atherosclerosis (hardening of the arteries, which can lead to heart disease,
stroke, kidney failure, etc.) than the proportion of atherosclerotic patients
in the general population. Since cancer and atherosclerosis can both lead
to death, the lower prevalence of atherosclerosis among cancer patients
is explained by the fact that cancer patients die of their cancer before
they acquire significant atherosclerosis. Before Berkson made this
observation, there was a lot of nonsense in the medical literature
that cancer somehow protected you against atherosclerosis,
and therefore it was somehow good for your heart if you got cancer.
Albert Einstein (1879-1955).
Swiss-American Physicist, and winner of the 1921 Nobel Prize in Physics.
Einstein's theory of relativity revolutionized our concepts of space and time
in physics, but Einstein was always suspicious of non-deterministic,
i.e., probabilistic, methods to describe physical observations.
His statement, in a letter to Max Born:
Der Herrgott würfelt nicht.
German: The Lord God does not play dice.
I have heard four interpretations of this famous saying.
First, that Einstein deeply believed that every move made by the
Lord God was planned and pre-determined. Not a swallow falls
from the sky without knowledge of the Lord God.
Second, that Einstein did not understand probability theory very well,
and was trying to rescue physics from the evils of quantum mechanics,
a probabilistic-statistical theory of physics invented by Erwin
Schrödinger in the 1930s.
Third, that Einstein lined up with the 19th-20th century religious Christian
fundamentalists in Britain and USA who believe that games of chance are
morally wrong (see below).
Fourth, that Einstein, not a man with a small ego, was giving
marching orders to the Lord God.
See:
http://en.wikipedia.org/wiki/Einstein
William S. Gossett (Student).
An employee of the Guinness Brewery in Dublin, Ireland,
who wrote the ground-breaking papers in the British journal, Nature,
about the Student t test. Gossett was a student of Karl Pearson,
but because Gossett did his work as an employee, he concealed his identity
because of his commercial ties. His papers were signed, simply,
Student.
The Guinness Book of World Records
was written by the Guinness Brewery as an aid to settle arguments
in British bars were Guinness products were served.
See:
http://en.wikipedia.org/wiki/William_Sealey_Gossett
Sir Ronald A. Fisher.
Greatest British statistician of the twentieth century.
Sir Ronald corrected a small error in a formula for variance
that had originally been promulgated by Karl Pearson.
Fisher proved that the correct formula for the sample variance
is:
s2 = (∑ni=1
(xi) - x)2)/(n-1), not
s2 = (∑ni=1
(xi) - x)2)/n, as Pearson had thought.
Sir Ronald was the scientist who demonstrated statistically
that Mendel had probably fudged his data.
The F-test for the analysis of variance is named in honor of Fisher.
However, Sir Ronald sold out to the tobacco industry.
When the news first emerged that tobacco use was bad for your health,
Fisher defended the tobacco industry by asserting that the
cause-effect relationship was not conclusively demonstrated.
Fisher developed the concept of CONFOUNDING,
in which he argued that tobacco users might have some other
mysterious quality that caused them to develop tobacco-related
illnesses, apart from the tobacco use. Fisher's prominence
in the field of statistics helped the tobacco industry hide
from its responsibilities for a number of years.
Fisher's assertion was eventually rebuffed by the fact
that tobacco users who quit experienced subsequent decrease
in tobacco-related illnesses.
See:
http://en.wikipedia.org/wiki/Ronald_Fisher
11. REFERENCES.
PREVIOUS PAGE.
RETURN TO TABLE OF CONTENTS.
1. Freedman D, Pisani R, Purves R.
Statistics. Third Edition.
New York: W.W. Norton & Company. 1998.
ISBN 0-393-97083-3, 578 pages.
2.
Livio M.
The Golden Ratio. The Story of Phi, the World's Most Astonishing Number.
New York: Broadway Books. 2003.
ISBN 0-7679-0816-3, 290 pages.
3.
Huntley HE.
The Divine Proportion. A Study of Mathematical Beauty.
New York: Dover Publications, Inc. 1970.
ISBN 486-22254-3, 186 pages.
4.
Brown D.
The Da Vinci Code.
New York: Doubleday. 2003.
ISBN 0-385-50420-9, 454 pages.
A best-seller murder mystery.
p. 91, ch 20.
Nice discussion of the Golden Ratio (1.618....)
and the Fibonacci Sequence. Repeats the legend
that the ratio of the height to the umbilicus-to-ground-height
of a beautiful woman is the Golden Ratio, phi.
p. 199, ch. 45.
Mention of cryptographers Bruce Schneier and Philip K. Zimmerman.
5.
Singh S.
Fermat's Enigma.
The Epic Quest to solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Divsion of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.
p. 62.
"Cuius rei damonstrationem mirabilem sane detexi hanc marginis
exiguitas non caperet."
Latin: I have a sanely miraculous demonstration of this thing,
which the tightness of this margin of text might not capture."
p. 52.
Photograph, Frontispiece of Claude Gaspar Bachet's French translation
of Diophantus's Arithemetica (originally in Latin). Published 1621.
Found in Fermat's literary estate.
"Diophanti Alexandrini Arithmeticorum Libri Sex.
Et de Numeris Multangulis Liber Unus."
Latin: Six Books of Arithemetic by Diophantus of Alexandria.
Book One of multangular numbers. Six books extant from a total
of thirteen books. Other seven books lost in the tragic burning
of the Library of Alexandria in 389 CE, by order of Emperor Theodosius,
certainly one of the least distinguished Roman Emperors in an altogether
very undistinguished line of rulers.
6.
Bühler WK.
Gauss : A Biographical Study .
Berlin: Springer Verlag; ISBN: 0387106626.
Hardcover (April, 1981) .
7.
Croxton FE.
Elementary Statistics with Applications.
in Medicine and the Biological Sciences.
New York: Dover Publications, Inc. 1953.
8.
Murphy EA.
A Companion to Medical Statistics.
Baltimore: The Johns Hopkins University Press. 1985.
9.
Edwards AL.
Statistical Analysis. Revised Edition.
New York: Rinehart & Company, Inc. 1946.
10.
Afifi AA, Azen SP.
Statistical Analysis.
A Computer Oriented Approach.
Second Edition.
New York: Academic Press. 1979.
11.
Lombard OM.
Biostatistics for the Health Professions.
New York: Appleton-Century-Crofts. 1975.
12.
Hines WW, Montgomery DC.
Probability and Statistics.
In Engineering and Management Science.
New York: The Ronald Press Company. 1972.
ISBN not stated, 509 pages.
13.
Downing D, Clark J.
Statistics. The Easy Way.
New York: Barron's Educational Series, Inc. 1989.
ISBN 0-8120-4196-8, 330 pages.
14.
Staff of Research and Education Association, Fogiel M, director.
The Statistics Problem Solver®.
A Complete Solution Guide to Any Textbook.
Piscataway, NJ: Research and Education Association. 1994;:.
ISBN 0-87891-515-X, 1045 pages.
15.
Mood AM, Graybill FA.
Introduction to the Theory of Statistics. Second Edition.
New York: McGraw-Hill Book Company. 1963.
16.
Lilienfeld DE, Stolley PD.
Foundations of Epidemiology. Fifth Edition.
New York: Oxford University Press. 1994.
17.
MacMahon B, Dimitrios T.
Epidemiology. Principles and Methods. Second Edition.
New York: Little, Brown and Co. 1996.
18.
Barker DJP, Cooper C, Rose G.
Epidemiology in Medical Practice. Fifth Edition.
New York: Churchill Livingstone. 1998.
19.
Gordis L.
Epidemiology.
Philadelphia: W. B. Saunders Co. 1996.
20.
Farmer R, Miller D, Lawrenson R.
Lecture Notes on Epidemiology and Public Health Medicine.
Fourth Edition.
Oxford: Blackwell Science. 1996.
21.
Moore GW, Boitnott JK, Miller RE, Eggleston JC, Hutchins GM.
Integrated pathology reporting, indexing, and retrieval system
using natural language diagnoses.
Mod Pathol. 1988 Jan;1(1):44-50.
PMID: 3070549; UI: 89184449.
22.
Pascal B.
Traité du Triangle Arithmétique. 1653.
As cited in: Huntley HE.
The Divine Proportion. A Study of Mathematical Beauty.
Discussion of Pascal's Triangle, first discovered
by the 13th c. Chinese. chap 10, pp. 131-140.
23.
Kendall MG.
Rank Correlation Methods. Third Edition.
New York: Hafner Publishing Co. 1962.
ISBN not stated, 199 pages.
24.
Johnson RR.
Elementary Statistics. Second Edition.
North Scituate, MA: Duxbury Press. 1976;:.
ISBN 0-87872-102-9, 550 pages.
25.
Bernstein PL.
Against the Gods. The Remarkable Story of Risk.
New York: John Wiley & Sons, Inc. 1996.
ISBN 0-471-29563-9, 383 pages.
A fantastic excursion through the history of probability and chance,
starting with the ancient Egyptians and ending with modern
worldwide business practices.
Probability was originally studied in order to INCREASE BENEFITS,
as in winning at gambling or staying alive longer.
Now, probability has its most important applications
26.
DeCew JW.
In Pursuit of Privacy.
Law, Ethics, and the Rise of Technology.
Ithaca, NY: Cornell University Press. 1997.
ISBN 0-8014-3380-0, 199 pages.
27.
Sandritter W.
Histopathologie.
Lehrbuch und Atlas fuer Studierende und Aerzte.
Sechste, verbesserte Auflage.
Stuttgart: F. K. Schattauer Verlag. 1975.
ISBN 3-7945-0454-2, 309 pages.
28.
Asimov I.
Isaac Asimov: The Complete Stories.
New York: Doubleday.
ISBN 038541627X, pages.
Begins with a tale of time travel......
29.
Sternberg SS, ed. Antonioli DA, Carter D, Eggleston JC, Mills SE,
Oberman H, assoc eds.
Diagnostic Surgical Pathology.
New York: Raven Press. 1989;:.
ISBN 0-88167-442-7, 1776 pages, 2 vols.
Surgical pathology with a strong emphasis on diagnosis
and differential diagnosis from clinical and morphologic findings.
Rich in differential diagnosis tables and photographs.
30.
Lever W, Schaumburg-Lever G.
Histopathology of the Skin. Seventh edition.
Philadelphia: J.B.Lippincott Company. 1990;:.
ISBN 0-397-50868-9, 940 pages.
The seventh edition is a vast improvement on previous editions,
which lacked many diseases commonly seen in dermatopathologic practice.
An eighth edition is now available.
31.
Enzinger FM, Weiss SW.
Soft Tissue Tumors. Second Edition.
St Louis: C.V.Mosby Company. 1988;:.
ISBN 0-8016-1902-5, 989 pages.
The definitive text on soft tissue tumors.
32.
Lemay L, Tyler D.
SAMS Teach Yourself Web Publishing with HTML4 in 21 Days.
Indianapolis, IN: SAMS. A division of Macmillan Computer Publishing.
1998;:.
ISBN 0-672-31345-6, 795 pages.
33.
Owen DA, Kelly JK.
Atlas of Gastrointestinal Pathology.
Philadelphia: W.B.Saunders Company.
A division of Harcourt Brace & Company. 1994;:.
ISBN 0-7216-6730-9, 258 pages.
34.
Percy C, Van Holten V, Muir C.
International Classification of Diseases for Oncology. Second Edition.
Geneva: World Health Organization. 1990;:.
ISBN 92-4-154414-7, 144 pages.
35.
Rothwell DJ, Cote RA, Brochu L.
The systematized Nomenclature of Human and Veterinary Medicine.
SNOMED International. Microglossary for Pathology.
Northfield, IL: College of American Pathologists. 1993;:.
ISBN not stated, 475 pages.
"Arguments for not making the switch to SNOMED International
are principally familiarity with the old system
and the cost of conversion.
Although many of the current systems have been extended and
modified to meet individual user needs, they lack the standardization
and depth of SNOMED and are unsuitable for data exchange between
individual institutions or individual units.
"Specific guidelines must be
established by each institution to define how an entity with more than one
possible SNOMED code will be coded.... The recommendation is to establish
a convention for your own institution and adhere to it. p. 8.
GWM's note: This is a remarkable statement, considering that
SNOMED is first recommended for inter-institutional data exchange,
and then each institution is advised to use its own local standards
for coding!!
36.
von Neumann J.
The Computer and the Brain.
New Haven: Yale University Press. 1958;:.
ISBN not stated, 82 pages.
37.
Zalman JF.
Biostatistics. Experimental Design and Statistical Inference.
New York: Oxford University Press. 1993;:.
ISBN 0-19-507810-1, 343 pages.
38.
Walker EA.
Introduction to Abstract Algebra.
New York: Random House.
The Random House/Birkhaeuser Mathematics Series. 1987;:.
ISBN 0-394-35611-X, 355 pages.
39.
Collins KA, Hutchins GM, eds. Tursky CL, CAP editor and designer.
Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists (CAP). 2003:;.
ISBN 0-930304-78-0, 397 pages.
40.
Hutchins GM, Berman JJ, Moore GW, Hanzlick RL, Collins KA,
Members of the Autopsy Committee of the College of American Pathologists.
Autopsy Reporting. Chapter 28.
in: Collins KA, Hutchins GM, eds. Tursky CL, CAP editor and designer.
Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists (CAP). 2003:;265-274.
ISBN 0-930304-78-0, 397 pages.
41.
Moore GW.
Computer-based Indexing. Chapter 32.
in: Collins KA, Hutchins GM, eds. Tursky CL, CAP editor and designer.
Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists (CAP). 2003:;313-323.
ISBN 0-930304-78-0, 397 pages.
42.
McWhirter ND, McWhirter AR.
Guinness Book of World Records.
Toronto: Bantam Books. 1984;:.
ISBN 0-533-23900-2, 702 pages.
43.
Arabie P, Carroll JD, DeSarbo WS.
Three-way scaling and clustering.
Quantitative Applications in the Social Sciences.
A Sage University Paper. 07-065.
Newbury Park: Sage Publications. 1987;:.
ISBN 0-8039-3068-2, 92 pages.
44.
Kleene SC.
Mathematical Logic.
Mineola, NY: Dover Publications, Inc. 1967;:.
ISBN 0-486-42533-9, 398 pages.
45.
Kirsten W, Klar R, eds.
Dokumentation und Informationsaufbereitung für den Arzt.
Beiträge zur Medizinischen Informatik der Wolfgang Giere.
Darmstadt: epsilon Verlag. 1996;:.
ISBN 3-9803214-7-9, 437 pages.
46.
Moore GW, Wakai I, Satomura Y, Giere W.
TRANSOFT: Medical translation expert system.
In: Kirsten W, Klar R, eds.
Dokumentation und Informationsaufbereitung für den Arzt.
Beiträge zur Medizinischen Informatik der Wolfgang Giere.
Darmstadt: epsilon Verlag. 1996;:.
ISBN 3-9803214-7-9, 437 pages.
pp. 161-178.
Reprinted from: Artificial Intelligence in Medicine 1989;1:
47.
Giere W, Wakai I.
Transpro: natural language to Prolog translation.
In: Kirsten W, Klar R, eds.
Dokumentation und Informationsaufbereitung für den Arzt.
Beiträge zur Medizinischen Informatik der Wolfgang Giere.
Darmstadt: epsilon Verlag. 1996;:.
ISBN 3-9803214-7-9, 437 pages.
pp. 179-
Reprinted from: Artificial Intelligence in Medicine 1991;3:
48.
Angermeyer J, Fahringer R, Jaeger K, Shafer D, The Waite Group.
Tricks of the MS-DOS® Masters.
Indianapolis, IN: Howard W. Sams & Company. 1987;:.
ISBN 0-672-22525-5, 542 pages.
49.
Lewkowicz J.
The Complete MUMPS. An Introduction and Reference Manual
for the MUMPS Programming Language.
Englewood Cliffs, NJ: Prentice Hall. 1989;:.
ISBN 0-13-162125-4, 404 pages.
50.
Hayslett HT jr.
Statistics Made Simple.
New York: Doubleday. 1968;:.
ISBN 0-385-02355-3, 192 pages.
51.
Mendenhall W, Ott L.
Understanding Statistics. Second Edition.
Belmont, CA: Duxbury Press.
A Division of Wadsworth Publishing Company. 1976;:.
ISBN 0-87872-101-0, 387 pages.
52.
Noether GE.
Introduction to Statistics. A Nonparametric Approach. Second Edition.
Boston, MA: Houghton Mifflin Company. 1976;:.
ISBN 0-395-18578-5, 292 pages.
53.
Hill B.
Principles of Medical Statistics. Fifth Edition.
New York: Oxford University Press. 1952;:.
ISBN not stated, 282 pages.
54.
Arkin H, Colton RR.
Statistical Methods. With Lists of Formulae and Symbols; Tables.
New York: Barnes & Noble Books. A Division of Harper & Row,
Publishers. 1970;:.
ISBN 389-00119-8, 344 pages.
55.
Steen LA, ed.
Mathematics Today. Twelve Informal Essays.
New York: Springer Verlag. 1978;:.
ISBN 0-387-90305-4, 367 pages.
56.
Derbyshire J.
Prime Obsession. Bernhard Reimann and the greatest unsolved problem
in mathematics.
New York: The Penguin Group. A Plume Book. 2004;:.
ISBN 0-452-28525-9, 422 pages.
57.
Searle JR.
Mind, Language, and Society. Philosophy in the Real World.
New York: Basic Books. A member of the Perseus Books Group. 1998;:.
ISBN 0-465-04521-9, 175 pages.
58.
Searle JR.
Intentionality. An essay in the philosophy of mind.
Cambridge: Cambridge University Press. 1983;:.
ISBN 0-521-27302-1, 278 pages.
59.
Mayr E.
What evolution is.
New York: Basic Books. A member of the Perseus Books Group. 2001;:.
ISBN 0-465-04426-3, 318 pages.
60.
Wilson EO.
Consilience. The Unity of Knowledge.
New York: Vintage Books. A division of Random House, Inc. 1999;:.
ISBN 0-679-76867-X, 367 pages.
61.
Gonick L, Smith W.
The cartoon guide to statistics.
New York: HarperResource. An imprint of HarperCollinsPublishers. 1993;:.
ISBN 0-06-273102-5, 230 pages.
62.
Diamond J.
Guns, Germs, and Steel. The fates of human societies.
New York: W. W. Norton & Company. 1999;:.
ISBN 0-393-31755-2, 494 pages.
63.
Voelker DH, Orton PA, Adams SV.
Statistics. CliffsQuickReview.
New York: Wiley Publishing, Inc. 2001;:.
ISBN 0-7645-6388-2, 154 pages.
64.
Huff D.
How to lie with statistics.
New York: W. W. Norton & Company. 1954;:.
ISBN 0-393-31072-8, 142 pages.
"In the space of one hundred seventy-six years, the Lower Mississippi
has shortened itself two hundred and forty-two miles. That is an average
of a trifle over one mile and a third per year. Therefore, any calm person,
who is not blind or idiotic, can see that in the Old Oölitic
Silurian Period, just a million years ago next November,
the Lower Mississippi River was upward of one million three hundred thousand
miles long, and stuck out over the Gulf of Mexico like a fishing-rod.
And by the same token, any person can see that seven hundred and forty-two
years from now, the Lower Mississippi will be only a mile and three-quarters
long, and Cairo [Illinois] and New Orleans [Louisiana] will have joined
their streets together, and be plodding comfortably along
under a single mayor and a mutual board of aldermen. There is something
fascinating about science. One gets such wholesale returns of conjecture
out of such a trifling investment of fact."
Cited in: Huff D. How to lie with statistics. New York:
W. W. Norton & Company. 1954;:. ISBN 0-393-31072-8, 142 pages.
Page 142.
COMMENT. Mark Twain's classic book, Life on the Mississippi,
is the first book in the world ever submitted by an author to a publisher
as a typewritten manuscript, in 1883. The inventor of the typewriter
was ... Howe, who was born on June 23, 18..
The (mechanical) typewriter was invented in 1868.
Source: Garrison Keillor, Author's Corner, Maryland Public Radio,
June 23, 2004.
65.
Twain M.
Life on the Mississippi.
New York: Signet Classics, Reissue edition. 2001;:.
(November 7, 2001). Twain M, Kaplan J.
ISBN: 0451528174, 359 pages.
66.
Awde N, Samano P.
The Arabic Alphabet. How to read and write it.
New York: Lyle Stuart. Kensington Publishing Corp. 1986;:.
ISBN 0-8184-0430-2, 95 pages.
67.
Wilcox HJ, Myers DL.
An Introduction to Lebesgue Integration and Fourier Series.
New York: Dover Publications, Inc. 1978;:.
ISBN 0-486-68293-5, 159 pages.