“ adolphe quetelet: statistics and social science in the early 19 th century”
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“ Adolphe Quetelet: Statistics and Social Science in the Early 19 th Century”. Evan Brott February 3, 2003. Quetelet: 1796-1874. Today, Quetelet is nearly unknown But, he made major contributions to statistics Also one of his era’s greatest social scientists. Main Works. - PowerPoint PPT PresentationTRANSCRIPT
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“Adolphe Quetelet:Statistics and Social Science in the Early
19th Century”
Evan Brott
February 3, 2003
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Quetelet: 1796-1874
• Today, Quetelet is nearly unknown
• But, he made major contributions to statistics
• Also one of his era’s greatest social scientists
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Main Works• 1835- Publishes Physique Sociale: A Treatise on
Man, and the Development of His Faculties which introduces the concept of the ‘Average Man,’
a basic concept in the Social Sciences. (That’s him on the right)
• 1846 – Is the first to fit a normal curve to a distribution of human traits
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Outline1) Science in the 1830’s2) The Early Life of Quetelet3) The Average Man: a Study of Mortality4) Comparisons of Average Men:
a Look at European Sex Ratios5) Statistical Morality and Early ANOVA:
Crime and Punishment in 1820’s France6) Fitting a Normal Curve:
the Chest Size of a Scotsman7) Quetelet’s Legacy
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Part I: Science in the 1830’s
or: They Thought WHAT!?
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State of the Arts
• Quetelet’s research was from 1820-1850.
• MANY theories we take for granted were not yet developed.
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Biology
• 1859 – Darwin publishes The Origin of Species
• 1860s – Pasteur develops Germ Theory of Disease
• 1865 – Mendel discovers basics of Genetics
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Quetelet’s Environment
• Spontaneous Generation not disproved
• Quetelet believes Miasmic Theory of Disease
• Many results seemed strange without understanding heredity
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Social Science
• Quetelet one of the first mathematical social scientists
• 1830’s beliefs seem very strange today
• Ex: Phrenology: Personality read by the shape of the skull
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Early Statistical History
• Beginnings in 17th century
• Studied Laws of Probability through Gambling
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More Proto-Statistics
• 1680s: Newton and Leibniz independently develop Theory of Calculus
• 1689: Bernoulli first states the
Law of Large Numbers
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Normal Distribution
• 1733: De Moivre finds Normal Distribution arises as a limit of the Binomial
• 1778-1812: Laplace develops the
Central Limit Theorem
• 1809: Gauss finds that most random errors are distributed normally
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Future Statistical Knowledge
• 1890s: Pearson develops his correlation coefficient
• 1904: Gosset (a.k.a. ‘Student’) develops the t-distribution
• 1920s: Fischer’s work starts the modern era of statistics
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Part II: The Early Life of Quetelet
or: how to build an observatory without really trying
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Origins
• Born on 2/22/1796 in Ghent, Belgium
• Doctorate in conic sections from University of Ghent in 1819
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Astronomy
• Initial post-doctoral work in astronomy under Arago and Bouvard
• Famous story about founding Belgium’s first observatory: traveled to France at age 26, and got funding despite having NO experience at all.
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Astronomical Statistics
• Galileo first showed astronomical measurement errors were:
- random
- symmetric
- small errors occur more often than large
errors.
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Hypothesized Error Distributions
• Thomas Simpson (1756)
• Daniel Bernoulli (1777)
• Karl Freidrich Gauss (1809)
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More Statistical Exposure
• Met the 75-year old Laplace while getting funding for his observatory
• Post-doctoral mathematical work with Fourier
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The Census
• 1826: began work with the Belgian Department of the Census- was in charge by 1829.
• All censuses at that time were total population counts; Laplace thought of a simpler method
• Count the number of births in several regions; then multiply by ratio of births/population
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Quetelet’s Plan• Quetelet was interested in Laplace’s method
• Received a letter from Baron de Keverberg
• Letter said far too many variables in social science for random sampling
• Quetelet was convinced- conducted full census anyway
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PART III:THE
AVERAGE MAN
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Physique Sociale• Newton’s mechanical physics was highly
esteemed in Quetelet’s time
• Quetelet envisioned a similar Social Physics
• Central to this was the idea of
The Average Man – which was likened to a social ‘center of gravity’
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What is the Average Man?
• It’s exactly what you think it is
• Consider human size:
Small AVERAGE Large
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Influential
• Quetelet was obviously not the first to think of this sort of thing
• He popularized it, and as we will see carried the concept much further though
• It is a VERY common concept today
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Nutritional Example
“The average man needs 250g of carbohydrates each day”
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Common Example
• “The Average Family has 2.4 Children”
(Here, we see the Average man doesn’t necessarily exist)
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- “The Average American will save $278 dollars with my tax plan”
- “But 50% goes to the top 1% of Americans”
- “The bottom 20% pays no taxes”
- “The top 1% makes over $300,000 already”
- And so on . . .
Political Example
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Silly Example• “The Average Man has less than 2 legs”
(Out of the worlds 6 billion people at least 10,000 have only 1 leg . . .)
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What Quetelet Thought
“If an individual at any given epoch of society possessed all the qualities of the average man, he would represent all that is great,
good, or beautiful.”
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Cournot’s Critique
• “A totally average man, if forced to exist, would be an unviable monstrosity: just as the averages of several different right triangles will not be a right triangle.”
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Quetelet’s First Example
• The beginnings of Survival Analysis came from Mortality Tables
• These listed the expected times of death
• In short, the Age of the Average Man
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Quetelet’s Work
• Mortality- P(dying this year)*10,000
• Viability- 1/P(dying this year)
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Part IV: Many Average Men
Or:Where Male Babies Come From
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Categories
• Quetelet did not only envision the Average Man as a ‘global average’
• Rather, there was: An Average Man – and Woman – for every
“race, location, age, and epoch – and all combinations of these”
• Allowed between group comparisons
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Categories• This was also understood before his time
• The mortality tables were divided by gender, location, and occupation
• Still, Quetelet popularized and greatly refined the notion
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• It is a biological fact that 1.06 male babies are born for every female baby.
• Known as early as the 17th Century
• Why?
1.06 : 1.00
The Sex Ratio
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Current Thought
• Evolutionary: men are more expendable
• Sources of variation:
- Prenatal diseases disproportionately effect boys
- First birth, younger women have more boys
- Effects of family planning
• Quetelet noticed most of these!
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The Mind of God• 1710: John Aurbuthnot believes probability
evidences the Divine Mind:
• Sees sex ratio as evidence – more men die in war, but still enough left to evenly match with women
• One of the first applications of probability outside of pure math / gaming
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Quetelet: by Country
• Shows global average; evidence of variation
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Sources of Variation
• Tried to explain why different countries had different ratios
• Decided on racial differences (e.g. Russians naturally have more boys than Swedes)
• Showed many other possible causes
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South Africa
• Climate, Race, Lifestyle, Small Samples
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Legitimacy• The following page shows a table of births
by marital status
• Quetelet never said WHY this effect was there – surely he didn’t think church sanction ‘blessed’ the couple with more boys?
• Proxy for age? Or social status?
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Legitimacy
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Age
• Quetelet presented other theories, this one from Hofacker:
• Overstates effect
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Other Theories
• Dismisses Bicke’s family planning theory
• Shows first marriages (not births) lead to more boys
• Town vs. Country also considered
• Decides on Race
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Still Births• Several Chapters later, demonstrates that
Stillbirths are predominately male
• Does not realize that differing levels of healthcare can exaggerate this effect- accounting for variation
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Part V: Analysis of Crime
or: “If you must murder, try to be a well-educated woman over 30”
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Victorian STAT 410• Ordinary Least Squares had been known for
centuries
• ‘Regression’ would not be called such until Galton in the 1870’s
• Hypothesis Testing, ANOVA still in extremely vague state
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Criminology• Data collected from the French Courts of
Assize from 1825-1830
• Avg. Probability of Conviction: 0.614
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Question
• P(conviction) = 0.614 for THE average man.
• Is this probability different for different groups of people (different “ average ‘men’ ”)?
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Answer: YES!
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New Question: How can we Explain this Variation?
• From the table, it appears that gender, age, type of crime, appearance at trial, and educational status are important.
• How can we tell which of these are significantly different from 0.614?
• Which of these variations are more significant than the yearly variation?
• Can we make multiple comparisons?
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Quetelet’s Paradigm
• 3 sources of variation
- Constant
(e.g. women always have a lower rate)
- Variable
(e.g. conviction rate decreases w/ time)
- Accidental
(e.g. a change in alcohol policy at the university causes more arrests, but not convictions, in 1828.)
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Analysis of Variation
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Relative Degree of Influence
• Calculated as
• For instance- for crimes against property we get |0.655-0.614|/0.614 = 0.067
• Thus, property crimes are ‘average crimes’
614.0
|614.0)|(| statusconvictionP
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How to Assess Variability without knowledge of 2
• Quetelet used (xmax-xavg)/xavg and
(xavg-xmin)/xavg
to give limits on variability.
• Hence for superior education we get a range of (0.40-0.35)/0.40 = 0.125 and
(0.48-0.40)/0.40 = 0.200
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What does all this mean?
• Higher ‘relative degree of influence’ means the cause is more likely to be constant, i.e. P(conviction|status) P(conviction)
• If ‘variability’ is less than R.D.I., then variation by year (variable cause) is less important than the constant cause
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Example: No Shows• Average Conviction Rate = 0.960
• Relative Degree of Influence = 0.563
• Lower Variability = 0.031
• Upper Variability = 0.010
• High RDI -> significant
• Small variability
-> same across years
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Comparisons• Can we compare groups’ conviction rates?
• No, not really. We have a very poor grasp on variability, and cannot conduct hypothesis testing.
• Nevertheless, Quetelet states that the best position to be in was “a well-educated female over thirty, appearing voluntarily to answer a crime against persons.”
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Primitive ANOVA• Can we decide which causes are more
variable or influential?
• Well, sort of. Quetelet has the basic framework of ANOVA set up
• Lacks consistency and optimality properties; ANOVA will be refined by Fischer in early 20th century
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Multiple Comparisons• Many data groupings highly dependent
(e.g. gender and higher education in the 1820’s)
• Basic, modern ANOVA would fail in these circumstances too!
• So the ‘well-educated, voluntarily appearing woman over 30’ comment is not valid
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Poisson• Quetelet’s most famous contemporary (by
today’s standards, anyway) was Poisson.• Poisson also analyzed this same dataset• Summary: - Using corrected data for 1825, refutes
Quetelet’s claim of decreasing rates - Modeling jury selections as a binomial
random variable, gets a rate distribution - Comes up with pseudo-Bayesian
probabilities on conviction.
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Part VI: Fitting a Normal Curve
or: Statistics and the 48-inch chest
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What is Normally Distributed?• Laplace’s CLT (1778-1812) showed that
the Normal is the limit of many distributions
• Gauss (1809) shows it is a very common error distribution
• Quetelet is the first to show human physiology can be normally distributed
• Thinks ALL natural variables are normal
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Scottish Army Uniforms in 1819
• Data on the following page collected by Scottish army
• Needed to fit shirts to soldiers – so tried to estimate soldier’s shirt sizes
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Average Soldier?
• Can’t just clothe the ‘Average Soldier’ – gotta clothe ‘em all.
• Possibility – Average solider of each height
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1846• Instead, decides to fit a normal curve to his
data.
• Did not have a normal table – used a binomial with n=999 (1,000 outcomes)
• Created a table by realizing
yn+1 = yn * (999-n)/(n+1)
for the binomial
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Odd fit
• 1) Split data at median
• 2) Find upper/lower cumulative frequencies
• 3) Transform to rank scale through inverse binomial
• 4) ‘Match ranks to transformed ranks through trial and error’ (???)
• 5) Transform fitted ranks through inverse normal.
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Influence• This gave Quetelet mathematical
justification for the average man
• He asks: can we tell the difference between these measurements, and very inaccurate measurements on a single soldier?
• Normal can only arise through Accidental causes: All is NORMAL!
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Part VII: Quetelet’s Fallout
or: The Good, the Bad, and the Statistical
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Francis Galton (1822-1911)• Primary work in the 1870s• Discovered Genetics independently of
Mendel• Coined the phrase ‘regression to the mean’• Developed several intelligence tests• Mentor to Karl Pearson; Cousin to Darwin• Found direct precursor to Pearson’s r2
• Often considered the father of social science
• Often mistakenly credited for Quetelet’s work on the Normal
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Theory of Heredity
• Firmly believed that performance was based solely on genetics
• Severely discounted education/life experience
• Concerned with intelligence, strength and beauty- thought all were dependent on each other
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Fallacy• Armed with:
- his belief in heredity
- Darwin’s theory of evolution
- Quetelet’s many Average Men
• Reached startling conclusion: groups of people can be mathematically shown to be inferior to others!
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Eugenics
• Therefore, we must ‘improve the human stock’
• Galton’s methods:
- encourage matings between desirable people
- forced sterilization of the truly unfit
(criminals, the insane, etc.)
• Science largely accepted in late 19th century England
• Pearson was Chair of Eugenics at Oxford!
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Theory to Practice
• Most infamously adopted in Germany,
1930-1945
• Justified concept of ‘Aryan Master Race’
• Sterilization upgraded to genocide
• Obviously, today Eugenics is widely condemned
• Galton’s Eugenics merely ‘bad’, not ‘monstrous’
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Quetelet’s Fault?
• Made few value judgments in comparison (e.g. only found one highly qualified mention of racial intelligence)
• Considered the Average Man to be ‘beautiful,’ not ‘mediocre’
• Advocated social reform (education, increased government spending) – not the gradual breeding out of the inferiors
• That is: NO!!!
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Florence Nightingale (1820-1910)
• Studied statistics extensively under her friends Quetelet and William Farr
• Strong believer that statistics was evidence of the Divine Mind: Statistics
was her religion
• Worked extensively in wartime hospitals, saving many lives
• Used statistics to do so!
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Hospital Sanitation• Germ Theory of disease not understood
• Hospitals – especially at war – lacked even basic methods of sterilization
• Demonstrated that Dr. Lister’s antiseptic surgical implements saved many lives- using Quetelet’s statistical methods
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Eulogy for Quetelet• “Quetelet has shown us the path we must go on if
we are to discover the laws of the Divine Government of the Moral World.”
• “It is not understood that human actions are – not subordinate, but – reducible to general laws . . . Of these at present, we know hardly any. Our object in life is to ascertain what they are.”
• “A fitting memorial to Quetelet would therefore be the introduction of his science in the studies of Oxford”.
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Overview of Quetelet’s Statistical Contributions
• Did much to firmly establish statistics as a reputable science, and to mathematicize the Social Sciences
• The Average Man is an enduring paradigm for statistical and social reasoning
• Showed basics of data analysis, hypothesis testing, and analysis of variance
• Demonstrated that natural human traits are normally distributed
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THE END