1

1
Inductive Generalization

• An inductive (as opposed to deductive) argument
  appeals to some features of a sample  a subset
  of a population  to draw a conclusion about the
  whole population.
• Electron 1 is negatively charged.
• Electron 2 is negatively charged.
• Electron 3 is negatively charged.
• ...
• ? All electrons are negatively charged.

2
Samples

• The central question one has to consider about an
  inductive argument is
• Is the sample a good one?
• That is, is the sample an accurate representation
  of what the population is like?
• This is a problem because by chance one may
  select samples that are very different from the
  rest of the population.

3
Sample Size

• A sample has to be big enough to avoid runs of
  luck that produce unusual results.
• If you flip a coin and it comes up heads three
  times, that may just be luck.
• If you flip it three million times and it comes
  up heads each time, you are reasonable in
  concluding that the coin is two-headed (or
  something else is non-standard).
• How large a sample is large enough?
• That depends on various factors, and your
  friendly neighborhood statistician can tell you
  the answer.

4
Hasty Generalization

• If you draw a conclusion about a population from
  a sample that is too small, you are committing
  the fallacy of hasty generalization.
• Aside Fallacy
• A defective inference is said to be fallacious.
• A fallacy is a common pattern of defective
  reasoning.

5
Hasty Generalization

• ?Of course your columnist Michele Slatalla was
  joking when she wrote about needing to talk with
  her 58-year-old mother about going into a nursing
  home. While I admire Slatalla's concern for her
  parents, and agree that as one approaches 60 it
  is wise to make some long-term plans, I hardly
  think that 58 is the right age at which to talk
  about a retirement home unless there are some
  serious health concerns. In this era, when people
  are living to a healthy and ripe old age,
  Slatalla is jumping the gun. My 85-year-old
  mother power-walks two miles each day, drives her
  car (safely), climbs stairs, does crosswords,
  reads the daily paper and could probably beat
  Slatalla at almost anything.?
• (Nancy Edwards, Letters to the Editor, Time,
  6/26/00.)

6
Biased Sample

• A sample is biased if it does not represent the
  population at large.
• In 1936 the Literary Digest predicted a landslide
  victory for the Republican Alf Landon on the
  basis of 2.5 million questionnaires  a huge
  sample.
• Nevertheless, Roosevelt was elected by a
  landslide.
• Editors used directories of automobile owners and
  telephone users to create a list of recipients
  for the questionnaires rather than choose
  recipients at random.
• These were more affluent voters who tended to
  vote Republican. This biased the sample.

7
Statistical Syllogism

• An inductive or statistical generalization is a
  process of reasoning from properties of a sample
  to properties of the population at large.
• A statistical syllogism moves from properties of
  the population to properties of a sample or
  member of that population.
• X percent of Fs have the feature G.
• A is an F.
• ? A has the feature G.
• 93 percent of professional bike riders have
  exceptionally low resting heart rates.
• Lance is a professional bike rider.
• ? Lance has an exceptionally low resting heart
  rates.

8
Important Features

• Two features of statistical syllogisms are
  important for evaluating their quality.
• First, the percentages of the population are
  crucial.
• Only close to 100 or 0 will be enough to draw a
  confident conclusion about a sample or
  individual.
• Second, the reference class. This is the class
  of Fs from which A is drawn.
• The choice of reference can change the conclusion
  one reaches by means of a statistical syllogism.

9
Reference Classes

• Only four percent of professional bike riders
  with a history of serious illness have
  exceptionally low resting heart rates.
• Lance is a professional bike rider with a
  history of serious illness.
• ? Lance does not have an exceptionally low
  resting heart rates.
• 97 percent of professional bike riders with a
  history of non-cardiac serious illness have
  exceptionally low resting heart rates.
• Lance is a professional bike rider with a
  history of non-cardiac serious illness.
• ? Lance has an exceptionally low resting heart
  rates.

10
Heuristics

• Many of the judgements we have to make in
  everyday life involve evaluating probabilities
  How likely is it that?
• But human beings are demonstrably terrible at
  dealing with probabilities.
• One way we get around this limitation is by using
  heuristics  rules of thumb that typically or
  often produce the right answer.

11
Exercise

• Suppose you are asked to guess which mental
  illness particular psychiatric patients have
  based on a picture drawn by the patient.
• One of the pictures is of a man. He is
  well-dressed, about medium height, and has very
  peculiar looking eyes.
• Which of the following diagnoses is most likely?
• A. Depression
• B. Anxiety
• C. Paranoia
• D. Sleep disorder

12
Biases

• There will be circumstances in which a heuristic
  is unsuccessful because the case about which we
  are reasoning doesnt satisfy the rule of thumb.
• In such cases, we are said to manifest a bias.

The most important work in this area was done
collaboratively by Amos Tversky (who died in
1996) and Daniel Kahneman who just received the
Nobel prize in Economics.
13
Exercise

• Dick Cheney
• Britney Spears
• Kofi Anan
• Tony Blair
• Kenneth Branagh
• Julia Roberts
• Gwynyth Paltrow
• David Duchovny
• Clarence Thomas
• Prince Charles

• Colin Powell
• Peter Carey
• Serena Williams
• Nicole Kidman
• John Lennon
• Elle McPherson
• Jiang Zemin
• Queen Elizabeth
• Peter Gabriel
• Marian Jones

14
Exercise

• Were there more men or more women on the list?

15
Exercise

• Dick Cheney
• Britney Spears
• Kofi Anan
• Tony Blair
• Kenneth Branagh
• Julia Roberts
• Gwynyth Paltrow
• David Duchovny
• Clarence Thomas
• Prince Charles

• Colin Powell
• Peter Carey
• Serena Williams
• Nicole Kidman
• John Lennon
• Elle McPherson
• Jiang Zemin
• Queen Elizabeth
• Peter Gabriel
• Marian Jones

16
Another Bias

• The availability heuristic can lead to bias.
• The heuristic used says, in effect, things that
  more easily come to mind are likely to be more
  numerous.
• This means, however, that when a set of things
  comes to mind for other reasons (e.g. fame), we
  will be led erroneously to think that the set is
  larger than some other.

17
Avoiding Bias

• In order to apply heuristics successfully, we
  need to know whether the situation at hand is a
  standard one  one where the heuristic will give
  the right answer.
• In practice, this is sometimes hard to do.
• To guarantee success, we must use formal
  probabilistic methods.