POLS 7012

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POLS 7012

• Introduction to Political Methodology
• Jan. 10, 2007

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The Scientific Approach to Politics

• Its not rocket science, but
• The scientific method works well for studying
  rocks and medicine, but what about behavior of
  individuals?
• Notwithstanding the challenges of creating a good
  research design, we can come up with testable
  hypotheses about how the world works
• How do we test them?

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How to test?

• Qualitative
• Easy to do (badly)
• Hard to do (well)
• Quantitative
• Hard to learn
• Once the skills are acquired, many of the facets
  of a good research design are built-in to the
  quantitative approach.
• We will work this semester to lay the foundations
  for this kind of work

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Where to beginQuantification

• Quantitiative reasoning requires measurement
• To measure, we must understand (or define)
  something about what we are measuring
• What can we measure? What cant we measure?
• The Better we define the concepts we are
  measuring, the better we can measure them

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Concept Truck

• What do we need to know to answer this question
• What proportion of people in the United States
  are truck owners?
• What is a truck?

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Truck

• What do we need to know to answer this question
• What proportion of people in the United States
  are truck owners?
• What is a truck owner?
• Lease Trucks?
• What about people with car loans?
• What about corporate owners / company cars?
• Do we count kids as being non-owners when its
  impossible for them to own?

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Clear Definitions Clear Measures

• Our questions and descriptions of results must be
  clear about what we are measuring!
• Bad
• What is the proportion of truck owners in the
  U.S.?
• Better
• What is the proportion of American adults (18 and
  up) who owns a registered, drivable pick-up truck
  (as defined by the manufacturers description)?

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What about politics?

• Definitions of all things regulated matter to us
• If truck is hard to measure, what about
• John Kerry is a Liberal
• Bush is a Compassionate Conservative
• Mexico is a Democratic country
• Americans have liberty
• Taiwan is a country

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Conceptual vs. Operational

• Concept How we describe, or how we think about,
  a concept. Platos World of Forms
• Conceptual Definition States the properties of a
  concept and the subject(s) to which it applies
• Operational Definition Instrument of Measurement
• Variable The Actual Measurement

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What is a Concept?

• Concepts do not physically exist
• Platos World of Forms
• Exist in shared understanding of Language
  (Heidegger)
• Somehow, we still know what they are (kind of)

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Conceptual Definition

• We may know what it is, but we often cant say
  exactly what it is
• Conceptual Definitions try to Define a concept
• The Subject to which the concept applies
• Variation within a characteristic
• How the characteristic is to be measured (what
  the characteristic is)

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What is the concept of Democracy

• Democracies
• Competitive Elections
• Open entry into candidacy
• Unfettered participation of citizens in elections

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Conceptual Definition

• We can conceptually define our characteristic as
  one or more of these things.
• Lets say
• The concept of democracy is defined as the extent
  to which countries exhibit the characteristic of
  competitive elections
• This definition includes everything we need
• The Subject to which the concept applies (the
  unit of analysis)
• Variation within a characteristic
• How the characteristic is to be measured (what
  the characteristic is)

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Side note Unit of Analysis

• Suppose we wanted to measure the concept of Party
  Identification to see if people change their
  party ID
• We sample 5 people, and then ask them again two
  years later
• At the first measurement, 3 out of 5 people are
  Democrats. At the second measurement, again, 3
  out of the 5 are Democrats. Can we conclude that
  people dont change their party ID?
• Ecological Fallacy and individual vs. aggregated

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Operational Definitions

• Conceptual Definitions specify a measurable
  characteristic of the concept
• The concept of democracy is defined as the extent
  to which countries exhibit the characteristic of
  competitive elections.
• Operational Definition specifies how we measure
  that characteristicWhat do you think?
• Percentage of people who, when called on the
  phone, think their country has competitive
  elections.
• Percentage of experts who think a country has
  competitive elections
• Mail Survey
• Percentage of the vote in the Executive Office
  race
• Percentage of candidates who run unopposed

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Variable

• The actual measurement, classified as

Variable
Quantitative
Categorical
Ratio
Interval
Ordinal
Nominal
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Now You Try!

• Suppose you wanted to measure smoking.
• How often do you smoke?
• Never
• 2-3 per day
• 1 pack per day
• gt 1 pack per day
• What is the level of measurement?
• What about this one?
• How many cigarettes do you smoke each day?

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Distributions

• The distribution of a variable tells us what
  values it takes and how often it takes those
  values
• Distributions allow us to summarize the main
  features of data, often graphically

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Graphs of Categorical Variables

• Lets begin with these data
• Education Count(mil.) Percent
• Less the H.S 4.6 11.8
• High School only 11.6 30.6
• Some College 7.4 19.5
• Associate Deg. 3.3 8.8
• BA/BS 8.6 22.7
• Advanced Degree 2.5 6.6 _

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Pie Charts vs. Bar Charts

• Pie Chart is most useful for emphasizing parts of
  a whole. But
• Requires that you include all possible categories
• More errors in reading (humans dont gauge areas
  very well)
• Bar Graphs are easier to read and more flexible
• In the end, these are less important because
  understanding the original table wasnt that hard!

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Graphs of Quantitative Variables Histograms

• IQ Scores for 60 randomly chosen fifth-grade
  students

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Summarizing Distributions

• Central Tendancy
• What is in the Middle?
• What is most common?
• What would we use to predict (best guess)?
• Dispersion
• How Spread out is the distribution?
• What Shape is it?

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Appropriate Measures of Central Tendency

• Nominal variables Mode
• Ordinal variables Median or Mode
• Interval Level variables Mean
• if the distribution is symmetric, otherwise
  consider median

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Mode

• Most Common Outcome

Male Female
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Median

• Middle-most Value
• 50 of observations are above the Median, 50 are
  below it
• The difference in magnitude between the
  observations does not matter
• Therefore, it is not sensitive to outliers

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Median

• Find the Median
• 4 5 6 6 7 8 9 10 12
• 7
• Find the Median
• 5 6 6 7 8 9 10 12
• 7.5
• Find the Median
• 5 6 6 7 8 9 10 100,000
• 7.5

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Aside on Sigma (S) Notation

• In statistics, we deal with many individuals
  being measured on the same characteristic
• We need a shortcut to help us
• Sigma (S) says that we add up every observed
  measurement.

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Example

• Let x 4, 5, 7, 9
• Sx x1 x2 x3 x4
• 4 5 7 9
• 25
• S(x 1) (x1 1) (x2 1) (x3 1) (x4 1)
• (4 1) (5 1) (7 1) (9 1)
• 5 6 8
  10
• 29

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Example

• Let x 4, 5, 7, 9 and y 2
• Sxy x1y x2y x3y x4y
• 42 52 72 92
• 8 10 14 18
• 50
• S(x y) (x1 y) (x2 y) (x3 y) (x4 y)
• (4 2) (5 2) (7 2) (9 2)
• 6 7 9
  11
• 33

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Mean

• Most Common Measure of Central Tendency
• Best for making predictions
• Also known as average
• Symbolized as
• X for the mean of a sample
• µ for the mean of a population

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Finding the Mean

• If X 3, 5, 10, 4, 3
• x (3 5 10 4 3) / 5
• 25 / 5
• 5

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Measures of Dispersion

• Percentiles The pth percentile of a distribution
  is the value such that p percent of the
  observations fall at or below it.
• Quartiles are the most commonly used percentiles.
• For the IQ variable, we have
• 89.4 99.9 105.7
• Note that the 1st and 3rd quartiles are the
  medians of the data below and above the median,
  respectively

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Five-number Summary and the boxplot

• Can summarize spread and central tendency with 5
  numberslowest value, highest value, median, and
  the first and third quartiles (25th and 75th
  percentiles)
• These can be combined graphically into a
  box-and-whisker plot (or boxplot)

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Outliers

• An outlier is a value of a variable that is far
  away from the other values of the variable
• One rule of thumb for determining outliers
• Compute the Inter-quartile range (IQR) as the
  difference between the first quartile and the
  third quartile.
• An observation is an outlier if it falls more
  than 1.5 IQR above the third quartile or below
  the first quartile.

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How do we describe this?

• Measures of variability (Dispersion)
• Mean Deviation
• Variance
• Standard Deviation

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Mean Deviation

• We could just calculate the average distance
  between each observation and the mean.

Problem This always sums to Zero!
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Mean Deviation

• We must take the absolute value of the distance
  (absolute deviation), otherwise they would just
  cancel out to 0!
• Formula

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Mean Deviation An Example
Data X 6, 10, 5, 4, 9, 8
X 42 / 6 7

• Compute X (Average)
• Compute X X and take the Absolute Value to get
  Absolute Deviations
• Sum the Absolute Deviations
• Divide the sum of the absolute deviations by N

12 / 6 2
Total 12
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What Does it Mean?

• On Average, each observation is two units away
  from the mean.

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Is it Really that Easy?

• No.
• Absolute values are difficult to manipulate
  algebraically
• Absolute values cause enormous problems for
  calculus (Discontinuity)
• We need something else(but what?)

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Variance and Standard Deviation

• Instead of taking the absolute value, we square
  the deviations from the mean. This yields a
  positive value.
• This will result in measures we call the Variance
  and the Standard Deviation
• Sample- Population-
• s Standard Deviation s Standard Deviation
• s2 Variance s2 Variance

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Calculating the Variance and/or Standard Deviation

• Formulae
• Variance Std. Dev.
• An Example Follows. . .

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GAINS LOSSES BY PRESIDENTS PARTY
IN MIDTERM ELECTIONS
MEAN S X/N -313/1226.08
S2 3716.9/12 309.7

S
17.6
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What Does it Mean?

• Interpretation is not quite as straightforward
  (but it is much more useful). It requires the
  use of the normal distribution

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Density Curve

• A density curve is like a smooth curve drawn over
  a histogram. It describes what the histogram
  would look like if it involved an infinite number
  of cases with infinitely small bins.

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Density Curves

• Are always on the horizontal axis
• Always have an area of exactly 1 underneath the
  curve.
• The familiar bell-shaped normal distribution is a
  density curve that represents many distributions
  of real data, particularly those involving chance
  outcomes.

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The Normal Distribution

• If a variable is normally distributed, the
  standard deviation has a special meaning
• Whatever the mean and std dev
• 68 of all the cases fall within 1 standard
  deviation of the mean,
• 95 of the cases within 2 standard deviation of
  the mean
• 99.7 of the cases within 3 sd of the mean.

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The Normal Distribution and the 68-95-99.7 rule
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Practice

• SAT scores are normally distributed with a mean
  of 500 and a std. dev. of 100.
• What percentage of students score between 400 and
  600?

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Practice

• SAT scores are normally distributed with a mean
  of 500 and a std. dev. of 100.
• What percentage of students score between 400 and
  500?

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Practice

• SAT scores are normally distributed with a mean
  of 500 and a std. dev. of 100.
• What percentage of students score less than 300?

2.5
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It works with different and s

• IQ scores are normally distributed with a mean of
  100 and a std. dev. of 15.
• What percentage of people have an IQ between 85
  and 115?

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Practice

• IQ scores are normally distributed with a mean of
  100 and a std. dev. of 15.
• What percentage of people have an IQ between 85
  and 100?

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Practice

• IQ scores are normally distributed with a mean of
  100 and a std. dev. of 15.
• What percentage of people have an IQ less than 70?

2.5
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Problem

• By manipulating probabilities, we can only handle
  situations where we are 1, 2, or 3 Std. Devs.
  Away from the mean.
• What happens when we want to know the probability
  of scoring between 100 and 105
• We need to convert the IQ score (or SAT score or
  whatever) into units of the standard Deviation.
• Example Distance between 100 and 105 is .333
  Standard Deviations

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Think of the Std. Dev. As a Unit

• How many inches are in a foot?
• 12
• How many cups are in a pint?
• 2
• How many IQ points are there in a standard
  deviation for IQ?
• 15
• How many SAT points are there in a standard
  deviation for SAT scores?
• 100

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How do you convert inches to feet?

• Distance in feet Distance in inches
• 12
• Distance in IQ std devs Distance in IQ points
• 100
• Distance in IQ std. devs

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Consider this problem

• Party-time employee salaries in a company are
  normally distributed with mean 20,000 and
  Standard Dev. 1,000
• How many Std. Devs. Is 18,500 away from the mean?

• Intuitively, we see that 1,500 is 1.5 Std. Devs.
  from
• Using the formula, we get
• -1.5 (negative specifies direction)

?
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Consider this problem

• How many Std. Devs. Is 19,371 away from the mean?

• Intuitively, we cant do this
• Using the formula, we get
• -.269 Std. devs. away

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X 19,371
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Z Scores

• We call these standard deviation values
  Z-scores
• Z score is defined as the number of standard
  units any score or value is from the mean.
• Z score states how many standard deviations the
  observation X falls away from the mean and in
  which direction plus or minus.

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What Good does this do???

• Someone figured out that 68 are within 1 s.d.
  and about 95 are within 2 s.d.
• Someone did this to show that 74.16 are within
  1.13 s.d. in the normal distribution
• 1.14 s.d 74.58
• 1.15 s.d 74.98
• 1.16 s.d 75.4
• It goes on and on and on.

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These results appear in a Z-table

• You calculate a Z score, and the Z-table will
  tell you
• The probability of getting a score between your
  Z-score and the mean (column B)
• The probability of getting a score greater than
  your Z-score, that is, from your Z-score out to
  the end of the normal distribution (column C)
• This Table can be downloaded from my web site

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It Looks like this

• Suppose you find a Z-score of .12
• Column B says that 4.78 of cases lie between the
  mean and your Z-score

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It Looks like this

• Suppose you find a Z-score of .12
• Column C says that 45.22 of cases lie beyond
  your Z-score

Column C
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IQ is normally distributed with a mean of 100 and
sd of 15. How do you interpret a score of
109? Use Z score
What does this Z-score .60 mean? Does not mean
60 percent of cases below this score BUT rather
that this Z score is .60 standard units above
the mean, We need the Z-table to interpret
this!
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Using the Z table

• Look at Column C for .60
• Only 27.43 of people have an IQ higher than
  this.
• If your IQ is 109 (.6 s.d. above the mean), you
  are smarter than almost 75 of people in the
  world!
• 72.57 of people have an IQ less than this.