PS601 Quantitative Methods

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PS601 Quantitative Methods

• Dr. Robert D. Duval
• Course Introduction
• Presentation Notes and Slides
• Version of January 15, 2008

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Overview of Course

• Syllabus
• Texts
• Grading
• Assignments
• To be turned in a single Web page
• Built in stages Keep up.
• Instruction provided
• Software
• Excel
• Stata
• NVu

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Prerequisites

• An fundamental understanding of calculus
• An informal but intuitive understanding of the
  mathematics of Probability
• A sense of humor

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Statistics is an innate cognitive skill

• We all possess the ability to do rudimentary
  statistical analysis
• in our heads
• intuitively.
• The cognitive machinery for stats is built in to
  us, just like it is for calculus.
• This is part of how we process information about
  the world
• It is not simply mysterious arcane jargon
• It is simply the mysterious arcane way you
  already think

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The First Two Weeks

• Review and Setting
• The Logic of Research
• Logic
• Microcomputers
• Statistics

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Overview of Statistics

• Descriptive Statistics
• Frequency Distributions
• Probability
• Statistical Inference
• Statistical tests
• Contingency Tables
• Regression Analysis

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The Logic of Research

• A quick review of the research process
• Theory
• Hypothesis
• Observation
• Analysis

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Sample Theories

• IR - Balance of Power
• Wars erupt when there are shifts in the balance
  of power
• Domestic Policy
• The crime rate is affected by the economy
• Democratic Peace
• Nations with democratic regimes engage in war
  less than authoritarian regimes.

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Examples of Research

• Sewage treatment plants
• Energy conservation goals
• Converting Coal fired Powerm plants to oil
• Air Quality
• Sex discrimination in raises

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• Theory
• many are normatively driven.

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Theory Hypothesis
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Theory Hypothesis Observatio
n
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Theory Analysis Hypothesis
Observation
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Theory Analysis Hypothesis
Observation
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Theory Deduction Analysis Hypoth
esis Induction Operationalization
Observation
Confirmation/ rejection
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Statistics
A Philosophical Overview

• Methods as Theory
• Doesnt see statistics as a tool
• It is the embodiment of the ideas we express
• Methods as Language
• We articulate an implicit structure when we
  ascribe causation or systematic patterns.
• This structure may be
• logical hence mathematical
• Relational equality/inequality
• Algebraic or geometric

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Principle organizing concepts

• The Nature of the Problem
• What are we asking?
• Measurement
• How do we have awareness of the phenomenon
• Standards for comparison
• How can we infer if what we are seeing is what we
  expect to see?

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Mathematical notation

• Important mathematical notation the student needs
  to know.
• Summation
• For instance, the sum of all Xi from i1 to n
  means beginning with the first number in your
  data set, add together all n numbers.
• The ? is a symbolic representation of the process
  of adding up a specified series or collection of
  numbers

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Mathematical notation (cont.)

• Square Roots and Exponents
• e - the base of natural logarithms
• Exponential and Logarithmic Equations

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The Base of Natural Logarithms

• Where does e come from?
• e is the base of natural logarithms
• Invented by John Napier in 1618 as a concept,
  but actually calculated/derived by Jacob
  Bernoulli
• It is the number such that the derivative of ax
  equals 1.0
• It is derived from

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Compound Interest and e

• If 1.00 is put in the bank at 100 interest,
  compounded annually, its future value is 2.00.
• What about compounded semi-annually?

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Demystifying e (sort of)

• So how does this translate to real life?
• Compound interest
• Where
• PV Present Value (amount deposited)
• FV future value (amount accrued)
• i interest rate (e.g .06 for 6 interest)
• k number of periods/year
• n number of years

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Levels of Measurement

• Nominal
• Dichotomous (two values)
• Ordinal
• Interval
• Ratio many times handled like interval
• For instance Levels of Measurement

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Nominal Measurement

• Nominal variables are those which can be named,
  but not quantified
• Religion (Protestant, Catholic, Hebrew, Buddhist,
  etc)
• Race (Caucasian, African-American, Hispanic,
  Asian, etc)
• Linguistic Group
• Marital Status (Married, Single, Divorced)

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Ordinal Measurement

• With ordinal variables, there is a rough
  quantitative sense to their measurement, but the
  differences between scores are not necessarily
  equal.
• They are thus in order, but not fixed

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Examples of Ordinal Measures

• Rankings (1st, 2nd, 3rd, etc)
• Grades (A, B, C, D, F)
• Education (High School, College, Adv degree)
• Evaluations
• Hi, Medium, Low
• Likert Scales
• 5 pt (Strongly Agree, Agree, Neither Agree nor
  Disagree, Disagree, Strongly Disagree)
• 7 pt liberalism scale (Strongly Liberal, Liberal,
  Weakly Liberal, Moderate, Weakly Conservative,
  Conservative, Strongly Conservative)

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Interval Measurement

• Variables or measurements where the difference
  between values is measured by a fixed scale.
• Money
• People
• Education (in years)
• Age
• Constructed Scales

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Dichotomous Measurement

• Variables that only have two values.
• May be treated as nominal, however, sometimes an
  ordinal quality may exist.
• Gender - male, female
• Race - black, white
• Agreement - yes, no
• T/F - true, false
• Value - high, low
• and others less easy to name
• war, no war
• vote, no vote

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Ratio Measurement

• Ratio Variables have fixed zero points.
• The percentage is a ratio variable
• part/whole\
• Feeling Thermometers
• We treat ratio and interval variables the same
• Also need an upper bound
• Although not an absolute constraint

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Statistics

• Induction about the Observable World
• A statistic is a number that provides information
  about some variable of interest.
• Descriptive Statistics
• Numbers that describe some aspect of the world
• Inferential Statistics
• We use inferential statistics to take information
  from a sample and make some inference about a
  population.

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Descriptive Statistics

• There are two main ways we describe collections
  of data.
• Measures of Central Tendency
• Measures of Dispersion
• These two approaches give us the ability to
  describe the distribution of the data what the
  data looks like.

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Statistical Tools for Describing the World -
Distributions

• Intuitive Definition
• A bunch of numbers that measure a characteristic
  for a group of cases.
• May be represented by a set of numbers, a graph
  or picture, or even a mathematical equation.

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Measures of Central Tendency

• Measures which provide some indication of the
  typical value or the 'middle' of the distribution

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Measures of Central Tendency The Arithmetic Mean
(or Average)

• The sum of all of the numbers in a set, divided
  by the number in the set
• Most appropriate for symmetric distributions
• Influenced by extreme values

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Measures of Central Tendency The Median

• The middle number in the data set.
• (Sort the Data...)
• The Median is the middle value if there are an
  odd number of cases.
• The Median is the average of the two middle
  values if there are an even number of cases.
• Best measure for skewed distributions
• Not very tractable mathematically!

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Measures of Central Tendency The Mode

• The most frequently occurring value.
• Used primarily for nominal data.
• The peak value of a frequency distribution is
  also referred to as the mode.

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Common terms for Measures of Central Tendency

• We use the idea of measures of central tendency a
  great deal in everyday language.
• Average, accordance, bread-and-butter,
  commonplace, Commensurate, congruent, consistent,
  conventional, customary, day-to-day, everyday,
  frequent, garden variety, general, habitual,
  humdrum, invariably, likeness, mean, median,
  medium, mediocrity, middle, middling,
  nondescript, normal, ordinary, popular,
  prevailing, regular, the same, standard,
  stereotypical, stock, typical, unexceptional,
  uniform, usual
• From The Elementary Forms of Statistical Reason
  by R. P. Cuzzort and James S. Vrettos)

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Measures of Dispersion

• The Range
• Range Highest value - lowest value
• Uses only two pieces of information
• Strongly influenced by the particular
  observations used.
• A single outlier gives a very misleading view
• For instance
• The range in the length of term in office for a
  President of the United States is 30 days to 12
  years.

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Measures of Dispersion

• Percentiles the point on a distribution below
  which that percent fall
• The 95th percentile means that you are in the top
  5
• The Inner Quartile range is between the 25th and
  the 75th - hence the middle 50 of the data.

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The Deviation about the Mean

• The Deviation about the Mean
• Indicates how far a value is from the center.
• Note that in looking at how a distribution
  spreads out, we are using the measure of the
  center as our conceptual foundation.

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The average of the deviations

• So it would seem to make sense to calculate all
  of the deviations and find their average.
• This would seem to give us a measure of the
  typical amount any given data point might vary.

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The Average Deviation

• Does the average of the deviations make sense?

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Calculating the Average Deviation
Xi
1 1-3-2
2 2-3-1
3 3-30
4 4-31
5 5-32
?15 3.0 ??
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The average absolute deviation.

• Can we find the average of the absolute value of
  the deviations?
• Yes, but difficult to use.

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Calculating the average absolute deviation
Xi
1 1-32
2 2-31
3 3-30
4 4-31
5 5-32
?15 3.0 ?6 ABD6/5 1.2
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Fixing these deviant measures

• In order to represent variation about the mean,
  we must get rid of the minus signs in a
  mathematically acceptable manner.

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The standard deviation

• Square the deviations to remove minus signs
• Take the square root to return to the original
  scale

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Calculating the standard deviation
Xi ( )2
1 1-3-2 4
2 2-3-1 1
3 3-30 0
4 4-31 1
5 5-32 4
?15 3.0 ?6 ABD6/5 1.2 ?10 ?(10/5) s1.414
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The Variance

• The mean of the squared deviations has some
  utility as well.
• Variance is what we seek to explain!

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Calculating the Standard Deviation

• The best way to calculate the standard deviation
  is to use a computer.
• If one is not available, try the table method.
• StDevdemo.xls (Excel)

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Population measures

• OKI lied. The formula for the standard
  deviation is not quite as I described.
• It turns out that the Standard Deviation is
  biased in small samples.
• The estimate is a little too small in small
  samples.
• Thus we designate whether we are using population
  or sample data.

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Population vs. Sample Means
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Population vs. Sample Standard Deviations
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Frequency Distributions

• A frequency distribution is a graph or chart that
  shows the number of observations of a given
  value, or class interval.

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The Frequency Histogram

• To create a frequency histogram
• Determine the class interval width.
• Determine the number of intervals desired.
• Tally number of observations in each range.
• Create bar chart from class totals.
• Note that
• The X-axis represents the class interval values
• The Y-axis represents the of cases

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Example Frequency Distribution

• Develop a frequency histogram for the following
  crime rate data for the 50 states.
• Use the data provided in class from the US
  Statistical Abstract on (US Crime Rate p.5)
• A brief aside on following computer
  demonstrations in class
• Follow the general conceptual process the
  details will come later.
• Most of the detail is provided on the screen
  scan it and all menu items.

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Frequency Polygon

• Same as a frequency histogram except the
  midpoints of the class intervals are used
• Points are connected with a line graph
• A large number of classes will make the
  distribution a smooth curve if there is a large
  sample size.

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Frequency DistributionsShape

• Modality
• The number of peaks in the curve
• Skewness
• An asymmetry in a distribution where values are
  shifted to one extreme or the other.
• Kurtosis
• The degree of Peakedness in the curve
• Continuity
• Discrete versus continuous

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Frequency Distributions - Modality

• Unimodal
• Bimodal
• See ADA scores
• Multimodal

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Frequency Distributions - Skewness

• The Third Moment about the Mean
• Right Skew (Positive Skew)
• Left Skew (Negative Skew)

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Frequency DistributionsMeasuring Skewness

• Measuring skewness alternate formula
• Normal distribution has skewness 0.0
• (Normal ranges between 3.0)

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Frequency Distributions - Kurtosis

• The Fourth Moment about the Mean
• Platykurtic
• Leptokurtic
• Mesokurtic

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Frequency DistributionsMeasuring Kurtosis

• Alternate measure of kurtosis
• Normal distributions have kurtosis 3.0

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Continuity

• Discrete distribution
• Values can take on only discrete specific values
• e.g. role of a die x ? 1, 2, 3, 4, 5, 6
• Continuous distributions
• Takes on infinitely fine values.
• Sometimes the distinction is meaningless
• i.e.

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Frequency Distributions - Types

• The Normal
• Characterized by Mean and SD
• Developed by Abraham de Moivre to describe errors
  in observations in astronomy
• Also called the Gaussian distribution, since
  Gauss discovered the use of 2 parameter
  exponential functions for distributions
• The normal curve is the most ubiquitous of this
  class
• Describes the distribution of an infinite sum or
  mean of independent randomly generated variables
• You should think of the normal curve as a
  fundamental law of the universe!

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Frequency Distributions - Types

• The Uniform

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Frequency Distributions - Types

• The Normal
• The Uniform
• The Log-normal
• The Exponential
• Statistical Distributions
• t
• ?-Square
• F

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Freuency Distributions Types (cont.)

• Hyper-geometric
• Poisson
• Binomial
• Gamma
• Weibull
• Logarithmic
• Benford

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A partial list of distributions

• Discrete
• Univariate
• Benford  Bernoulli  binomial  Boltzmann 
  categorical  compound Poisson  discrete
  phase-type  degenerate  Gauss-Kuzmin 
  geometric  hypergeometric  logarithmic 
  negative binomial  parabolic fractal  Poisson 
  Rademacher  Skellam  uniform  Yule-Simon 
  zeta  Zipf  Zipf-Mandelbrot
• Multivariate
• Ewens  multinomial  multivariate Polya
• Continuous
• Univariate
• Beta  Beta prime  Cauchy  chi-square  Dirac
  delta function  Coxian  Erlang   exponential 
  exponential power  F  Fermi-Dirac  Fisher's
  z  Fisher-Tippett  Gamma  generalized extreme
  value  generalized hyperbolic  generalized
  inverse Gaussian  Half-logistic  Hotelling's
  T-square  hyperbolic secant  hyper-exponential 
  hypoexponential  inverse chi-square (scaled
  inverse chi-square)  inverse Gaussian  inverse
  gamma (scaled inverse gamma)  Kumaraswamy 
  Landau  Laplace  Lévy  Lévy skew
  alpha-stable  logistic  log-normal 
  log-logistic  Maxwell-Boltzmann  Maxwell
  speed  Nakagami  normal (Gaussian) 
  normal-gamma  normal inverse Gaussian  Pareto 
  Pearson  phase-type  polar  raised cosine 
  Rayleigh  relativistic Breit-Wigner  Rice 
  RosinRammler  shifted Gompertz  Student's t 
  triangular  truncated normal  Tweedie  type-1
  Gumbel  type-2 Gumbel  uniform 
  Variance-Gamma  Voigt  von Mises  Weibull 
  Wigner semicircle  Wilks' lambda
• Multivariate
• Dirichlet  Generalized Dirichlet 
  inverse-Wishart  Kent  matrix normal 
  multivariate normal  multivariate Student  von
  Mises-Fisher  Wigner quasi  Wishart
• Miscellaneous
• bimodal  Cantor  conditional  equilibrium 
  exponential family  Infinite divisibility
  (probability)  location-scale family 
  marginal  maximum entropy  posterior  prior 
  quasi  sampling  singular   unimodal

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Charts and Graphs

• A picture is worth
• Charts convey a large amount of specialized
  information in a compact way
• They do not require the same type of cognitive
  processing that words and numbers do
• Learn to use them!

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Graphs Charts - Types

• Descriptive Graphs
• Bar Chart
• Pie Graph
• Line Graph
• Distributions
• Histogram
• Box Plot
• Steam and Leaf

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Bar Charts

• Best for displaying actual values.
• Can handle moderate of cases (bars)
• Excel calls it a column chart

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Bar Chart An Example
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Pie Charts

• Best used with small number of categories or
  cases to display
• Good for showing relative distribution
• Percentages, proportions
• Use only one column of data
• Plus one column of labels

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Pie Chart Examples
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Line graphs

• Best for showing data across time
• Always give dates
• Label X axis
• Indicate units on Y axis
• Use legend for multiple lines

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Line Graphs - Example
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Box Plot

• Quick picture of a distribution
• Parts of box give distribution characteristics
• Your Text is not quite accurate!

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Stem and Leaf Plot

• Good for showing distribution while preserving
  data
• Figuring out stems can be tricky

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Probability

• Before starting a discussion of the normal curve,
  a couple of brief points about probability is in
  order
• Probability is essentially How likely is it that
  some event x will occur?
• All Probabilities range between 0.0 and 1.0
• Probabilities outside that range are impossible
  they are not probabilities
• We symbolize probability as P(x)
• The Probability of some event x is P
• 0.0 ? P(x) ? 1.0

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Probability Density Functions

• A probability density function is a frequency
  distribution whose area is set equal to 1.0.
• Most distributions are PDFs.
• They let us assess the likelihood or probability
  of cases taking on particular values.

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The Normal Distribution

• The normal distribution is one of the most
• Popular
• Ubiquitous
• Useful
• distributions that we have.
• It gives great predictive ability when we can
  apply it to data.

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The Normal Distribution the Formula

• The normal curve is described by the following
  formula.

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The Normal Distribution (cont.)

• This formula will give us the following
  distribution

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Using the Normal Distribution

• The normal distribution is a tool for examining
  how values are distributed.
• Think of the normal distribution as an underlying
  physical process that generates values according
  to a general pattern.
• Because these values have this pattern, we have
  information about them that translates to
  probabilities.

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Standard Normal Distribution

• The standard normal distribution is specialized
  example of the Normal Dist.
• The Standard Normal distribution has ? 0.0 and
  ? 1.0
• We say this symbolically as
• Z ?N(0,1)
• (or Z is normally distributed with a mean of zero
  and a standard deviation of one)

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The Normal PDF

• Because the standard normal curve is a PDF, we
  can use it to make probability assessments about
  values in the distribution.
• We can, in essence, convert our regular normal
  distribution to the Standard Normal, and examine
  areas under the curve to calculate probabilities.

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Using the Normal PDF

• We know the following facts
• Area under the curve 1.0
• Its symmetric, so the probability of Xi being
  greater that 0.0 is .5
• Symbolically,
• P(Xi gt 0.0) .5

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Using the Normal PDF

• We can use this information in the following
  fashion
• The P(0.0 ? Xi ? 1.0) .3413
• Thus 68 of the Xis will fall between ?1?
• Thus 96 of the Xis will fall between ?2?

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Standard Normal Variables

• A Standard Normal Variable is one that has been
  transformed by the following formula
• All Z-scores, as they are called, will have a
  mean 0.0 and s 1.0

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Creating indices

• Z-score transformations are useful for adding
  apples and oranges
• They transform similar variables to the same
  scale.
• As a result, we can add together things that
  would otherwise be impossible to combine.

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The assumption of normality

• As a result of the application of the standard
  normal distribution, we can make probability
  statements about data.
• (A probability statement is one that is of the
  general How likely is it that? format.)
• If we can assume that the data is normally
  distributed, then we know how likely it is that
  individual cases are above or below selected
  values expressed in standard deviation units.

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The Central Limit Theorem

• The Normal Distribution pops up in one very
  important context
• The Central Limit Theorem
• This is a fundamental concept that allows us to
  infer the characteristics of a population based
  upon a sample.

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Sampling Distributions

• The probability distribution of a statistic is
  called its sampling distribution.
• If we collect a sample and calculate the mean,
  that is one data point in the sampling
  distribution of the sample mean.
• If we do this many times, we have a sampling
  distribution, which we can then describe.

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The Central Limit Theorem

• The CLT tells us
• As the sample size n gets larger, the sampling
  distribution of the sample mean can be
  approximated by a normal distribution with mean
  of ?, and a standard deviation equal to
• where ? and ? are the population
  characteristics.

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The Implications of the Central Limit Theorem

• We can use the CLT to make probability statements
  about the sample mean because we know its
  distributional characteristics.
• Even if the original variable X is not normally
  distributed, the sampling dist of the sample mean
  is!

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Statistical inference

• We can use information about the way variables
  are distributed to make assessments of
  probability about them.
• Many of these questions are phrased as Is A
  greater (or less) than B?
• This may also be phrased
• Does A belong to the same population as B?

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Assessing probabilities

• Take income
• Would you expect a doctor to have a higher income
  than the population at large?
• Would dog-catchers be lower?
• Would you expect males to have a higher income
  than females
• Is WV income lower than the national average?
  What about Oklahoma?

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Statistical Decision-making

• Many of these questions are best answered with a
  statement of statistical confidence a
  probability assessment.
• This statistical confidence places a decision
  within an objective framework.
• If we define the criteria for making decisions
  according to some reasonable standards, then we
  can remove (or certainly reduce the subjectivity
  of the researcher.

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Statistical D-M (cont.)

• If you collected the following information, would
  you conclude that males had a higher income than
  females?
• MeanMales 55.5K, MeanFemales 54.9K
• MeanMales 55.5K, MeanFemales 50.9K
• MeanMales 55.5K, MeanFemales 34.9K
• Where would you draw the line?
• Does sample size matter?

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Statistical Decision-making problem setup

• The Mon river