Describing Univariate Variables

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Describing Univariate Variables

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Stages of empirical research

• - derive a hypothesis from theory
• - identify epistemic relationships
• - test relationship with data

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Agenda

• Foundations of Hypothesis Testing
• How do we measure concepts of interest?
• How do we describe our measures?

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Topic OneWhat is a variable?

• Definition A variable is any characteristic or
  attribute of an object under investigation that
  takes on numerical values.
• For example, variables associated with employees
  may be their talent, work-ethic, wage, gender,
  age, productivity level, etc.

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Latent vs. Manifest Variables

• A manifest variable can be observed.
• e.g. age, gender, productivity-level, tenure and
  wage
• A latent variable is not observed and can only be
  measured indirectly.
• e.g. talent, work ethic

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Dependent vs. Independent Variables

• An independent variable has an antecedent or
  causal role.
• e.g. talent, work-ethic, age, tenure
• A dependent variable plays a consequent, or
  affected, role in relation to the independent
  variable.
• e.g. productivity

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Dependent vs. Independent Variables cont.

• Sometimes a variable can be both an independent
  and a dependent variable.
• e.g. productivity
• talent, work-ethic, age gt productivity
• productivity gt wage
• It is even possible to have a system of equations
  such as
• talent, work-ethic, age, wage gt productivity
• productivity, tenure gt wage

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Discrete vs. Continuous Variables

• A discrete variable classifies persons, objects,
  or events according to the kind or quality of
  their attributes.
• e.g. gender, race
• A continuous variable can, in theory, take on all
  possible numerical values.
• e.g. age, income

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Types of Discrete Variables

• A non-orderable discrete variable does not have
  an intrinsic order from high to low that can be
  imposed on categories.
• e.g. East, North, South, West
• e.g. Asian, Black, Latino, White
• An orderable discrete variable can be arranged in
  a meaningful ascending or descending sequence.
• e.g. Responses to a Likert Scale survey
  question such as
• Do you strongly agree, agree, neither agree
  nor disagree, disagree, or strongly disagree with
  the statement that
• A dichotomous variable classifies observations
  into two mutually exclusive categories
• e.g. Gender

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Desirable Properties of Variables

• Validity is the degree to which a variables
  operationalization accurately reflects the
  concept it is intended to measure.
• This is a property related to the strength of the
  epistemic relationship (the linkage between data
  and theory)
• e.g. For simple tasks, productivity may be a
  valid measure of work ethic.
• For harder tasks, productivity may not be a valid
  measure of work ethic.

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Desirable Properties of Variables cont.

• Reliability is the extent to which different
  operationalizations of the same concept produces
  consistent results.
• - It is desirable to find that multiple measures
  of the same concept yield consistent inferences.
• e.g. Reliable measures of a countrys level of
  industrialization are the kilowatt hours of
  electricity per capita and the proportion of GNP
  in manufacturing.

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Data Collection

• Data collection is the activity of constructing
  primary data records for a given set of
  observations.
• Two Basic Forms of Data Collection
• 1) A population refers to a set of variables
  collected for the entire set of objects of
  analysis
• e.g. The census, collections of roll call votes
• 2) A sample refers to a data collection that
  contains a subset of cases or elements selected
  from a population.
• e.g. A public opinion survey

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Desirable Properties of Samples

• Most social scientific research is based on a
  sample of the entire population.
• A desirable properties of a samples is
  representativeness, which refers to the selection
  of units of analysis that accurately capture the
  characteristics of the larger population.
• A standard way to ensure representativeness is
  through the collection of a random sample, which
  refers to a sample in which each member of the
  population is equally likely to be selected for a
  sampel.

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Topic Two-How do we summarize variables?

• It is impossible to consider piece by piece all
  of the data that we collect
• Instead we summarize the data in ways that
  facilitate inferences.

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Tabular Representation of a Variable

• A Frequency Distribution is a table of the
  outcomes of a variable and the number of times
  each outcomes is observed.

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Distribution of Income of People 15 Years and
Over in 2001
Source Current Population Survey, March 2002
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Graphical Representation of Data

• A Bar Chart is a type of diagram for discrete
  variables in which the numbers of percentages of
  cases in each outcome are displayed.
• Note the number of people is represented by the
  height of the bar.

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Graphical Representation of Data

• A Histogram is fundamentally similar to a bar
  chart, except that it is used for its cases are
  associated with an interval of outcomes of a
  variable.

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Mathematical or Statistical Representation of the
Data

• A more succinct way to present data is through
  statistics.
• Interest focuses on two primary quantities
• - the center of a distribution
• - the dispersion of observations around that
  center

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Measuring the Center of a Distribution

• The mode identifies the most common value of the
  variable being analyzed.
• The median is the middle value of the data when
  ordered from highest to lowest value.
• The mean of the data is the simple arithmetic
  average.

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Different formulas for the Mean

• F1. Mean of X X1 X2 Xn / n
• F2. Mean of X ?? Xi / n
• For grouped data, such as the income data above,
  we use
• Mean of X
• ?categories midpointnumber in category / n

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The Dispersion of a Distribution

• The center of mass obviously does not provide a
  complete description of a variable.
• For example, we would not want to draw the same
  conclusion if we observed these two
  distributions.

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Measures of Dispersion

• Range The difference between the largest and
  smallest observed values for a variable.
• Variance the mean squared deviation from the
  mean of a continuous distribution.

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Variance

• Sample Variance of X
• ??(Xi - Mean(x) )2 / (n-1)
• Notice that the denominator is n-1, not n.
• - This is because this is a unbiased estimate
  of the true variance population variance.
• - If we were to divide by n, we would
  underestimate the population variance.
• The standard deviation of X is the square root of
  the sample variance.

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Symmetry of Distribution

• The mean and variance do not summarize all of the
  information about a distribution.
• We also want to be able to distinguish between
  the following situations

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How do we measure symmetry?

• Skewness is a measure of how asymmetric a
  variables distribution is around its median
  value.
• Skewness of x 3(Mean(x) Median) / ??X
• Positive Skew ? the tail of a skewed distribution
  is to the right of the median
• Negative Skew ? the tail of a skewed distribution
  that is to the left of the median