Ch 2 Relationships Between 2 Variables

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Ch 2Relationships Between 2 Variables

• What are Relationships?
• A relationship between two variables can be
  thought as how one variable is linked to the
  other.
• We want to look at the relationship among these
  variables.
• Consider the following
• An insurance group reports that heavier cars have
  fewer accident
• deaths per 100,000 vehicles registered than do
  lighter cars.
• - Here the variables are weight of
  cars and number of accident
• deaths.

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• To study the relationship between two variables
• we measure both the variables on the same
  individual
• we try to answer how one variable influences
  the other?
• Is there an association between these two
  variables?
• Two variables measured on the same individuals
  are associated if some values tend to occur more
  often with some values of the second variable
  than with other values of that variable.

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Examining Relationships

• When you examine the relationship between two
  variables, first ask the preliminary questions
  (see Chapter 1)
• What are the individuals?
• What variables are present and how are they
  measured?
• Which variables are quantitative and which are
  categorical?

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• Be sure of what you actually want
• Is your purpose simply to explore the nature of
  the relationship?
• Do you hope to show that one variable can
  explain variation in the other? That is, some
  variables are explanatory variables and others
  are response variables.
• It is easiest to identify explanatory and
  response variables when we actually set values of
  one variable in order to see how it affects
  another variable.

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Relationships Between 2 Variables

• If we expect one variable to influence another,
  we call it the ___________ variable.
• Explains or influences changes in the response
  variable
• The variable that is influenced is called the
  ____________ variable.
• Measures an outcome of a study

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EXAMPLES

• In each of the following situation, is it more
  reasonable to
• simply explore the relationship between the two
  variables
• or to view one of the variables as an explanatory
  variable
• and the other as a response variable? In the
  later case,
• which is the explanatory variable and which is
  the response
• variable?
• The amount of time spent studying for a
  statistics exam and the grade on that exam.
• The weight in kilograms and the height in
  centimeters of a person.

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• 3. Inches of rain in growing season and the yield
  of corn per acre
• 4. A students score on SAT math exam and SAT
  verbal exam.
• 5. A familys income and the year of the
  education their eldest child completes.

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Relationships Between 2 Variables

• We may be interested in relationships of
  different types of variables.
• Categorical and Numeric
• Categorical and Categorical
• Numeric and Numeric

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Relationships between Categorical and Numeric
Variables

• We are interested in comparing the numerical
  variable across each of the levels of the
  categorical variable.
• Examples
• Compare high speeds for 4 different car brands
• Compare sucrose levels for 5 different types of
  fruit
• Compare GPR for 20 different majors

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Relationships Between Two Categorical Variables

• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• In this case, we look at the percentages of one
  variable for each level of the other variable.
• Examples
• Gender and Soda Preference
• Country of Origin and Marital Status
• Smoking Habits and Socioeconomic Status

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Graphical display of relationship between 2
numeric variables

• Scatterplots
• Look for overall pattern and any striking
  deviations from that pattern.
• Look for outliers, values falling outside the
  overall pattern of the relationship
• You can describe the overall pattern of a
  scatterplot by the form, direction, and strength
  of the relationship.
• Form
• Direction
• Strength

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Interpreting Scatterplots

• Form Linear relationships, where the points show
  a straight-line pattern, are important form of a
  relationship between two variables. Curved
  relationships and clusters are other forms to
  watch for.

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• When a scatterplot shows distinct clusters, it is
  often useful to describe the overall pattern
  separately within each cluster.

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• Direction We see whether the relationship has a
  clear trend visible in the scatterplot. We try to
  answer if x increases what happens to y? or
  if x decreases what happens to y?
• - Two variables are POSITIVELY ASSOCIATED
  when above-average values of one tend to
  accompany above-average values of the other and
  likewise below-average values also tend to occur
  together.

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• -Two variables are NEGATIVELY ASSOCIATED
  when above-average values of one variable
  accompany below-average values of the other
  variable, and vice-versa.
• Strength The strength of a relationship is
  determined by how close the points in the scatter
  plot lie to a simple form such as a line.

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Adding Categorical Variables to Scatterplots