Scatterplots and Correlation

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Chapter 4

• Scatterplots and Correlation

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Chapter outline

• Explanatory and response variables
• Displaying relationships Scatterplots
• Interpreting scatterplots
• Adding categorical variables to scatterplots
• Measuring linear association correlation r
• Facts about correlation

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Explanatory and Response Variables

• Response variable measures an outcome of a study.
• An explanatory variable explains, influences or
  cause changes in a response variable.
• Independent variable and dependent variable.
• Be careful!! The relationship between two
  variables can be strongly influenced by other
  variables that are lurking in the background.

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Explanatory and response variables

• Note There is not necessary to have a
  cause-and-effect relationship between explanatory
  and response variables.
• Example 4.1(P. 80)
• Example. Cigarette smoking and lung cancer
• Example. Sales of personal computers and athletic
  shoes

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Displaying relationships Scatterplots

• A scatterplot displays the relationship between
  two quantitative variables measured on the same
  individuals.
• It is the most common way to display the relation
  between two quantitative variables.
• It displays the form, direction, and strength of
  the relationship between two quantitative
  variables.
• The values of one variable appear on the
  horizontal axis, and the values of the other
  variable appear on the vertical axis. Each
  individual in the data appears as the point in
  the plot fixed by the values of both variables
  for that individual.

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Example 4.3 (P.82)
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Interpreting scatterplots

• How to examine a scatterplot
• An overall pattern showing
• The form, direction, and strength of the
  relationship
• Outliers or other deviations from this pattern.

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Interpreting scatterplots

• Overall Pattern
• Form Linear relationships, where the points show
  a straight-line pattern, are an important form of
  relationship between two variables. Curved
  relationships and clusters (a number of similar
  individuals that occur together) are other forms
  to watch for.
• Directions If the relationship has a clear
  direction, we speak of either positive
  association (the more the x, the more the y) or
  negative association (the more the x, the less
  the y).
• Strength The strength of a relationship is
  determined by how close the points in the
  scatterplot lie to a line.

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Example 4.5 (P.84)
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Example 4.5 (P.84)
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Adding categorical variables to scatterplots
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Scatterplot Correlation

• Scatterplots provide a visual tool for looking at
  the relationship between two variables.
  Unfortunately, our eyes are not good tools for
  judging the strength of the relationship. Changes
  in the scale or the amount of white space in the
  graph can easily change our judgment of the
  strength of the relationship.
• Correlation is a numerical measure we use to show
  the strength of linear association.

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Measuring linear association correlation r(The
Pearson Product-Moment Correlation Coefficient or
Correlation Coefficient)

• The correlation r measures the strength and
  direction of the linear association between two
  quantitative variables, usually labeled X and Y.

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Facts about correlation

• What kind of variables do we use?
• 1. No distinction between explanatory and
  response variables.
• 2. Both variables should be quantitative
• Numerical properties
• 1.
• 2. rgt0 positive association between variables
• 3. rlt0 negative association between variables
• 4. If r 1or r - 1, it indicates perfect linear
  relationship
• 5. As r is getting close to 1, much stronger
  relationship
• 6. Effected by a few outliers ?not resistant.
• 7. It doesnt describe curved relationships
• 8. Not easy to guess the value of r from the
  appearance of a scatter plot

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