Scatterplots, Association, and Correlation

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Chapter 7

• Scatterplots, Association, and Correlation

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Examining Relationships

• Relationship between two variables
• Examples
• Height and Weight
• Alcohol and Body Temperature
• SAT Verbal Score and SAT Math Score
• High School GPA and College GPA

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Two Types of Variables

• Response Variable (Dependent)
• Measures an outcome of the study
• Explanatory Variable (Independent)
• Used to explain the response variable.
• Example Alcohol and Body Temp
• Explanatory Variable Alcohol
• Response Variable Body Temperature

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Two Types of Variables

• Does not mean that explanatory variable causes
  response variable
• It helps explain the response
• Sometimes there are no true response or
  explanatory variables
• Ex. Height and Weight
• SAT Verbal and SAT Math Scores

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Graphing Two Variables

• Plot of explanatory variable vs. response
  variable
• Explanatory variable goes on horizontal axis (x)
• Response variable goes on vertical axis (y)
• If response and explanatory variables do not
  exist, you can plot the variables on either axis.
• This plot is called a scatterplot
• This plot can only be used if explanatory and
  response variables are both quantitative.

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Scatterplots

• Scatterplots show patterns, trends, and
  relationships.
• When interpreting a scatterplot (i.e., describing
  the relationship between two variables) always
  look at the following
• Overall Pattern
• Form
• Direction
• Strength
• Deviations from the Pattern
• Outliers

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Interpreting Scatterplots

• Form
• Is the plot linear or is it curved?
• Strength
• Does the plot follow the form very closely or is
  there a lot of scatter (variation)?

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Interpreting Scatterplots

• Direction
• Is the plot increasing or is it decreasing?
• Positively Associated
• Above (below) average in one variable tends to be
  associated with above (below) average in another
  variable.
• Negative Associated
• Above (below) average in one variable tends to be
  associated with below (above) average in another
  variable.

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Example Scatterplot

• The following survey was conducted in the U.S.
  and in 10 countries of Western Europe to
  determine the percentage of teenagers who had
  used marijuana and other drugs.

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Example Scatterplot
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Example Scatterplot
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Example Scatterplot

• The variables are interchangeable in this
  example.
• In this example, Percent of Marijuana is being
  used as the explanatory variable (since it is on
  the x-axis).
• Percent of Other Drugs is being used as the
  response since it is on the y-axis.

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Example - Scatterplot

• The form is linear
• The strength is fairly strong
• The direction is positive since larger values on
  the x-axis yield larger values on the y-axis

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Example - Scatterplot

• Negative association
• Outside temperature and amount of natural gas used

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Correlation

• The strength of the linear relationship between
  two quantitative variables can be described
  numerically
• This numerical method is called correlation
• Correlation is denoted by r

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Correlation

• A way to measure the strength of the linear
  relationship between two quantitative variables.

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Correlation

• Steps to calculate correlation
• Calculate the mean of x and y
• Calculate the standard deviation for x and y
• Calculate
• Plug all numbers into formula

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Correlation
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Calculating r.

• Femur (x) 38 56 59 63 74
• Humerus (y) 41 63 70 72 84
• Set up a table with columns for x, y, ,
• , , , and

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Calculating r.
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Calculating r

• Recall
• So,

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Calculating r

• Recall
• So,

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Calculating r.

• Put everything into the formula

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Properties of r

• r has no units (i.e., just a number)
• Measures the strength of a LINEAR association
  between two quantitative variables
• If the data have a curvilinear relationship, the
  correlation may not be strong even if the data
  follow the curve very closely.

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Properties of r

• r always ranges in values from 1 to 1
• r 1 indicates a straight increasing line
• r -1 indicates a straight decreasing line
• r 0 indicates no LINEAR relationship
• As r moves away from 0, the linear relationship
  between variables is stronger

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Properties of r

• Changing the scale of x or y will not change the
  value of r
• Not resistant to outliers
• Strong correlation ? Causation
• Strong linear relationship between two variables
  is NOT proof of a causal relationship!

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Reading JMP Output

• The following is some output from JMP where I
  considered Blood Alcohol Content and Number of
  Beers. The explanatory variable is the number of
  beers. Blood alcohol content is the response
  variable.

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Reading JMP Output
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Reading JMP Output
Summary of Fit
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Reading JMP Output

• RSquare r2
• This means
• I know this is positive because the scatterplot
  has a positive direction.
• The Mean of the Response is the mean of the ys
  or