Scatterplots, Association, and Correlation

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Chapter 7

• Scatterplots, Association, and Correlation

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Looking at Scatterplots

• Scatterplots may be the most common and most
  effective display for data.
• Look for patterns, trends, relationships, and
  possible outliers
• Best way to picture between two
  variables

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Looking at Scatterplots (cont.)

• When looking at scatterplots, we will look for
  direction, form, , and unusual features
• Direction
• A pattern that runs from the upper left to the
  lower right is said to have a direction
• A trend running the other way has a
• direction

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Looking at Scatterplots (cont.)

• This example shows a negative association between
  central pressure and maximum hurricane wind speed
• As the central pressure , the maximum wind
  speed

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Looking at Scatterplots (cont.)

• Form
• If there is a straight line
• ( ) relationship, it will appear as a cloud or
  swarm of points stretched out in a generally
  consistent, straight form.

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Looking at Scatterplots (cont.)

• Form
• If the relationship isnt straight while still
  increasing or decreasing steadily we can usually
  find ways to make it more straight

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Looking at Scatterplots (cont.)

• Form
• If the relationship curves sharply, the methods
  of this book cannot really help us

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Looking at Scatterplots (cont.)

• Strength
• At one extreme, the points appear to follow a
  stream
• At the other extreme, the points appear as a
  vague cloud with no discernable trend or

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Looking at Scatterplots (cont.)

• Unusual features
• Look for the unexpected
• Look for any outliers standing away from the
  overall pattern of the scatterplot
• Clusters or subgroups should also raise questions

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Roles for Variables

• Need to determine which of the two quantitative
  variables goes on the x-axis and which on the
  y-axis
• This determination is made based on the roles
  played by the variables
• When the roles are clear, the explanatory or
  variable goes on the x-axis, and the
  variable goes on the y-axis

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Correlation

• Data collected from students in Statistics
  classes included their heights (in inches) and
  weights (in pounds)
• There is a
  positive association
  
• Fairly straight
  form
• Seems to be
  a high outlier

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Correlation (cont.)

• How strong is the association between weight and
  height of Statistics students?
• Units should not matter when quantifying strength
• A scatterplot of heights
  
  (in centimeters) and
  weights (in
  kilograms)
  doesnt change the
  
  

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Correlation (cont.)

• both variables and write the coordinates of
  a point as (zx, zy)
• Removes the units from the data
• Here is a scatterplot of the standardized weights
  and heights

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Correlation (cont.)

• The linear pattern seems steeper in the plot
  than in the scatterplot
• Thats because we made the scales of the axes the
  
• Equal scaling gives a neutral way of drawing the
  scatterplot and a fairer impression of the

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Correlation (cont.)

• A numerical measurement of the strength of the
  linear relationship between the explanatory and
  response variables
• For the students heights and weights, the
  correlation is 0.644
• Formula

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Correlation Conditions

• Correlation measures the strength of the linear
  association between two quantitative variables.
• Before you use correlation, you must check
  several conditions
• Quantitative Variables Condition
• Straight Enough Condition
• Outlier Condition

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Correlation Conditions (cont.)

• Quantitative Variables Condition
• Correlation applies only to variables
• Dont apply correlation to categorical data
• Need to know the variables units and what they
  measure

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Correlation Conditions (cont.)

• Straight Enough Condition
• You can calculate a correlation coefficient for
  any pair of variables
• Correlation measures the strength only of the
  linear association
• Results will be misleading if the relationship is
  not linear

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Correlation Conditions (cont.)

• Outlier Condition
• Outliers can distort the correlation
• It can even change the direction of the
  correlation coefficient
• Switch from negative to positive, or
• When you see an outlier report the correlations
  with and without that point

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Correlation Properties

• The sign of a correlation coefficient gives the
  direction of the association
• Correlation is always between
• Correlation close or equal to -1 or 1 indicates
  a linear relationship
• A correlation near zero corresponds to a
• linear association.

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Correlation Properties (cont.)

• Correlation treats x and y
• The correlation of x with y is the same as the
  correlation of y with x
• Correlation has
• Correlation is not affected by changes in the
  center or scale of either variable
• Correlations depend only on

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Correlation Properties (cont.)

• Correlation measures the strength of the linear
  association between the two variables
• Variables can have a strong association but still
  have a small correlation if the association isnt
  linear
• Correlation is sensitive to

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Correlation ? Causation

• Whenever we have a strong correlation, it is
  tempting to explain it by imagining that the
  predictor variable has the response
• Scatterplots and correlation coefficients
• prove causation
• Watch out for
• A hidden variable that stands behind a
  relationship and determines it by simultaneously
  affecting the other two variables

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Correlation Tables

• Compute the correlations between every pair of
  variables in a dataset and arrange these
  correlations in a table

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Straightening Scatterplots

• If a scatterplot shows a bent form that
  consistently increases or decreases, we can often
  straighten the form of the plot by
• one or both variables
• Transforming the data can straighten the
  scatterplots form

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What Can Go Wrong?

• Dont say correlation when you mean
  association
• Dont confuse correlation with causation
• Dont correlate variables
• Be sure the association is linear
• Beware of outliers

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What Can Go Wrong? (cont.)

• Dont assume the relationship is linear just
  because the correlation coefficient is high
• R 0.979, but the relationship is actually bent

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What have we learned?

• We examine scatterplots for direction, form,
  strength, and unusual features
• Although not every relationship is linear, when
  the scatterplot is straight enough, the
• is a useful numerical summary
• The sign of the correlation tells us the of the
  association
• The magnitude of the correlation tells us the
• of a linear association
• Shifting, or scaling the data, standardizing, or
  swapping the variables has no effect on the
  numerical value

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Exercise 7.38

• Fast food is often considered unhealthy because
  much of it is high in both fat and calories. But
  are the two related? Here are the fat contents
  and calories of several brands of burgers.
  Analyze the association between fat content and
  calories.

Fat (g) 19 31 34 35 39 39 43
Calories 410 580 590 570 640 680 660
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Exercise 7.38 (cont.)

• Let us first plot the data
• Y-axis?
• X-axis?

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Exercise 7.38 (cont.)
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Exercise 7.38 (cont.)
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Exercise 7.38 (cont.)
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Exercise 7.38 (cont.)

• From what we learned in Chapter 4

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Exercise 7.38 (cont.)
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Exercise 7.38 (cont.)
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Exercise 7.38 (cont.)
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