Chapters 8 and 9: Correlations Between Data Sets

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Chapters 8 and 9 Correlations Between Data Sets

• Math 1680

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Overview

• Scatter Plots
• Associations
• The Correlation Coefficient
• Sketching Scatter Plots
• Changes of Scale
• Summary

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Scatter Plots

• Often, we are interested in comparing two related
  data sets
• Heights and weights of students
• SAT scores and freshman GPA
• Age and fuel efficiency of vehicles
• We can draw a scatter plot of the data set
• Plot paired data points on a Cartesian plane

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Scatter Plots

• Scatter plot for the heights of 1,078 fathers and
  their adult sons
• From HANES study

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Scatter Plots

• What does the dashed diagonal line represent?
• Find the point representing a 5'3¼" father who
  has a 5'6½" son

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Scatter Plots

• What does the vertical dashed column represent?
• Consider the families where the father was 72"
  tall, to the nearest inch
• How tall was the tallest son?
• Shortest?

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Scatter Plots

• Was the average height of the fathers around 64,
  68 or 72?
• Was the SD of the fathers heights around 3", 6"
  or 9"?

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Scatter Plots

• The points form a swarm that is more or less
  football-shaped
• This indicates that there is a linear association
  between the fathers heights and the sons heights

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Scatter Plots

• Short fathers tend to have short sons, and tall
  fathers tend to have tall sons
• We say there is a positive association between
  the heights of fathers and sons
• What would it mean for there to be a negative
  association between the heights?

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Scatter Plots

• Does knowing the fathers height give a precise
  prediction of his sons height?
• Does knowing the fathers height let you better
  predict his sons height?

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Scatter Plots

• We will generally assume the scatter plots are
  football-shaped
• Association is linear in nature
• Each data set is approximately normal

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Scatter Plots

• Key features of scatter plots
• Given two data sets X and Y,
• The point of averages is the point (?x, ?y)
• The average of a data set is denoted by µ (Greek
  mu, for mean)
• The subscript indicates which set is being
  referenced
• It will be in the center of the cloud
• Due to the normal approximation, the vast
  majority (95) of the cloud should fall within 2
  SDs less than and greater than average for both
  X and Y

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Scatter Plots
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Associations

• When given a value in one data set, we often want
  to make a prediction for the other data set
• We call our given value the independent variable
• We call the value we are trying to predict the
  dependent variable

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Associations

• If there is indeed a relationship between the two
  data sets, we can say various things about their
  association
• Strong Knowing X helps you a lot in predicting
  Y, and vice versa
• Weak Knowing X doesnt really help you predict
  Y, and vice versa
• Positive X and Y are directly proportional
• The higher in one you look, the higher in the
  other you should be
• Negative X and Y are inversely proportional
• The higher in one you look, the lower in the
  other you should be

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Associations

• Positive associations
• Study time/final grade
• Height/weight
• SAT score/GPA
• Clouds in sky/chance of rain
• Bowling practice/bowling score
• Age of husband/age of wife

• Negative associations
• Age of car/fuel efficiency
• Golfing practice/golf score
• Dental hygiene/cavities formed
• Pollution/air quality
• Speed/mile time

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Associations

• What kind of association is this?

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Associations

• What kind of association is this?

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Associations

• Remember that even a very strong association does
  not necessarily imply a causal relationship
• There may be a confounding influence at play

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The Correlation Coefficient

• While strong/weak and positive/negative give a
  sense of the association, we want a way to
  quantify the strength and direction of the
  association
• The correlation coefficient (r) is the statistic
  which accomplishes this

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The Correlation Coefficient

• The correlation coefficient is always between 1
  and 1
• A positive r means that there is a positive
  association between the sets
• A negative r means that there is a negative
  association between the sets
• If r is close to 0, then there is only a weak
  association between the sets
• If r is close to 1 or 1, then there is a strong
  association between the sets

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The Correlation Coefficient

• The following plots have and
  , with 50 points in them
• The only difference between them is the
  correlation coefficient
• Note how the points fall into a line as r
  approaches 1 or 1

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The Correlation Coefficient

• To calculate r
• Find the average and SD of each data set
• Multiply the data sets pairwise and find the
  average
• The correlation is the average of the product
  minus the product of the averages, all divided by
  the product of the SDs

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The Correlation Coefficient
X Y
1 5
3 9
4 7
5 1
7 13
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The Correlation Coefficient

• Compute r for the following data

X Y
1 2
2 1
3 4
4 3
5 7
6 5
7 6
X Y
1 3
3 7
4 9
5 11
7 15
1
0.8214
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The Correlation Coefficient

• Estimate the correlation

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The Correlation Coefficient

• Estimate the correlation

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Sketching Scatter Plots

• The SD line is the line consisting of all the
  points where the standard score in X equals the
  standard score in Y
• zX zY
• To sketch the SD line, draw a line bisecting the
  long axis of the football shape
• Note that the SD line always goes through the
  point of averages

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Sketching Scatter Plots

• Given the five-statistic summary (averages, SDs,
  and correlation) for a pair of data sets, we can
  sketch the scatter plot
• Plot the point of averages in the center
• Mark two SDs in both directions, on both axes
• Plot the point 1 SD above average for both data
  sets
• draw a line connecting this point and the point
  of averages
• This is the SD line
• Draw an ellipse with the SD line as its long axis
  
• Ellipse should go just beyond the 2 SD marks in
  all directions
• The value of r determines how oblong the ellipse
  is

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Sketching Scatter Plots

• A study of the IQs of husbands and wives obtained
  the following results
• Husbands average IQ 100, SD 15
• Wives average IQ 100, SD 15
• r 0.6
• Sketch the scatter plot

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Changes of Scale

• The correlation coefficient is not affected by
  changes of scale
• Moving adding the same number to all of the
  values of one variable
• Stretching multiplying the same positive number
  to all the values of one variable
• Would r change if we multiplied by a negative
  number?
• The correlation coefficient is also unaffected by
  interchanging the two data sets

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Changes of Scale
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Changes of Scale
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Changes of Scale

• Compute r for each of the following data sets

X Y
0 8
4 9
6 10
8 12
12 6
X Y
0 2
2 3
3 4
4 6
6 0
r -0.15
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Summary

• The relationship between two variables, X and Y,
  can be graphed in a scatter plot
• When the scatter plot is tightly clustered around
  a line, there is a strong linear association
  between X and Y
• A scatter plot can be characterized by its
  five-statistic summary
• Average and SD of the X values
• Average and SD of the Y values
• Correlation coefficient

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Summary

• When the correlation coefficient gets closer to 1
  or 1, the points cluster more tightly around a
  line
• Positive association has a positive r-value
• Negative association has a negative r-value
• Calculating the correlation coefficient
• Take the average of the product
• Subtract the product of the averages
• Divide the difference by the product of the SDs

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Summary

• The correlation coefficient is not affected by
  changes of scale or transposing the variables
• Correlation does not measure causation!