AGR EDUC 387 Data Analysis in Social Sciences

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AGR EDUC 387Data Analysis in Social Sciences

• Objective
• Describe relationships (correlations)
• April 21, 2004

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Relationships (Correlations)

• Correlation
• When one variable varies with another variable
• Positive Correlation
• When pairs of observations tend to occupy similar
  relative positions (high with high, low with low)
  in their respective distributions
• Negative Correlation
• When pairs of observations tend to occupy
  dissimilar relative positions (high with low, low
  with high) in their respective distributions

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Greeting Cards Given and Received by Friends
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Positive Relationship
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Negative Relationship
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Little or No Relationship
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Exercises

• Positive or Negative Relationship?
• High school students with lower IQs have lower
  GPAs
• Increasing rates of unemployment accompany
  decreasing rates of inflation
• More densely populated areas have higher crime
  rates
• Better-educated people have higher incomes

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Scatterplots

• X and Y axis
• Each pair of observations located with a dot
  within the scatterplot

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Scatterplot for Greeting Cards
20

15


Number of Cards Received
10


5
0
5
10
15
20
Number of Cards Given
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Linear Relationship

• A relationship that can be described best with a
  straight line

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Curvilinear Relationship

• A relationship that can be described best with a
  curved line

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Scatterplot for GRE Scores
800
700
E
C
F
600
H
Math Scores
B
G
500
400
A
D
300
I
0
300
400
500
600
700
800
Verbal Scores
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Exercises

• Which students scored about the same on both
  tests?
• Which students scored higher on the verbal test
  than on the math test?
• Which students scored above 600 on both tests?
• Which students will be eligible for an honors
  program that requires minimum scores of 700
  (verbal) and 500 (math)?
• True or False? Generally, students who perform
  well on the verbal test also perform well on the
  math test.
• True or False? There is a negative relationship
  between verbal and math scores.

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Correlation Coefficient (r)

• A number between 1 and 1 that describes the
  relationship between pairs of variables
• Pearson correlation coefficient is a number
  between 1 and 1 that describes the linear
  relationship between pairs of quantitative
  variables

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r

• The sign of r indicates the type of linear
  relationship, whether positive or negative
• The value of r without regard to sign, indicates
  the strength of the linear relationship
• Number without a sign (or no sign) indicates a
  positive relationship
• The closer a value of r approaches either 1 or
  1, the stronger the relationship
• The closer the value of r approaches 0, the
  weaker the relationship

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Verbal Description of r

• Examples
• An r of .70 for height and weight could be
  translated into taller students tend to weigh
  more
• An r of -.42 for time spent taking an exam and
  subsequent exam score could be translated into
  students who take less time tend to make a
  higher score
• An r near 0 for shoe size and IQ could be
  translated into little, if any, relationship
  exists between shoe size and IQ

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Exercises

• Supply a verbal description for the following
• An r of -.84 between total mileage and automobile
  resale value
• A r of .89 between gross annual income and total
  dollar value of claimed tax deductions
• A r of .03 between anxiety level and college GPA
• An r of -.05 between height and IQ

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Cause - Effect

• A correlation coefficient, regardless of size,
  NEVER provides information about whether an
  observed relationship reflects a simple
  cause-effect relationship or some more complex
  state of affairs.

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z Score for r

• Patterns among pairs of z scores can be used to
  anticipate the value of r
• If pairs of z scores are similar in both
  magnitude and sign, the value of r will tend
  toward 1.00, indicating a strong positive
  correlation
• If pairs of z scores are similar in magnitude,
  but different on sign, the value of r will tend
  towards 1.00, indicating a strong negative
  correlation
• As pairs of z scores become less apparent, the
  value of r tends towards 0, indicating a weak or
  nonexistent correlation

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Correlation Coefficient(computational formula)
n?xy (?x)(?y)
r
?n?x2 (?x)2 ?n?y2 (?y)2
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Exercise
Calculate a value for r using the computational
formula.
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Exercise
n?xy (?x)(?y)
r
?n?x2 (?x)2 ?n?y2 (?y)2
(5)(484) (35)(60)

?(5)(325) (35)2 ?(5)(800) (60)2
320


0.80
400
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Outliers

• Defined as very extreme observations that require
  special attention because of their potential
  impact on a summary of data
• Focus on dots on scatterplots that deviate
  conspicuously from the main dot cluster

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Scatterplot for Greeting Cards
r .80
20
B
15
A

Number of Cards Received
10


5
0
5
10
15
20
Number of Cards Given
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Dealing with Outliers

• The greater the number of observations, the less
  effect an outlier has on the value of r
• The most defensible strategy is to report the
  values of r both with and without any outliers

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Correlation Matrix