Topics: Correlation

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Topics Correlation

• The road map
• Examining bi-variate relationships through
  pictures
• Examining bi-variate relationships through
  numbers

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Correlational Research

• Exploration of relationships between variables
  for better understanding
• Exploration of relationships between variables as
  a means of predicting future behavior.

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Correlation Bi-Variate Relationships

• A correlation describes a relationship between
  two variables
• Correlation tries to answer the following
  questions
• What is the relationship between variable X and
  variable Y?
• How are the scores on one measure associated with
  scores on another measure?
• To what extent do the high scores on one variable
  go with the high scores on the second variable?

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Types of Correlation Studies

• Measures of same individuals on two or more
  different variables
• Measures of different individuals on the same
  variable
• Measures of the same individuals on the same
  variable(s) measured at different times

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Representations of Relationships

• Tabular Representation arrangement of scores in
  a joint distribution table
• Graphical Representation a picture of the joint
  distribution
• Numerical Represenation a number summarizing the
  relationship

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Scatter Plot SAT/GPA(Overachievement Study)
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Creating a Scatter Plot

• Construct a joint distribution table
• Draw the axis of the graph
• Label the abscissa with name of units of the X
  variable
• Label the ordinate with the name of the units of
  the Y variable
• Plot one point for each subject representing
  their scores on each variable
• Draw a perimeter line (fence) around the full
  set of data points trying to get as tight a fit
  as possible.
• Examine the shape
• The tilt
• The thickness

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Reading the Nature of Relationship

• Tilt The slope (or slant) of the scatter as
  represented by an imaginary line.
• Positive relationship The estimated line goes
  from lower-left to upper right (high-high,
  low-low situation)
• Negative relationship The estimated line goes
  from upper left to lower right (high-low,
  low-high situation)
• No relationship The line is horizontal or
  vertical because the points have no slant

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Examples of Various Scatter Plots Demontrating
Tilt
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Reading the Strength of Relationship

• Shape the degree to which the points in the
  scatter plot cluster around the imaginary line
  that represents the slope.
• Strong relationship If oval is elongated and
  thin.
• Weak relationship If oval is not much longer
  than it is wide.
• Moderate relationship Somewhere in between.

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Examples of Various scatter plots Demontrating
Shape (Strength)
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Numerical Representation The Correlation
Coefficient

• Correlation Coefficient numerical summary of
  scatter plots. A measure of the strength of
  association between two variables.
• Correlation indicated by r (lowercase)
• Correlation range -1.00 0.00
  1.00
• Absolute magnitude is the indicator of the
  strength of relationship. Closer to value of 1.00
  ( or -) the stronger the relationship closer to
  0 the weaker the relationship.
• Sign ( or -) is the indication of the nature
  (direction,)tilt) of the relationship
  (positive,negative).

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Types of Correlation Coefficients
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Influences on Correlation Coefficients

• Restriction of range
• Use of extreme groups
• Combining groups
• Outliers (extreme scores)
• Curvilinear relationships
• Sample size
• Reliability of measures

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Restriction of Range Example
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Using Extreme Groups Example
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Combining Groups Example
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Outliers (Extreme Scores) Example
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Curvilinear Examples
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Coefficient of Determination

• Coefficient of Determination the squared
  correlation coefficient
• The proportion of variability in Y that can be
  explained (accounted for) by knowing X
• Lies between 0 and 1.00
• r2 will always be lower than r
• Often converted to a percentage

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Coefficient of DeterminationGraphical Display
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Some Warnings

• Correlation does not address issue of cause and
  effect correlation ? causation
• Correlation is a way to establish independence of
  measures
• No rules about what is strong, moderate,
  weak relationship