SATGPA Scatter

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					         Topics: Correlation

• The road map
• Examining “bi-variate” relationships
  through pictures
• Examining “bi-variate” relationships
  through numbers
        Correlational Research

• Exploration of relationships between
  variables for better understanding
• Exploration of relationships between
  variables as a means of predicting future
         Bi-Variate Relationships
• A correlation describes a relationship between two

• 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?
    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
   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
                  Scatter Plot: SAT/GPA
                  (Overachievement Study)





        900          1000   1100    1200    1300

            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
• 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”
  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
Examples of Various Scatter Plots
      Demontrating Tilt
 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
  – Weak relationship: If oval is not much longer
    than it is wide.
  – Moderate relationship: Somewhere in between.
Examples of Various scatter plots
 Demontrating Shape (Strength)
         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).
Types of Correlation Coefficients
Scale of         Interval, Ratio    Ordina l       Nominal      Dichoto mous
Measurement                                                     Artificial
                                                                Dichoto my
Interval,Ratio   Pearson P roduct
                 Momen t

Ordina l                            Spearman Rho
                                    Kendal l Tau

Nominal                                            Cramer's V

Dichoto mous,    Point Biserial                                 Phi
Artificial       Biserial                                       Tetrachoric
Dichoto my
Influences on Correlation Coefficients

•   Restriction of range
•   Use of extreme groups
•   Combining groups
•   Outliers (extreme scores)
•   Curvilinear relationships
•   Sample size
•   Reliability of measures
Restriction of Range: Example
Using Extreme Groups Example
Combining Groups Example
Outliers (Extreme Scores) Example
Curvilinear Examples
      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
Coefficient of Determination:
     Graphical Display
              Some Warnings
• Correlation does not address issue of cause and
  effect: correlation ≠ causation
• Correlation is a way to establish independence of
• No rules about what is “strong”, “moderate”,
  “weak” relationship

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