# SATGPA Scatter

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```					         Topics: Correlation

• 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
behavior.
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?
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
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)
4.0

3.5

3.0

2.5

2.0
GPA

1.5
900          1000   1100    1200    1300

SAT
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”
• 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

• 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.
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
measures
• No rules about what is “strong”, “moderate”,
“weak” relationship

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 views: 20 posted: 8/21/2012 language: English pages: 22