1. Topics: Correlation The road map
Examining “bi-variate” relationships through pictures
Examining “bi-variate” relationships through numbers

2. Correlational Research
Exploration of relationships between variables for better understanding
Exploration of relationships between variables as a means of predicting future behavior.

3. 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?

4. 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

5. 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

6. Scatter Plot: SAT/GPA (Overachievement Study)

7. 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”

8. 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

9. Examples of Various Scatter Plots Demontrating Tilt

10. 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.

11. Examples of Various scatter plots Demontrating Shape (Strength)

12. 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:
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).

13. Types of Correlation Coefficients

14. Influences on Correlation Coefficients
Restriction of range
Use of extreme groups
Combining groups
Outliers (extreme scores)
Curvilinear relationships
Sample size
Reliability of measures

15. Restriction of Range: Example

16. Using Extreme Groups Example

17. Combining Groups Example

18. Outliers (Extreme Scores) Example

19. Curvilinear Examples

20. 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

21. Coefficient of Determination: Graphical Display

22. 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