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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

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Topics: Correlation

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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 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)


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”


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 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


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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