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Notes on Topic 15:
Correlation: The Relationship of Two Variables

Correlation Indices and Scatterplots

    Definition of Correlation:
    Correlation is a statistical technique that is used to measure and describe the STRENGTH and DIRECTION of the relationship between two variables.

    Correlation requires two scores from the SAME individuals. These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot. Usually the two variables are observed, not manipulated.

    Definition of a Scatterplot:
    A scatterplot is a statistical graphic that displays the STRENGTH, DIRECTION and SHAPE of the relationship between two variables.

    A scatterplot requires two scores from the SAME individuals. These scores are normally identified as X and Y. A scatterplot displays the X variable on the horizontal (X) axis, and the Y variable on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X and Y scores.
    Example: Consider the correlation between the SAT-M scores and GPA of the 1997 Psych 30 class. Here are the Math SAT scores and the GPA scores of 13 of the students in the class, and the scatterplot for all 41 students:

    Scatterplot: The scatterplot has the X variable (GPA) on the horizontal (X) axis, and the Y variable (MathSAT) on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X (GPA) and Y (MathSAT) scores.

    For example, the student named "Obs5" (in the sixth row of the datasheet) has GPA=2.30 and MathSAT=710. This student is represented in the scatterplot by high-lighted and labled ("5") dot in the upper-left part of the scatterplot. Note that is to the right of MathSAT of 710 and above GPA of 2.30.

    Pearson Correlation:The Pearson correlation (explained below) between these two variables is .32.

    Correlations and Scatterplots:
    Correlations can tell us about the direction, and the degree (strength) of the relationship between two variables. Scatterplots can also tell us about the form (shape) of the relationship.

    1. The Direction of a Relationship The correlation measure tells us about the direction of the relationship between the two variables. The direction can be positive or negative.

      1. Positive: In a positive relationship both variables tend to move in the same direction: If one variable increases, the other tends to also increase. If one decreases, the other tends to also.

        In the example above, GPA and MathSAT are positively related. As GPA (or MathSAT) increases, the other variable also tends to increase.

      2. Negative: In a negative relationship the variables tend to move in the opposite directions: If one variable increases, the other tends to decrease, and vice-versa.

      The direction of the relationship between two variables is identified by the sign of the correlation coefficient for the variables. Postive relationships have a "plus" sign, whereas negative relationships have a "minus" sign.

    2. The Degree (Strength) of a Relationship

      A correlation coefficient measures the degree (strength) of the relationship between two variables. The Pearson Correlation Coefficient measures the strength of the linear relationship between two variables. Two specific strengths are:

      1. Perfect Relationship: When two variables are exactly (linearly) related the correlation coefficient is either +1.00 or -1.00. They are said to be perfectly linearly related, either positively or negatively.

      2. No relationship: When two variables have no relationship at all, their correlation is 0.00.

      There are strengths in between -1.00, 0.00 and +1.00. Note, though. that +1.00 is the largest postive correlation and -1.00 is the largest negative correlation that is possible.

      Examples: Here are three examples. These examples concern variables measuring characteristics of automobiles. The variables are their weight, miles-per-gallon, horsepower and drive ratio (number of revolutions of the engine per revolution of the wheels).

      Weight and Horsepower
      The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect.

      The Pearson correlation coefficient is +.92.

      Drive Ratio and Horsepower
      The relationship between drive ratio and Horsepower is weekly negative, though not zero.

      The Pearson correlation coefficient is -.59.

      Drive Ratio and Miles-Per-Gallon
      The relationship between drive ratio and MPG is weekly positive, though not zero.

      The Pearson correlation coefficient is .42.

    3. Scatterplots and The Form (Shape) of a Relationship: The form or shape of a relationship refers to whether the relationship is straight or curved.

      1. Linear: A straight relationship is called linear, because it approximates a straight line. The GPA, MathSAT example shows a relationship that is, roughly, a linear relationship.

      2. Curvilinear: A curved relationship is called curvilinear, because it approximates a curved line. An example of the relationship between the Miles-per-gallon and engine displacement of various automobiles sold in the USA in 1982 is shown below. This is curvilinear (and negative).

      Miles-per-gallon and engine displacement
      The relationship between Miles-per-gallon and engine displacement is strongly positive, but curvilinear.

      The Pearson correlation coefficient is not appropriate.

      The Pearson correlation coefficient is only appropriate as a measure of linear relationship. We will see other correlation coefficients that measure curvilinear relationship.

    Where & Why we use Correlation: Correlations are used for Prediction, Validity, Reliability, and Verification.

    1. Prediction: Correlations can be used to help make predictions. If two variables have been known in the past to correlate, then we can assume they will continue to correlate in the future. We can use the value of one variable that is known now to predict the value that the other variable will take on in the future.

      For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college. Thus, we predict high SAT scores will lead to high GPA scores, and conversely.

    2. Validity: Suppose we have developed a new test of intelligence. We can determine if it is really measuring intelligence by correlating the new test's scores with, for example, the scores that the same people get on standardized IQ tests, or their scores on problem solving ability tests, or their performance on learning tasks, etc.

      This is a process for validating the new test of intelligence. The process is based on correlation.

    3. Reliability: Correlations can be used to determine the reliability of some measurement process. For example, we could administer our new IQ test on two different occasions to the same group of people and see what the correlation is. If the correlation is high, the test is reliable. If it is low, it is not.

    4. Theory Verification: Many Psychological theories make specific predictions about the relationship between two variables. For example, it is predicted that parents and children's intelligences are positively related. We can test this prediction by administering IQ tests to the parents and their children, and measuring the correlation between the two scores.