The Pearson Product-Moment Correlation Coefficient (r), or correlation coefficient for short is a measure of the degree of linear relationship between two variables. usually labeled X and Y. While in regression the emphasis is on predicting one variable from the other, in correlation the emphasis is on the degree to which a linear model may describe the relationship between two variables. In regression the interest is directional, one variable is predicted and the other is the predictor; in correlation the interest is non-directional, the relationship is the critical aspect.
The computation of the correlation coefficient is most easily accomplished with the aid of a statistical calculator. The value of r was found on a statistical calculator during the estimation of regression parameters in the last chapter. Although definitional formulas will be given later in this chapter, the reader is encouraged to review the procedure to obtain the correlation coefficient on the calculator at this time.
The correlation coefficient may take on any value between plus and minus one.
The sign of the correlation coefficient (+. -) defines the direction of the relationship, either positive or negative. A positive correlation coefficient means that as the value of one variable increases, the value of the other variable increases; as one decreases the other decreases. A negative correlation coefficient indicates that as one variable increases, the other
decreases, and vice-versa.
Taking the absolute value of the correlation coefficient measures the strength of the relationship. A correlation coefficient of r=.50 indicates a stronger degree of linear relationship than one of r=.40. Likewise a correlation coefficient of r=-.50 shows a greater degree of relationship than one of r=.40. Thus a correlation coefficient of zero (r=0.0) indicates the absence of a linear relationship and correlation coefficients of r=+1.0 and r=-1.0 indicate a perfect linear relationship.
UNDERSTANDING AND INTERPRETING THE CORRELATION COEFFICIENT
The correlation coefficient may be understood by various means, each of which will now be examined in turn.
The scatterplots presented below perhaps best illustrate how the correlation coefficient changes as the linear relationship between the two variables is altered. When r=0.0 the points scatter widely about the plot, the majority falling roughly in the shape of a circle. As the linear relationship increases, the circle becomes more and more elliptical in shape until the limiting case is reached (r=1.00 or r=-1.00) and all the points fall on a straight line.
A number of scatterplots and their associated correlation coefficients are presented below in order that the student may better estimate the value of the correlation coefficient based on a scatterplot in the associated computer exercise.
r = 1.00
r = -.54