The reader may verify this by computing the correlation coefficient using X and z Y or Y and z X. Since the z-score transformation is a special case of a linear transformation (X' = a + bX), it may be proven that the correlation coefficient is invariant (doesn't change) under a linear transformation of either X and/or Y. Next computing the correlation coefficient with z X and z Y yields the same value, r=0.85. There are two points to be made with the above numbers: (1) the correlation coefficient is invariant under a linear transformation of either X and/or Y, and (2) the slope of the regression line when both X and Y have been transformed to z-scores is the correlation coefficient.Ĭomputing the correlation coefficient first with the raw scores X and Y yields r=0.85. That is, the mean is subtracted from each raw score in the X and Y columns and then the result is divided by the sample standard deviation. The second two columns are the X and Y columns transformed using the z-score transformation. The raw score values of the X and Y variables are presented in the first two columns of the following table. This interpretation of the correlation coefficient is perhaps best illustrated with an example involving numbers. This is related to the difference between the intuitive regression line and the actual regression line discussed above. The larger the size of the correlation coefficient, the steeper the slope. The correlation coefficient is the slope (b) of the regression line when both the X and Y variables have been converted to z-scores. 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.Ī 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. When r=0.0 the points scatter widely about the plot, the majority falling roughly in the shape of a circle. The scatterplots presented below perhaps best illustrate how the correlation coefficient changes as the linear relationship between the two variables is altered. The correlation coefficient may be understood by various means, each of which will now be examined in turn. UNDERSTANDING AND INTERPRETING THE CORRELATION COEFFICIENT 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. Likewise a correlation coefficient of r=-.50 shows a greater degree of relationship than one of r=.40. A correlation coefficient of r=.50 indicates a stronger degree of linear relationship than one of r=.40. Taking the absolute value of the correlation coefficient measures the strength of the relationship. A negative correlation coefficient indicates that as one variable increases, the other decreases, and vice-versa. 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. The sign of the correlation coefficient (+, -) defines the direction of the relationship, either positive or negative. The correlation coefficient may take on any value between plus and minus one. 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 value of r was found on a statistical calculator during the estimation of regression parameters in the last chapter. The computation of the correlation coefficient is most easily accomplished with the aid of a statistical calculator. 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. 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. 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. Introductory Statistics: Concepts, Models, and Applications
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