z Score Definition of r

Note: This demonstration assumes that you are using Microsoft's Internet Explorer as your browser. You can download it for free from http://www.microsoft.com/ie/download/windows.htm. It also assumes that you are using Windows 95, 98 or NT. When asked if you want to install the control, click on Yes.

This demonstration deals with the z score definition of Pearson's product moment correlation coefficient:

This formula is based on the fact that the z scores transform the scale of the X and Y distributions so that both have a mean of zero and a standard deviation of 1. This allows us to more easily compare X scores to Y scores.

In a perfect, positive correlation, the largest X score is paired with the largest Y score. (If the largest X score was paired with some other Y score, we would not be able to predict the Y score given the X score.) Likewise, the second largest X score is paired with the second largest Y score, and so on. By transforming the X and Y scores to Z scores, we make the largest Z_X score identical to the largest Z_Y score. This is true for all the other pairs of scores. In other words, when there is a perfect, positive correlation Z_X = Z_Y. If we then substitute Z_X for Z_Y in the equation for the definition of r, we get that r equals the mean of the squared Z scores. But the sum of the squared Z scores always equals N, so r equals N divided by N or 1.

When two sets of scores have no relation, then knowing the value of X tells us nothing about the value of Y. That is, for any given value of X, the value of Y could be larger than X, smaller than X, or equal to X. The same is true for the Z scores of X and Y. When Z_X is larger than the mean of X (i.e. Z_X is positive), Z_Y is as likely to larger than the mean of Y (i.e. Z_Y is positive) as Z_Y is to be smaller than the mean of Y (i.e. Z_Y is negative.) Thus, the product of Z_X and Z_Y is as likely to be positive as it is to be negative. In general, the positive Z_X times Z_Ys will tend to cancel out the negative Z_X times Z_Ys and the sum will tend toward 0, making r = 0.

You can enter 10 pairs of X and Y values into the control. When there are at least two X scores and two Y scores, the Z scores for X and Y will be displayed. (Why can't the Z scores be displayed when there is only one X score or one Y score?) Watch the mean of the product of the Z scores -- this is the correlation coefficient. Try entering X and Y scores that are perfectly correlated (e.g. (1, 2), (2, 4), (3, 6), etc.) Look at the relation between the Z_X and Z_Y scores. Try entering X and Y scores at random. Look at the relation between the Z_X and Z_Y scores.