Since we are interested in finding out whether two variables describing our customers are related, the concept of covariation is a very useful idea. Covariation is defined as the amount of change in one variable that is consistently related to a change in another variable of interest. For example, if we know that DVD purchases are related to age, then we want to know the extent to which younger persons purchase more DVDs. Another way of stating the concept of covariation is that it is the degree of association between two items (e.g., the attitude toward Starbucks coffee advertising is more favorable among heavy consumers of Starbucks coffee than it is for light consumers). If two variables are found to change together on a reliable or consistent basis, then we can use that information to make predictions that will improve decision making about advertising and marketing strategies.

One way of visually describing the covariation between two variables is with the use of a **scatter diagram.** A scatter diagram plots the relative position of two variables using horizontal and vertical axes to represent the variable values. Exhibits 16.1 through 16.4 show some examples of possible relationships between two variables that might show up on a scatter diagram. In Exhibit 16.1, the best way to describe the visual impression left by the dots representing the values of each variable is probably a circle. That is, there is no particular pattern to the collection of dots. Thus, if you take two or three sample values of variable Y from the scatter diagram and look at the values for X, there is no predictable pattern to the values for X. Knowing the values of Y or X would not tell you very much (maybe nothing at all) about the possible values of the other variable. Exhibit 16.1 suggests that there is no systematic relationship between Y and X and that there is very little or no covariation shared by the two variables. If we measured the amount of covariation shared by these two variables, it would be very close to zero.

In Exhibit 16.2, the two variables present a very different picture from that of Exhibit 16.1. There is a distinct pattern to the dots. As the values of Y increase, so do the values of X. This pattern could be described as a straight line or an ellipse (a circle that has been stretched out from both sides). We could also describe this relationship as positive, because increases in the value of Y are associated with increases in the value of X. That is, if we know the relationship between Y and X is a linear, positive relationship, we would know that the values of Y and X change in the same direction. As the values of Y increase, so do the values of X. Similarly, if the values of Y decrease, the values of X should decrease as well. If we try to measure the amount of covariation shown by the values of Y and X, it would be relatively high. Thus, changes in the value of Y are systematically related to changes in the value of X.

Exhibit 16.3 shows the same type of distinct pattern between the values of Y and X, but the direction of the relationship is the opposite of Exhibit 16.2. There is a linear pattern, but now increases in the values of Y are associated with decreases in the values of X. This type of relationship is known as a negative relationship. The amount of covariation shared between the two variables is still high, because Y and X still change together, though in a direction opposite from that shown in Exhibit 16.2. The concept of covariation refers to the amount of shared movement, not the direction of the relationship between two variables.

Finally, Exhibit 16.4 shows a more complicated relationship between the values of Y and X. This pattern of dots can be described as curvilinear. That is, the relationship between the values of Y and the values of X is different for different values of the variables. Part of the relationship is positive (increases in the small values of Y are associated with increases in the small values of X), but then the relationship becomes negative (increases in the larger values of Y are now associated with decreases in the larger values of X).

This pattern of dots cannot be described as a linear relationship. Many of the statistics marketing researchers use to describe association assume the two variables have a linear relationship. These statistics do not perform well when used to describe a curvilinear relationship. In Exhibit 16.4, we can still say the relationship is strong, or that the covariation exhibited by the two variables is strong. But now we can’t talk very easily about the direction (positive or negative) of the relationship, because the direction changes. To make matters more difficult, many statistical methods of describing relationships between variables cannot be applied to situations where you suspect the relationship is curvilinear.