A mean, as defined in this chapter, is a pretty simple concept—it is the average of a set of numbers. But even this simple concept can cause confusion if you aren’t careful. The data in Table are typical of data presented by marketing researchers for a type of product, in this case beer.
Each value is an average of the number of six packs of beer purchased per customer during a month. For the individual brands, the value is the average only for the customers who purchased at least one six-pack of that brand. For example, the value for Miller is the average number of six-packs purchased of all of these brands for customers who purchased at least one six pack of Miller. In contrast, the “Any” average is the average number of six-packs purchased of these brands for all customers in the population.
Is there a paradox in these averages? On first glance, it might appear unusual, or even impossible, that the “Any” average is less than each brand average. Make up your own (small) data set, where you list a number of customers, along with the number of six-packs of each brand of beer each customer purchased, and calculate the averages for your data that correspond to those in Table. Do you get the same result (that the “Any” average is lower than all of the others)? Are you guaranteed to get this result? Does it depend on the amount of brand loyalty in your population, where brand loyalty is greater when customers tend to stick to the same brand, rather than buying multiple brands? Write up your results in a concise report.