In Part I of this post, we started with a typical problem: which of two advertising campaigns was “better” in terms of generating more sales. Campaigns A and B were each tested for 20 days, during which time sales data was collected. The mean sales during Campaign A was $421 and the mean sales during Campaign B was $440.
Campaign B looks better on this evidence, doesn’t it? But suppose instead of 20 days, we only ran the campaigns one day each, and that the sales for A was just $421 and that for B was $440. B is still better, but our intuition tells us that the evidence isn’t as strong because the difference might be due to something other than differences in the ad campaigns themselves. One day’s worth of data just isn’t enough to convince us that B is truly better. But is 20 days enough?
Maybe. How can we tell? This is the part that Statistics plays. And it turns out that this is no easy problem. But please stay with me, because failing to understand how to properly answer this question leads to the most common mistake made in statistics. If you routinely use statistical models to make decisions like this—“Which campaign should I go with?”, “Which drug is better?”, “Which product do customers really prefer?”—you’re probably making this mistake too.
In Part I, we started by assuming that the (observable) sales data could be described by probability models. A probability model gives the chance that the data can take any value. For example, we could calculate the probability that the sales in Campaign A was greater than $500. We usually write this using math symbols like this:
Pr(Sales in Campaign A > $500 | e)
Most of that formula should make sense to you, except for the right-hand side of it. The bar at the end, the “|”, is the “given” bar. It means that whatever appears to the right of it is accepted as true. The “e” is whatever evidence we might have, or think is true. We can ignore that part for the moment, because what we really want to know is
Pr(Sales in B > Sales in A | data collected)
But that turns out to be a question that is impossible to answer using classical statistics!