Figure 1.1: Pigeon about to tap for food (source: https://archive.org/details/controllingbehaviorthroughreinforcement).

      This kind of learning is not quite a weighted sum of past experience: for example, negative experiences typically count more than positive ones, and once a pattern is established it takes a lot to shift it. However, it is not so far from a probability estimate. We humans share these subconscious learning processes with other animals. They are powerful and lead to very rapid reactions, but need very large numbers of exposures to similar situations to establish memories.

      Of course we are not just our subconscious! In addition, we have conscious thinking and reasoning, which enable us to learn from a single experience. Retrospectively we are able to retrieve a relevant past experience, compare it to what we are encountering now, and work out what to do based on it. This is very powerful, but unlike our more unconscious sea of overlapping memories and associations, our conscious mind is linear and is normally locked into a single model of the world. Because of that single model, this form of thinking is not so good at intuitively grasping probabilities, as is repeatedly evidenced by gambling behaviour and more broadly our assessment of risk.

      One experiment used four packs of cards with different penalties and rewards to see how quickly people could assess the difference [5]. The experiment included some patients with prefrontal brain damage, but we’ll just consider the non-patients. The subjects could choose cards from the different packs. Each pack had an initial reward attached to it, but when they turned over a card it might also have a penalty, “sorry, you’ve lost $500.” Some of the packs, those with the higher initial per-card reward, had more penalties, and the other packs had a better balance of rewards. After playing for a while most subjects realised that the packs were different and could tell which were better. The subjects were also wired up to a skin conductivity sensor as used in a lie detector. Well before they were able to say that some of the card packs were worse than the others, they showed a response on the sensor when they were about to turn over a card from the disadvantageous pack—that is subconsciously they knew it was likely to be a bad card.

      Because our conscious mind is not naturally good at dealing with probabilities we need to use the tool of mathematics to enable us to reason explicitly about them. For example, if the subjects in the experiment had kept a tally of good and bad cards, they would have seen, in the numbers, which packs were better.

1.1.2 MATHS AND MORE

      Some years ago, when I was first teaching statistics, I remember learning that statistics education was known to be particularly difficult. This is in part because it requires a combination of maths and real-world thinking.

      In statistics we use the explicit tallying of data and mathematical reasoning about probabilities to let us do quite complex reasoning from effects (measurements) back to causes (the real word phenomena that are being measured). So you do need to feel reasonably comfortable with this mathematics. However, even if you are a whizz at maths, if you can’t relate this back to understanding about the real world, you are also stuck. It is a bit like the applied maths problems where people get so lost in the maths that they forget the units: “the answer is 42”—but 42 what? 42 degrees, 42 metres, or 42 bananas?

      On the whole, those who are good at mathematics are not always good at relating their thinking back to the real world, and those of a more practical disposition are not always best at maths—no wonder statistics is hard!

      However, knowing this we can try to make things better.

      It is likely that the majority of readers of this book will have a stronger sense of the practical issues, so I will try to explain some of the concepts that are necessary, without getting deep into the mathematics of how they are calculated—leave that to the computer!

1.2 DO YOU NEED STATS AT ALL?

      The fact that you have opened this book suggests that you think you should learn something about statistics. However, maybe the majority of your work is qualitative, or you typically do small-scale studies and you wonder if it is sufficient to eyeball the raw data and make a judgement.

      Sometimes no statistics are necessary. Perhaps you have performed a small user trial and one user makes an error; you look at the circumstances and think “of course lots of users will have the same problem.” Your judgement is based purely on past experience and professional knowledge.

      However, suppose you have performed a survey comparing two alternative systems and asked users which system they prefer. The results are shown in Fig. 1.2. It is clear that System A is far more popular than System B. Or is it?

      Notice that the left hand scale has two notches, but no values. Let’s suppose first that the notches are at 1000 and 2000: the results of surveying 3000 people. This is obviously a clear result. However, if instead the notches were at 1 and 2, representing a survey of 3 users, you might not be so confident in the results. As you eyeball the data, you are performing some informal statistics.