There are almost too many examples of the power and pervasiveness of mathematical ideas. For instance, this essay was written on a computer. The software of the computer, its mind and spirit, if you will, is a compilation of code that is based on the ideas of Claude Shannon, the father of information theory, and his article ‘A Mathematical Theory of Communication’ (1948). But perhaps this is both too obvious and too slight an example. The personal computer is hardly an essential part of human existence, even if most of us have structured our lives around it today. Let’s take something more basic and widespread, something most of us probably already apprehend, even if only dimly – like the idea of regression to the mean.
As the Oxford English Dictionary explains, regression to the mean is ‘the tendency for the values of any distributed variable to move towards the mean over repeated independent trials’. In other words, the more trials, the less random or mistaken the measure. For instance, say you’re in a race at school. You do surprisingly well and beat most of your classmates. All things being equal, the next time around, you’re actually not likely to do as well, relative to the other runners. Obviously, one’s actual rank depends on skill and talent – if you did well the first time, you probably are pretty fast – but each result also depends on luck as well as a host of other circumstances. Therefore, in order to mitigate against any selection effect, one has to run the experiment multiple times. In order to be able to see just where you actually place or rank, you have to be able to know the shape or form of the distribution of outcomes. The notion of regression to the mean informs how we think about a wide range of things, from the design of clinical trials, to gambling, to, well, the prosaic pep-talks we give ourselves after coming up short by saying: ‘OK, next time will be better.’ Actually, it probably will be.