Is It Normal?

We return to the two histograms at the beginning of the previous chapter to ask why such different processes give rise to similar distributions. We simulate sampling a number of values from an exponential distribution and consider their sample mean. We find that the distribution of such sample means looks like a normal distribution. To understand why, we take a digression into the topic of random walks. After considering a random walk in discrete space, we identify a differential equation describing the random walk in continuous time and space. Using some rather complex mathematics, we find that the solution to this differential equation is a normal distribution with variance proportional to the time spent in the walk. We reason that such a random walk can describe the behavior of the sample sum or the sample mean taken in an experiment. This observation is very valuable because it allows us to assess how far from the true population mean our sample mean is likely to be. We then use this idea to introduce the idea of a t -test: asking whether two samples with different means are likely to have been drawn from populations with the same mean, using data on the lifespan of fruit flies as an example.