Suspected serial killers and unsuspected statistical blunders

A whole branch of theoretical statistics devotes itself to the analysis of clusters, the aim being to distinguish an apparent cluster arising randomly from one that is more likely to have been produced as a result of some systematic influence. There are many examples in medicine and some that involve both medicine and the legal field; criminal law in particular. Observed clusters or a series of cases in a given setting can set off alarm bells, the recent conviction of Lucy Letby in England being an example. It was an observed cluster, a series of deaths among neonates, that prompted the investigation of Letby. There have been other similar cases in the past and there will be similar cases in the future. Our purpose is not to reconsider any particular trial but, rather, to work with similar, indeed more extreme numbers of cases as a way to underline the statistical mistakes that can be made when attempting to make sense of the data. These notions are illustrated via a made-up case of 10 incidents where the anticipated count was only 2. The most common statistical analysis would associate a probability of less than 0.00005 with this outcome: A very rare event. However, a more careful analysis that avoids common pitfalls results in a probability close to 0.5, indicating that, given the circumstances, we were as likely to see 10 or more as we were to see less than 10.