Enforced symmetry: the necessity of symmetric waxing and waning

A fundamental question in ecology is how the success of a taxon changes through time and what drives this change. This question is commonly approached using trajectories averaged over a group of taxa. Using results from probability theory, we show analytically and using examples that averaged trajectories will be more symmetric as the number of averaged trajectories increases, even if none of the original trajectories they were derived from is symmetric. This effect is not only based on averaging, but also on the introduction of noise and the incorporation of a priori known origination and extinction times. This implies that averaged trajectories are not suitable for deriving information about the processes driving the success of taxa. In particular, symmetric waxing and waning, which is commonly observed and interpreted to be linked to a number of different paleobiological processes, does not allow drawing any conclusions about the nature of the underlying process.