In 1873 Francis Galton had constructed a simple mechanical device where a ball is dropped vertically through a harrow of pins that deflect the ball sideways as it falls. Galton called the device a quincunx, although today it is usually referred to as a Galton board. Statisticians often employ (conceptually, if not physically) the quincunx to illustrate random walks and the central limit theorem. In particular, how a Binomial or Gaussian distribution results from the accumulation of independent random events, that is, the collisions in the case of the quincunx. But how valid is the assumption of "independent random events" made by Galton and countless subsequent statisticians? This paper presents evidence that this assumption is almost certainly not valid and that the quincunx has the richer, more predictable qualities of a low-dimensional deterministic dynamical system. To put this observation into a wider context, the result illustrates that statistical modeling assumptions can obscure more informative dynamics. When such dynamical models are employed they will yield better predictions and forecasts.
Time-evolution of the probability density under the action of a deterministic dynamical system
