The accelerometer data from mobile smart phones provide stochastic trajectories that change over time. This rate of change is unique to each person and can be well characterized by the continuous two-parameter family of Gamma probability distributions. Accordingly, on the Gamma plane each participant can be uniquely localized by the shape and the scale parameters of the Gamma probability distribution. The scatter of such points contains information that can unambiguously separate the normal controls (NC) from those patients with Parkinson’s disease (PD) that are at a later stage of the disease. In general normal aging seems conducive of more predictable patterns of variation in the accelerometer data. Yet this trend breaks down in PD where the statistical signatures seem to be a more relevant predictor of the stage of the disease. Those patients at a later stage of the disease have more random and noisier patterns than those in the earlier stages, whose statistics resemble those of the older NC. Overall the peak rates of change of the stochastic trajectories of the accelerometer are a good predictor of the stage of PD and of the age of a ‘normally’ aging individual.