We use a large single particle tracking data set to analyze the short time and small spatial scale motion of quantum dots labeling proteins in cell membranes. Our analysis focuses on the jumps which are the changes in the position of the quantum dots between frames in a movie of their motion. Previously we have shown that the directions of the jumps are uniformly distributed and the jump lengths can be characterized by a double power law distribution. Here we show that the jumps over a small number of time steps can be described by scalings of a single double power law distribution. This provides additional strong evidence that the double power law provides an accurate description of the fine scale motion. This more extensive analysis provides strong evidence that the double power law is a novel stable distribution for the motion. This analysis provides strong evidence that an earlier result that the motion can be modeled as diffusion in a space of fractional dimension roughly 3/2 is correct. The form of the power law distribution quantifies the excess of short jumps in the data and provides an accurate characterization of the fine scale diffusion and, in fact, this distribution gives an accurate description of the jump lengths up to a few hundred nanometers. Our results complement of the usual mean squared displacement analysis used to study diffusion at larger scales where the proteins are more likely to strongly interact with larger membrane structures.