Understanding the physics behind the dispersal of photo spheric magnetic flux is crucial to studies of magnetoconvection, dynamos, and stellar atmospheric activity. The rate of flux dispersal is often quantified by a diffusion coefficient, D. Published values of D differ by more than a factor of 2, which is more than the uncertainties allow. We propose that the discrepancies between the published values for D are the result of a correlation between the mobility and flux content of concentrations of magnetic flux. This conclusion is based on measurements of displacement velocities of Ca II K mottles using an uninterrupted 2 day sequence of filtergrams obtained at the South Pole near cycle minimum. We transform the Ca II K intensity to an equivalent magnetic flux density through a power-law relationship defined by a comparison with a nearly simultaneously observed magnetogram. One result is that, wherever the network is clearly defined in the filtergrams, the displacement vectors of the mottles are preferentially aligned with the network, suggesting that network-aligned motions are more important to field dispersal than deformation of the network pattern by cell evolution. The rms value of the inferred velocities, R = <|v|2>½, decreases with increasing flux, Φ, contained in the mottles, from R ≍ 240 m s-1 down to 140 s-1. The value of R(Φ) appears to be independent of the flux surrounding the concentration, to the extreme that it does not matter whether the concentration is in a plage or in the network. The determination of a proper effective diffusion coefficient requires that the function R(Φ) be weighted by the number density n(Φ) of mottles that contain a total flux. We find that n(Φ) decreases exponentially with Φ and propose a model of continual random splitting and merging of concentrations of flux to explain this dependence. Traditional methods used to measure D tend to be biased toward the larger, more sluggish flux concentrations. Such methods neglect or underestimate the significant effects of the relatively large number of the more mobile, smaller concentrations. We argue that the effective diffusion coefficient for the dispersal of photo spheric magnetic flux is ∼600 km2 s-1.