A simple model of the random Brownian walk of a spherical mesoscopic particle in viscous liquids is proposed. The model can be solved analytically and simulated numerically. The analytic solution gives the known Einstein-Smoluchowski diffusion law r[superscript 2] = 2Dt, where the diffusion constant D is expressed by the mass and geometry of a particle, the viscosity of a liquid and the average effective time between consecutive collisions of the tracked particle with liquid molecules. The latter allows us to make a simulation of the Perrin experiment and to verify in the detailed study the influence of the statistics on the expected theoretical results. To avoid the problem of small statistics causing departures from the diffusion law we introduce in the second part of the paper the idea of the so-called artificially increased statistics (AIS), and we prove that, within this method of experimental data analysis, one can confirm the diffusion law and get a good prediction for the diffusion constant even if trajectories of just a few particles immersed in a liquid are considered. (Contains 1 table and 6 figures.)