This work is focused on explicit finite size corrections in the calculation of the fluctuation in the number of hard disks and parallel (aligned) hard squares deposited on a finite flat surface through a random sequential adsorption process. Explicit size effects are made evident by using a finite-system pair correlation function for calculating the fluctuation. The method is based on the relation between this pair correlation function and its infinite-system counterpart. A diagrammatic density (coverage) expansion of the corresponding infinite-system pair correlation function is used to calculate the low-coverage behavior of the fluctuation. Results also include border effects due to consider a finite size region for evaluating the fluctuation. A comparison with Monte Carlo computer simulations shows an excellent agreement between theoretical and simulation results.