We investigate the ferromagnetic Ising model with spin-flip dynamics by Monte Carlo computer simulation. The system is prepared at time t=0 by deeply quenching from a high-temperature disordered state, to a low-temperature nonequilibrium state. We analyze the growth of domains of the ordered phase through two measures of the average size of these domains: the fluctuation in magnetization and the perimeter density. Systems of size 602, 752, 1052, 1502, and 2402 are studied over large numbers of quenches (from 48 to 450 on a given lattice). We find that domains grow self-similarly following the Allen-Cahn law (domain area proportional to time). The effects of different updating procedures, finite size, and varying number of runs on the evolution and the statistics of the data are studied. We find that the time evolution given by random updating or a multispin coding algorithm are the same. We estimate the percentage error in the observed size of domains from a simple zero-time sum rule, which is independent of system size. This is found to be a reasonable estimate of error throughout the self-similar scaling regime.