To describe the polypeptide helix-coil transition, while the Ising-based theory has been playing the principal role for 40 years, we can now make use of computer simulation using the so-called "all-atom model" that is far more precise than the Ising-based model. In this study, by conducting molecular dynamics (MD) simulations of helix-coil transition exhibited by a short polyalanine chain, we investigated how the MD simulation results and the Ising-based theoretical values coincide with each other, placing a focus on their equilibrium statistical mechanical properties. Several important physical properties, such as temperature-dependent helix ratio, distribution of the helix-residue number, position-dependent helix ratio, and pair-correlation between residue states were taken up as the proving grounds on which we made a comparison between the all-atom model simulation and the Ising-based theory. As an overall trend, we realized that the Ising-based theoretical results agreed with the all-atom simulation results at least qualitatively, suggesting that the Ising-based model, though very simple, extracts the essence of the phenomenon with respect to the equilibrium properties. On the other hand we found some quantitative disagreements between them. The origins of the observed disagreements are discussed by going into details of the all-atom model.