The local states and hypothetical scanning methods enable one to define a series of lower bound approximations for the free energy, F-A from a sample of configurations simulated by any exact method. F-A is expected to anticorrelate with its fluctuation sigma(A), i.e., the better (i.e., larger) is F-A the smaller is sigma(A), where sigma(A) becomes zero for the exact F. Relying on ideas proposed by Meirovitch and Alexandrowicz [J. Stat. Phys. 15, 123 (1976)] we best-fit such results to the function F-A = F-extp + C[sigma(A)](alpha) where C, and alpha are parameters to be optimized, and F-extp is the extrapolated value of the free energy. If this function is also convex (concave down), one can obtain an upper bound denoted F-up. This is the intersection of the tangent to the function at the lowest sigma(A) measured with the vertical axis at sigma(A) = 0. We analyze such simulation data for the square Ising lattice and four polymer chain models for which the correct F values have been calculated with high precision by exact methods. For all models we have found that the expected concavity always exists and that the results for F-extp and F-up are stable. In particular, extremely accurate results for the free energy and the entropy have been obtained for the Ising model.