Accuracy [(a) and (c)] and precision [(b) and (d)] of the dimensionless free energies, fi, from WHAM, BAR, ST-WHAM, UIM, and approximate formulae for the villin headpiece in solution. Two types of samples were used. A large sample [(a) and (b)] and a small sample [(c) and (d)] contain roughly 1% and 0.01% of all trajectory frames, respectively. The results were averaged over random samples. The MBAR results computed from all trajectory frames were used as the reference. fi at T = 300 K is fixed at zero. The lines are a guide for the eyes. (e) Energy histograms collected with bin width ΔE = 1.0.