A method is proposed for improving the accuracy and efficiency of free energy simulations. The essential idea is that the convergence of the relevant measure (e.g., the free energy derivative in thermodynamic integration) is monitored in the reverse direction starting from the last frame of the trajectory, instead of the usual approach, which begins with the first frame and goes in the forward direction. This simple change in the use of the simulation data makes it straightforward to eliminate the contamination of the averages by contributions from the equilibrating region. A statistical criterion is introduced for distinguishing the equilibrated (production) region from the equilibrating region. The proposed method, called reverse cumulative averaging, is illustrated by its application to the well-studied case of the alchemical free energy simulation of ethane to methanol. (C) 2004 American Institute of Physics.