Additionally, we repeated 1000 rounds of WHAM using umbrella windows along the reaction coordinate selected randomly from the independent repeats of the US runs. This "mixing" of the umbrella windows, or cross-WHAM, results in new pathways, which are combined from pieces of the original pathways. Figure shows that cross-WHAM can lead to lower and higher ∆W R values than for original repeats. We used the cross-WHAM replicas to estimate the error of each averaging approach. We randomly selected 5 replicas out of 1000 generated and computed averages for those 5 replicas. We repeated this procedure 10000 times and estimated the errors of various averaging approaches as the standard deviation of 10000 computed averages. The errors were 1.3, 1.8, and 1.7 kcal/mol for equal weight, exponential, and Boltzmann averaging respectively (Table ) . These errors were at least 0.9 kcal/mol higher than WHAM bootstrapping error (0.3-0.8 kcal/mol) for the trypsinbenzamidine system. Similar results for the errors were obtained for other systems (Table ).