where N is the total number of samples, n 0 is the number of the first sample used in the average (earlier samples are discarded), N n 0 = N -n 0 +1, and x n is the vector of phase-space coordinates for the nth sample of the system. Assuming ergodicity, in the limit of an infinite number of samples, N n 0 , ⟨A⟩ [n 0 ,N ] = ⟨A⟩ [n 0 ,∞] = ⟨A⟩ π . With finite sampling, ⟨A⟩ [n 0 ,N ] is an approximation of ⟨A⟩ π with an associated error. The expected root-mean-squared error (RMSE) of the estimate can be separated into contributions from bias and variance