  We analyze two theoretical approaches to ensemble averaging for integrable
systems in quantum chaos - spectral averaging and parametric averaging. For
spectral averaging, we introduce a new procedure - rescaled spectral averaging.
Unlike traditional spectral averaging, it can describe the correlation function
of spectral staircase and produce persistent oscillations of the interval level
number variance. Parametric averaging, while not as accurate as rescaled
spectral averaging for the correlation function of spectral staircase and
interval level number variance, can also produce persistent oscillations of the
global level number variance and better describes saturation level rigidity as
a function of the running energy. Overall, it is the most reliable method for a
wide range of statistics.
