  A statistical analysis of the eigenfrequencies of two sets of superconducting
microwave billiards, one with mushroom-like shape and the other from the
familiy of the Limacon billiards, is presented. These billiards have mixed
regular-chaotic dynamics but different structures in their classical phase
spaces. The spectrum of each billiard is represented as a time series where the
level order plays the role of time. Two most important findings follow from the
time-series analysis. First, the spectra can be characterized by two distinct
relaxation lengths. This is a prerequisite for the validity of the
superstatistical approach which is based on the folding of two distribution
functions. Second, the shape of the resulting probability density function of
the so-called superstatistical parameter is reasonably approximated by an
inverse chi-square distribution. This distribution is used to compute
nearest-neighbor spacing distributions and compare them with those of the
resonance frequencies of billiards with mixed dynamics within the framework of
superstatistics. The obtained spacing distribution is found to present a good
description of the experimental ones and is of the same or even better quality
as a number of other spacing distributions, including the one from Berry and
Robnik. However, in contrast to other approaches towards a theoretical
description of spectral properties of systems with mixed dynamics,
superstatistics also provides a description of properties of the
eigenfunctions. Indeed, the inverse chi-square parameter distribution is found
suitable for the analysis of experimental resonance strengths in the Limacon
billiards within the framework of superstatistics.
