  The spectral form factor as well as the two-point correlator of the density
of (quasi-)energy levels of individual quantum dynamics are not self-averaging.
Only suitable smoothing turns them into useful characteristics of spectra. We
present numerical data for a fully chaotic kicked top, employing two types of
smoothing: one involves primitives of the spectral correlator, the second a
small imaginary part of the quasi-energy. Self-averaging universal (like the
CUE average) behavior is found for the smoothed correlator, apart from noise
which shrinks like $1\over\sqrt N$ as the dimension $N$ of the quantum Hilbert
space grows. There are periodically repeated quasi-energy windows of
correlation decay and revival wherein the smoothed correlation remains finite
as $N\to\infty$ such that the noise is negligible. In between those windows
(where the CUE averaged correlator takes on values of the order ${1\over N^2}$)
the noise becomes dominant and self-averaging is lost. We conclude that the
noise forbids distinction of CUE and GUE type behavior. Surprisingly, the
underlying smoothed generating function does not enjoy any self-averaging
outside the range of its variables relevant for determining the two-point
correlator (and certain higher-order ones). --- We corroborate our numerical
findings for the noise by analytically determining the CUE variance of the
smoothed single-matrix correlator.
