Ensemble X-ray variability of Active Galactic Nuclei. II. Excess
  Variance and updated Structure Function
  Most investigations of the X-ray variability of active galactic nuclei (AGN)
have been concentrated on the detailed analyses of individual, nearby sources.
A relatively small number of studies have treated the ensemble behaviour of the
more general AGN population in wider regions of the luminosity-redshift plane.
We want to determine the ensemble variability properties of a rich AGN sample,
called Multi-Epoch XMM Serendipitous AGN Sample (MEXSAS), extracted from the
fifth release of the XMM-Newton Serendipitous Source Catalogue (XMMSSC-DR5),
with redshift between 0.1 and 5, and X-ray luminosities in the 0.5-4.5 keV band
between 10^42 and 10^47 erg/s. We urge caution on the use of the normalised
excess variance (NXS), noting that it may lead to underestimate variability if
used improperly. We use the structure function (SF), updating our previous
analysis for a smaller sample. We propose a correction to the NXS variability
estimator, accounting for the light curve duration in the rest frame on the
basis of the knowledge of the variability behaviour gained by SF studies. We
find an ensemble increase of the X-ray variability with the rest-frame time lag
tau, given by tau^0.12. We confirm an inverse dependence on the X-ray
luminosity, approximately as L_X^-0.19. We analyse the SF in different X-ray
bands, finding a dependence of the variability on the frequency as nu^-0.15,
corresponding to a softer when brighter trend. In turn, this dependence allows
us to parametrically correct the variability estimated in observer-frame bands
to that in the rest frame, resulting in a moderate shift upwards
(V-correction). Ensemble X-ray variability of AGNs is best described by the
structure function. An improper use of the normalised excess variance may lead
to an underestimate of the intrinsic variability, so that appropriate
corrections to the data or the models must be applied to prevent these effects.
