  Using deep Herschel and ALMA observations, we investigate the star formation
rate (SFR) distributions of X-ray selected AGN host galaxies at 0.5<z<1.5 and
1.5<z<4, comparing them to that of normal, star-forming (i.e., "main-sequence",
or MS) galaxies. We find 34--55 per cent of AGNs in our sample have SFRs at
least a factor of two below that of the average MS galaxy, compared to ~15 per
cent of all MS galaxies, suggesting significantly different SFR distributions.
Indeed, when both are modelled as log-normal distributions, the mass and
redshift-normalised SFR distributions of X-ray AGNs are roughly twice as broad,
and peak ~0.4 dex lower, than that of MS galaxies. However, like MS galaxies,
the normalised SFR distribution of AGNs in our sample appears not to evolve
with redshift. Despite X-ray AGNs and MS galaxies having different SFR
distributions, the linear-mean SFR of AGNs derived from our distributions is
remarkably consistent with that of MS galaxies, and thus with previous results
derived from stacked Herschel data. This apparent contradiction is due to the
linear-mean SFR being biased by bright outliers, and thus does not necessarily
represent a true characterisation of the typical SFR of X-ray AGNs.
