  A Low Humidity and Temperature Profiling (LHATPRO) microwave radiometer,
manufactured by Radiometer Physics GmbH (RPG), is used to monitor sky
conditions over ESO's Paranal observatory in support of VLT science operations.
The unit measures several channels across the strong water vapour emission line
at 183 GHz, necessary for resolving the low levels of precipitable water vapour
(PWV) that are prevalent on Paranal (median ~2.4 mm). The instrument consists
of a humidity profiler (183-191 GHz), a temperature profiler (51-58 GHz), and
an infrared camera (~10 {\mu}m) for cloud detection. We present, for the first
time, a statistical analysis of the homogeneity of all-sky PWV using 21 months
of periodic (every 6 hours) all-sky scans from the radiometer. These data
provide unique insight into the spatial and temporal variation of atmospheric
conditions relevant for astronomical observations, particularly in the
infrared. We find the PWV over Paranal to be remarkably homogeneous across the
sky down to 27.5{\deg} elevation with a median variation of 0.32 mm (peak to
valley) or 0.07 mm (rms). The homogeneity is a function of the absolute PWV but
the relative variation is fairly constant at 10-15% (peak to valley) and 3%
(rms). Such variations will not be a significant issue for analysis of
astronomical data. Users at ESO can specify PWV - measured at zenith - as an
ambient constraint in service mode to enable, for instance, very demanding
observations in the infrared that can only be conducted during periods of very
good atmospheric transmission and hence low PWV. We conclude that in general it
will not be necessary to add another observing constraint for PWV homogeneity
to ensure integrity of observations.
