A Low Humidity and Temperature Profiling (LHATPRO) microwave radiometer, manufactured by Radiometer Physics GmbH (RPG), is used to monitor sky conditions over ESOs 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.
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