  The probability distribution function of column density (PDF) has become the
tool of choice for cloud structure analysis and star formation studies. Its
simplicity is attractive, and the PDF could offer access to cloud physical
parameters otherwise difficult to measure, but there has been some confusion in
the literature on the definition of its completeness limit and shape at the low
column density end. In this Letter we use the natural definition of the
completeness limit of a column density PDF, the last closed column-density
contour inside a surveyed region, and apply it to a set of large-scale maps of
nearby molecular clouds. We conclude that there is no observational evidence
for log-normal PDFs in these objects. We find that all studied molecular clouds
have PDFs well described by power-laws, including the diffuse cloud Polaris.
Our results call for a new physical interpretation for the shape of the column
density PDFs. We find that the slope of a cloud PDF is invariant to distance
but not to the spatial arrangement of cloud material, and as such it is still a
useful tool to investigate cloud structure.
