  $Context.$ The traditional approach to characterize the structure of
molecular clouds is to map their line emission.
  $Aims.$ We aim to test and apply a stratified random sampling technique that
can characterize the line emission from molecular clouds more efficiently than
mapping.
  $Methods.$ We sampled the molecular emission from the Perseus cloud using the
H2 column density as a proxy. We divided the cloud into ten logarithmically
spaced column density bins, and we randomly selected ten positions from each
bin. The resulting 100 cloud positions were observed with the IRAM 30m
telescope, covering the 3mm-wavelength band and parts of the 2 and 1mm bands.
  $Results.$ We focus our analysis on 11 molecular species detected toward most
column density bins. In all cases, the line intensity is tightly correlated
with the H2 column density. For the CO isotopologs, the trend is relatively
flat, while for high-dipole moment species such as HCN, CS, and HCO+ the trend
is approximately linear. We reproduce this behavior with a cloud model in which
the gas density increases with column density, and where most species have
abundance profiles characterized by an outer photodissociation edge and an
inner freeze-out drop. The intensity behavior of the high-dipole moment species
arises from a combination of excitation effects and molecular freeze out, with
some modulation from optical depth. This quasi-linear dependence with the H2
column density makes the gas at low column densities dominate the
cloud-integrated emission. It also makes the emission from most high-dipole
moment species proportional to the cloud mass inside the photodissociation
edge.
  $Conclusions.$ Stratified random sampling is an efficient technique for
characterizing the emission from whole molecular clouds. It shows that despite
the complex appearance of Perseus, its molecular emission follows a relatively
simple pattern.
