$Context.$ 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.
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