The statistical properties of physical systems in thermal equilibrium are blatantly different from their far-from-equilibrium counterparts. In the latter, fluctuations often dominate the dynamics and might cluster in ordered patterns in the form of dissipative coherent structures. Here, we study the transition of a cold atomic cloud, driven close to a sharp electronic resonance, from a stable to a turbulent phase. From the atomic density distribution -- measured using a spatially-resolved pump-probe technique -- we have computed the Shannon entropy on two different basis sets. Information compression, corresponding to a minimum in the Shannon entropy, has been observed at criticality, where the system fluctuations organize into high-order (low-entropy) patterns. Being independent of the representation used, this feature is a property shared by a vast class of physical systems undergoing phase transitions.