We consider several stochastic processes corresponding to the same physical solid-on-solid deposition problem. Simplified models presenting the same (conditional) mean and variance for each process are also introduced as well as generalizations in terms of the deposition of blobs and probabilistic deposition rules. We compare the evolution of the roughness as a function of time for a three-parameter family that includes as limit cases the Family model and the Edwards-Wilkinson equation, showing that in all cases the derived models with the same mean and variance are indistinguishable from the originating models in terms of the evolution of the roughness. Finally, we show that although all the models studied belong to the same universality class, some relevant features such as the final surface roughness are reproduced only for models within a restricted class determined by sharing the same (conditional) mean and variance.
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