Traditional stochastic programs assume that the probability distribution of uncertainty is known. However, in practice, the probability distribution oftentimes is not known or cannot be accurately approximated. One way to address such distributional ambiguity is to work with distributionally robust convex stochastic programs (DRSPs), which minimize the worst-case expected cost with respect to a set of probability distributions. In this paper we analyze the case where there is a finite number of possible scenarios and study the question of how to identify the critical scenarios resulting from solving a DRSP. We illustrate that not all, but only some scenarios might have “effect” on the optimal value, and we formally define this notion for our general class of problems. In particular, we examine problems where the distributional ambiguity is modeled by the so-called total variation distance. We propose easy-to-check conditions to identify effective and ineffective scenarios for that class of problems. Computational results show that identifying effective scenarios provides useful insight on the underlying uncertainties of the problem.