In this paper, we propose a new concept of almost second-degree stochastic dominance (ASSD), which we term almost risk-averse stochastic dominance (ARSD). Compared with existing ASSD conditions, ARSD can exclude extremely risk-averse utility functions. Hence, ARSD is able to reveal clear preferences of most risk-averse decision makers in practice, which are otherwise unable to be revealed. The simple closed-form of ARSD not only makes it easy to use in practice but also provides a clear insight into the preferences of decision makers and the difference in expected values and stochastic dominance violations. Moreover, we show that ARSD can be inferred based on mean and variance alone, and thus it is applicable even when distribution information is incomplete.