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Stochastic ordering among distributions has been considered in a variety
of scenarios. However, it is often a restrictive model, not supported by the data even
in cases in which the researcher tends to believe that a certain variable is somehow
smaller than other. Alternatively, we propose to look at a more flexible version in
which two distributions satisfy an approximate stochastic order relation if they are
slightly contaminated versions of distributions for which stochastic order holds. The
minimal level of contamination required for stochastic order to hold is used as a
measure of deviation from exact stochastic order model. Our approach is based on
the use of trimmings of probabilities. We discuss their connection to approximate
stochastic order and provide theoretical support for its use in data analysis, proving
uniform consistency and giving non-asymptotic bounds for the error probabilities of
our tests. We provide simulation results and a case study for illustration.