Motivated by the importance of contextuality and a work on the robustness of the entanglement of mixed quantum states, the robustness of contextuality (RoC) R C (e) of an empirical model e against non-contextual noises was introduced and discussed in Science China Physics, Mechanics and Astronomy (59(4) and 59(9), 2016). Because noises are not always non-contextual, this paper introduces and discusses the generalized robustness of contextuality (GRoC) R g (e) of an empirical model e against

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... ral noises. It is proven that R g (e) = 0 if and only if e is non-contextual. This means that the quantity R g can be used to distinguish contextual empirical models from non-contextual ones. It is also shown that the function R g is convex on the set of all empirical models and continuous on the set of all no-signaling empirical models. For any two empirical models e and f such that the generalized relative robustness of e with respect to f is finite, a fascinating relationship between the GRoCs of e and f is proven, which reads R g (e)R g ( f ) ≤ 1. Lastly, for any n-cycle contextual box e, a relationship between the GRoC R g (e) and the extent ∆ e of violating the non-contextual inequalities is established.