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Open-ocean deep convection is a highly variable and strongly nonlinear process that plays an essential role in the global ocean circulation. A new view of its stability is presented here, in which variability, as parameterized by stochastic forcing, is central. The use of an idealized deep convection box model allows analytical solutions and straightforward conceptual understanding while retaining the main features of deep convection dynamics. In contrast to the generally abrupt stability changes in deterministic systems, measures of stochastic stability change smoothly in response to varying forcing parameters. These stochastic stability measures depend chiefly on the residence times of the system in different regions of phase space, which need not contain a stable steady state in the deterministic sense. Deep convection can occur frequently even for parameter ranges in which it is deterministically unstable; this effect is denoted wandering unimodality. The stochastic stability concepts are readily applied to other components of the climate system. The results highlight the need to take climate variability into account when analyzing the stability of a climate state. [ABSTRACT FROM AUTHOR]