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The ability of random environmental variation to stabilize competitor coexistence was pointed out long ago and, in recent years, has received considerable attention. Analyses have focused on variations in the log-abundances of species, with mean logarithmic growth rates when rare, $\mathbb{E}[r]$, used as metrics for persistence. However, invasion probabilities and the times to extinction are not single-valued functions of $\mathbb{E}[r]$ and, in some cases, decrease as $\mathbb{E}[r]$ increases. Here, we present a synthesis of stochasticity-induced stabilization (SIS) phenomena based on the ratio between the expected arithmetic growth $μ$ and its variance $g$. When the diffusion approximation holds, explicit formulas for invasion probabilities and persistence times are single valued, monotonic functions of $μ/g$. The storage effect in the lottery model, together with other well-known examples drawn from population genetics, microbiology and ecology (including discrete and continuous dynamics, with overlapping and non-overlapping generations), are placed together, reviewed, and explained within this new, transparent theoretical framework. We also clarify the relationships between life-history strategies and SIS, and study the dynamics of extinction when SIS fails.