We develop a new approach for quantifying uncertainty in finite populations, by using design distributions to calibrate sensitivity parameters in finite population identified sets. This yields uncertainty intervals that can be interpreted as identified sets, robust Bayesian credible sets, or uniform frequentist design-based confidence sets. We focus on quantifying uncertainty about the average treatment effect, where our approach (1) yields design-based confidence intervals which allow for heterogeneous treatment effects without using asymptotics, (2) provides a new motivation for examining covariate balance, and (3) gives a new formal analysis of the role of randomization. We illustrate our approach in three empirical applications.

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