  This paper establishes the functional average as an important estimand for
causal inference. The significance of the estimand lies in its robustness
against traditional issues of confounding. We prove that this robustness holds
even when the probability distribution of the outcome, conditional on treatment
or some other vector of adjusting variables, differs almost arbitrarily from
its counterfactual analogue. This paper also examines possible estimators of
the functional average, including the sample mid-range, and proposes a new type
of bootstrap for robust statistical inference: the Hoeffding bootstrap. After
this, the paper explores a new class of variables, the $\mathcal{U}$ class of
variables, that simplifies the estimation of functional averages. This class of
variables is also used to establish mean exchangeability in some cases and to
provide the results of elementary statistical procedures, such as linear
regression and the analysis of variance, with causal interpretations.
Simulation evidence is provided. The methods of this paper are also applied to
a National Health and Nutrition Survey data set to investigate the causal
effect of exercise on the blood pressure of adult smokers.
