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13.1 Potential Outcomes, Causal Effects, and Idealized Experiments 523

example, the effect of a drug could depend on your age, whether you smoke, or
other health conditions. The problem is that there is no way to measure the causal
effect for a single individual. Because the individual either receives the treatment
or does not, one of the potential outcomes can be observed, but not both.

Although the causal effect cannot be measured for a single individual, in
many applications it suffices to know the mean causal effect in a population. For
example, a job training program evaluation might trade off the average expendi-
ture per trainee against average trainee success in finding a job. The mean of the
individual causal effects in the population under study is called the average causal
effect or the average treatment effect.

The average causal effect for a given population can be estimated, at least in
theory, using an ideal randomized controlled experiment. To see how, first sup-
pose that the subjects are selected at random from the population of interest.
Because the subjects are selected by simple random sampling, their potential
outcomes, and thus their causal effects, are drawn from the same distribution, so
the expected value of the causal effect in the sample is the average causal effect
in the population. Next suppose that subjects are randomly assigned to the treat-
ment or the control group. Because an individual’s treatment status is randomly
assigned, it is distributed independently of his or her potential outcomes. Thus
the expected value of the outcome for those treated minus the expected value of
the outcome for those not treated equals the expected value of the causal effect.
Thus, when the concept of potential outcomes is combined with (1) random
selection of individuals from a population and (2) random experimental assign-
ment of treatment to those individuals, the expected value of the difference in
outcomes between the treatment and control groups is the average causal effect
in the population. That is, as was stated in Section 3.5, the average causal effect
on Yi of treatment (Xi = 1) versus no treatment (Xi = 0) is the difference in the
conditional expectations, E(Yi ͉ Xi = 1) - E(Yi ͉ Xi = 0), where E(Yi ͉ Xi = 1)
and E(Yi ͉ Xi = 0) are respectively the expected values of Y for the treatment and
control groups in an ideal randomized controlled experiment. Appendix 13.3
provides a mathematical treatment of the foregoing reasoning.

In general, an individual causal effect can be thought of as depending both
on observable variables and on unobservable variables. We have already
encountered the idea that a causal effect can depend on observable variables;
for example, Chapter 8 examined the possibility that the effect of a class size
reduction might depend on whether a student is an English learner. For most of
this chapter, we focus on the case that variation in causal effects depends only
on observable variables. Section 13.6 takes up unobserved heterogeneity in
causal effects.