Group and Total Average
Ecological fallacy also happens when the average for a group is approximated by the average in the total population divided by the group size. Suppose one knows the number of Protestants and the suicide rate in the USA, but one does not have data linking religion and suicide at the individual level. If one is interested in the suicide rate of Protestants, it is a mistake to consider as an unbiased estimate the total suicide rate divided by the number of Protestants. This estimate implicitly assumes that the suicidal rate of the other religions is zero.
Formally, denote the variable of interest and the mean of the group, we generally have:
However, the law of total expectation gives
This equation must be used to compute a more congent estimate. In this equation, the only things we don't know how to estimate are in blue. We know that, as a probability of suicide, is between 0 and 1. We can plug in this bound and our total estimate in the equation above to obtain an estimate of .
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