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 .
Read more about this topic: Ecological Fallacy
Famous quotes containing the words group and, group, total and/or average:
“The conflict between the need to belong to a group and the need to be seen as unique and individual is the dominant struggle of adolescence.”
—Jeanne Elium (20th century)
“The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.”
—Babette Deutsch (18951982)
“Never hug and kiss your children! Mother love may make your childrens infancy unhappy and prevent them from pursuing a career or getting married! Thats total hogwash, of course. But it shows on extreme example of what state-of-the-art scientific parenting was supposed to be in early twentieth-century America. After all, that was the heyday of efficiency experts, time-and-motion studies, and the like.”
—Lawrence Kutner (20th century)
“The average parent may, for example, plant an artist or fertilize a ballet dancer and end up with a certified public accountant. We cannot train children along chicken wire to make them grow in the right direction. Tying them to stakes is frowned upon, even in Massachusetts.”
—Ellen Goodman (b. 1941)