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:
“No group and no government can properly prescribe precisely what should constitute the body of knowledge with which true education is concerned.”
—Franklin D. Roosevelt (1882–1945)
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)
“I believe in the total depravity of inanimate things ... the elusiveness of soap, the knottiness of strings, the transitory nature of buttons, the inclination of suspenders to twist and of hooks to forsake their lawful eyes, and cleave only unto the hairs of their hapless owner’s head.”
—Katharine Walker (1840–1916)
“The average American is a good sport, plays by the rules. But this war is no game. And no secret agent is a hero or a good sport—that is, no living agent.”
—John Monks, Jr., U.S. screenwriter, Sy Bartlett, and Henry Hathaway. Robert Sharkey (James Cagney)