Malatesta Estimator
We frequently encounter competing estimates of politically salient magnitudes. One example would be the number of attendees at the 1995 “Million Man March”. Obviously, frequently the estimates emanate from biased observers seeking to create or dispel an impression of strength. Someone interested in generating a more neutral estimate might consider applying what I would call the Malatesta Estimator, which I have named after its formulator, the 14th Century Italian mercenary captain, Galeotto Malatesta of Rimini (d. abt. 1385). His advice was: “Take the mean between the maximum given by the exaggerators, and the minimum by detractors, and deduct a third” (Saunders 2004). This simplifies into: the sum of the maximum and the minimum, divided by three. It adjusts for the fact that the minimum is bounded below by zero, while there is no bound on the maximum. Of course, it only works if the maximum is at least double the minimum.
In the case of the Million Man March, supporters from the Nation of Islam claimed attendance of 1.5 to 2 million. The Park Service suggested initially that 400,000 had participated. The Malatesta Estimator therefore yields an estimate of 800,000. We can calibrate this by comparing it with an estimate by Dr. Farouk El-Baz and his team at the Boston University Remote Sensing Lab. Dr. El-Baz and his team used samples of 1 meter square pixels from a number of overhead photos to estimate the density per pixel, and then calculated an estimate for the entire area. Their estimate was 837,000, with 20% error bounds giving a range from 1 million to 670,000.
Saunders, Frances Stonor. 2004. The Devil’s Broker: Seeking Gold, God, and Glory in Fourteenth-Century Italy. (New York: HarperCollins), p. 93.
BU Remote Sensing Lab Press Release: http://www.bu.edu/remotesensing/Research/MMM/MMMnew.html
Accessed 14 December 2006.
As a mathematician, the obvious solution to this problem seems to me to be to take the geometric mean of the max and min, rather than either the arithmetic mean or the Malatesta estimate. That solves the problem of the minimum being bounded below without the arbitrariness of subtracting a third or the awkwardness of not working when the maximum and minimum are "too close". It also gives more intuitive estimates if the maximum and minimum differ greatly on a logarithmic scale, e.g. if the maximum is a million and the minimum is 10,000, then the estimate according to the geometric mean is 100,000, which seems like a reasonable estimate given those two bounds, unlike the Malatesta estimate of 337k, which seems too high. Finally, I should point out that in the example you give, the geometric mean estimate does approximately as well as Malatesta estimate (the geometric mean estimate would be 894k).
No reason to defer to an estimate just because it has a fancy name attached.
Anders, you are probably right if we can limit our Max and Min estimates to people who are acting rationally. There are complications though. One is pronouncements by persons who seek to be dramatic and have no concern with being believed. A second issue is the problem of "many" or "gazillions". In the Bible and the Middle East one finds frequent use of the number 40, e.g., Ali Baba and the Forty thieves, Moses wandering in the desert for 40 years, 40 paras equals one piastre, etc. Possibly or apparently in some ur-Semitic language the word for 40 and the word for myriad sounded alike, perhaps modulo vowels. Some size estimates may be nothing more than a loose way of saying many, many, many supporters. I suspect that in these cases something like a Benford's Law will be at work, so that we get disproportionately many estamiates that begin with the number 1, i.e., one hundred, one thousand, etc.