Herschel's Wrong Calculation
The mathamtics of error statistics is now familiar. But it must be remembered that even a first-rate scientist made an elementary mistake some 150 years ago, in calculating the probability of errors in shooting at a "bull's eye", mentioned in Mayo's chapter 7. As a telling evidence for this, let me quote from John Herschel's popular lecture; it is included in his Familiar Lectures on Scientific Subjects, Alexander Strahan, 1867, and quotations are from chapter XIV, "On the Estimation of Skill in Target-Shooting".
Needless to say, John Herschel was one of the most respected and influential scientists of his day, and he wrote a long influential review of Quetelet's work on probability and statistics; Maxwell, Darwin, Jevons and many other eminent scholars respected him. And, in all probability, Maxwell's kinetic theory of gas (introducing velocity distribution among gas molecules) owes an important idea to Herschel's review.
The proportional numbers of hits in the several areas distinguished as gold, red, blue, white, black, marked out by equidistant rings of 4.8 inches each in breadth, surrounding the gold circle of that radius, ought, according to the statement given by myself in my review of M. Quetelet's work on Probabilities, to run as follows:---In the gold (out of 500 hits), 107; in the red annulus, 106; in the blue, 101; in the black, 97; and in the white, 89; supposing the target (terminating with the white) to receive half the entire number (1000) of arrows discharged; which in the case observed was not far from the truth. Whereas by the actual record of that day's shooting, handed to me afterwards, the proportional numbers corresponding to a total of 500 hits were:---Gold, 31; red, 89; blue, 121; black, 140; white, 119. This discordance with observation, being far too great to be attributable to ordinary casualty (the whole number of arrows discharged on the day in question being upwards of 7000), led me, of course, to re-examine the reasoning on which the first expectation had been grounded. And so enlightened, I was at no loss to discover its fallacy,---affording, as it does, a good example of the necessity of close attention to the wording of all reasonings on questions of probability. [495-496]
All right, let me insert my illustration for this passage. Herschel's problem is the following.
And Herschel made a wrong calculation. Wrong values and actual values are summarized in the table.
ª@ Herschel then continues as follows:
Now, it is perfectly true that the deviation of the point of incidence from the mark is error. But it is something more special. It is error in that one particular direction in which the point of incidence lies from the mark aimed at. In estimating, therefore, the probability of striking a target at a certain distance from the centre aimed at, we must multiply the probability of striking a determinate point at that distance from the centre, by the number of points within the extent of the target which actually do lie at that distance from it, without regard to the directions in which they lie: [497]
Although the distribution in this example may be different from Perrin's displacements, you can easily see that the actual numbers closely follow a similar pattern of those figures in Mayo's Table 7.1. Notice that Herschel's data are a half of the total shots, so that they consitute the middle part of the whole distribution (its density distribution is bell-shaped, and the (cumulative) distribution itself looks like an oblique flat "S"-like shape). In order to make the nature of Herschel's error intuitively clear, I will add the following figure. The bell-curve of normal distribution applies to a probability density distribution, and "density" means a number (of shots) per unit area. Notice the difference between Herschel's wrong density distribution and the actual density distribution.
In a word, Herschel's wrong calculation was based on the wrong density distribution (because he neglected the difference of area of each ring).
Last modified Jan. 27, 2003. (c) Soshichi Uchii
