The geometric mean in vital and social statistics / by Francis Galton. The law of the geometric mean / by Donald McAlister [sic].

  • Galton, Sir Francis, 1822-1911.
Date:
[1879]
Catalogue details

Licence: Public Domain Mark

Credit: The geometric mean in vital and social statistics / by Francis Galton. The law of the geometric mean / by Donald McAlister [sic]. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

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    [From the ProcebiVimos nuua^YAE, S^ciktv, No. 198, 1879.] 1 c-mr2o “ The Geometric Mean, jin' Vi'thl aricl Social Statistics.” By Francis Galton, F.B.Kr Received October 21, 1879. My purpose is to show that an assumption which lies at the basis of the well-known law of “ Frequency of Error ” (commonly expressed by the formula // = e-/‘’j3), is incorrect in many groups of vital and social phenomena, although that law has been applied to them by sta- tisticians with partial success and corresponding convenience. Next, I will point out the correct hypothesis upon which a Law of Error suitable to these cases ought to be calculated ; and subsequently I will communicate a memoir by Mr. Donald McAlister, who, at my sugges- tion, has mathematically investigated the subject. The assumption to which I refer is, that errors in excess or in deficiency of the truth are equally probable; or conversely, that if two fallible measurements have been made of the same object, their arithmetical mean is more likely to be the true measurement than any other quantity that can be named. This assumption cannot be justified in vital phenomena. For example, suppose we endeavour to match a tint; Fechner’s law, in its approxi- mative and simplest form of sensation=log stimulus, tells us that a series of tints, in which the quantities of white scattered on a black ground are as 1, 2, 4, 8, 16, 32, &e., will appear to the eye to be sepa- rated by equal intervals of tint. Therefore, in matching a grey that contains 8 portions of white, we are just as likely to err by selecting one that has 16 portions as one that has 4 portions. In the first case there would be an error in excess, of 8; in the second there would be an error in deficiency, of 4. Therefore, an error of the same magnitude in excess or in deficiency is not equally probable in the judgment of tints by the eye. Conversely, if two person#, who are equally good judges, describe their impressions of a certain tint, and one says that it contains 4 portions of white and the other that it contains 16 portions, the most reasonable conclusion is that it really contains 8 portions. The arithmetic mean of the estimates is lil? 2 or 10, which is not the most probable value. It is the geometric mean 8 (4 : 8 : : 8 : 16) which is the most probable. Precisely the same condition characterises every determination by any of the senses ; for example, in judging of the weight of bodies and of their temperatures, of the loudness and of the pitch of tones, and of estimates of lengths and distances as wholes. Thus, three rods of the lengths a, b, e, when taken successively in the hand, appear to differ by equal intervals when a : h : : b : c, and not when a—b—b—c. In