Mr. F. Galton on the Geometric Mean. [Nov. 20 r 



by the formula y=e~ h '- l '°~), 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 vrhich 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, &c, 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 persons, 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 



4 + 16 



contains 8 portions. The arithmetic mean of the estimates is — 



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, c, when taken successively in the hand, appear to differ 

 by equal intervals when a : b : : b : c, and not when a — b = b — c. In 

 all physiological phenomena, where there is on the one hand a stimulus 

 and on the other a response to that stimulus, Fechner's law may be 

 assumed to prevail ; in other words, the true mean is the geometric. 



The same condition of the geometric mean appears to characterise 

 the majority of the influences, which, combined with those of purely 

 vital phenomena, give rise to the events with which sociology deals. 

 It is difficult to find terms sufficiently general to apply to the varied 

 topics of sociology, but there are two categories of causes, which are 



