1879.] 



On the Law of the Geometric Mean. 



367 



of common occurrence. The one is that of ordinary increase, as 

 exemplified by the growth of population, where an already large 

 nation tends to become larger than a small one under similar cir- 

 cumstances, or when a capital employed in a business increases in 

 proportion to its size. The other category is that of surrounding 

 influences, or " milieux " as they are often called, such as a period of 

 plenty in which a larger field or a larger business yields a greater 

 excess over its mean yield than a smaller one. Most of the causes 

 of those differences with which sociology are concerned, and which 

 are not purely vital phenomena, such as those already discussed, may 

 be classified under one or other of these two categories, or under such 

 as are in principle almost the same. In short, sociological phenomena, 

 like vital phenomena are, as a general rule, subject to the condition of 

 the geometric mean. 



The ordinary law of Frequency of Error, based on the arithmetic 

 mean, corresponds, no doubt, sufficiently well with the observed facts 

 of vital and social phenomena, to be very serviceable to statisticians, 

 but it is far from satisfying their wants, and it may lead to absurdity 

 when applied to wide deviations. It asserts that deviations in excess 

 must be balanced by deviations of equal magnitude in deficiency 

 therefore, if the former be greater than the mean itself, the latter 

 must be less than zero, that is, must be negative. This is an impossi- 

 bility in many cases, to which the law is nevertheless applied by statis- 

 ticians with no small success, so long as they are content to confine its 

 application within a narrow range of deviation. Thus, in respect of 

 stature, the law is very correct in respect to ordinary measurements, 

 although it asserts that the existence of giants, whose height is more 

 than double the mean height of their race, implies the possibility of 

 the existence of dwarfs, whose stature is less than nothing at all. 



It is, therefore, an object not only of theoretical interest but of 

 practical use, to thoroughly investigate a Law of Error, based on the 

 geometric mean, even though some of the expected results may 

 perhaps be apparent at first sight. With this view I placed the fore- 

 going remarks in Mr. Donald McAlister's hands, who contributes the 

 following memoir. 



XIII. " The Law of the Geometric Mean." By Donald 

 McAlister, B.A., B.Sc, Fellow of St. John's College, 

 Cambridge. Communicated by Francis Galton, F.R.S. 

 Received October 21, 1879. h 



Suppose we have before us a large number of measurements. They 

 may either be all approximations to the true value of a single unknown 

 quantity, or may refer to the several members of a large class. The 



