1886.] 



Family Likeness in Stature. 



observations, its axis is divided into 100 parts, the fiftieth division 

 being reckoned as 0°, then the deviates standing at the + graduations 

 of 10°, 20°, 25°, 30°, 40°, and 45° are measured. The mean of each 

 pair of lengths, not regarding signs, has then to be divided by the 

 mean lengths of the deviates at + 25°, that is by the quartiie deviate, 

 and so is made to yield a series that is directly comparable with 

 column B in Table I. The closeness with which it conforms to that 

 standard series is the test of the closeness with which the observations 

 conform to the law of frequency of error. 



Table II effects this comparison for all the series that I have to 

 deal with in the present paper. The values are entirely unsmoothed, 

 except in two named instances, being taken from measurements made 

 to the above-mentioned polygonal boundary. I thought it best to 

 give these interpolated values in this, their rudest form, leaving it to 

 be understood that with perfectly legitimate correction the accordance 

 would become still closer. I do not carry the comparison beyond 45°, 

 partly because my cases are not numerous enough to admit of a fair 

 comparison being made, and chiefly because I am well aware that 

 conformity is not to be expected towards the end of any series. I am 

 content to deal with nine-tenths of the observations, namely, those 

 between 0° and 45°, and to pay little heed to the remaining tenth, 

 between 45° and 50°. It will be seen that the conformity of more 

 than one half of each series is closer than to the first decimal place, 

 and that in absolute measurement it is closer than to one-tenth of an 

 inch. 



Arithmetic and Geometric Means. — I use throughout this inquiry the 

 ordinary law of frequency of error, which being based on the assump- 

 tion of entire ignorance of the conditions of variability, necessarily 

 proceeds on the hypothesis that plus and minus deviations of equal 

 amounts are equally probable. In the present subject of discussion our 

 ignorance is not so complete ; there is good reason to suppose that 

 plus and minus deviations, of which the probability is equal, are so 

 connected together that the ratio between the lower observed measure- 

 ment and the truth is equal to that between the truth and the upper 

 observed measurement. My reasons for this were explained some 

 years ago, and were accompanied by a memoir by Mr. Donald 

 Macalister, showing how the law of frequency of error would be 

 modified if based on the geometric, instead of on the arithmetic mean.* 

 Though in the present instance the former process is undoubtedly the 

 more correct of the two, the smallness of the error here introduced by 

 using the well known law is so insignificant that it is not worth 

 regarding. Thus the mean stature of the population is about 

 68*3 inches, and the quartiie of the stature-scheme (the probable 

 error) is 1*7 inch, or only about one-fortieth of its amount, and the 

 * "Proc. Boy. Soc," vol. 29 (1879), pp. 305, 367. 



VOL. XL. E 



