Jan, 24, 1889] 



NATURE 



299 



An ingenious traveller might obtain a great number 

 of approximate and interesting data by the method just 

 described, measuring various faculties of the natives, 

 such as their delicacy of eyesight and hearing, their swift- 

 ness in running, their accuracy of aim with spear, arrow, 

 boomerang, sling, gun, and so forth, either laterally or 

 else vertically, distance of throw, stature, and much else. 

 But he should certainly use three test objects, and not 

 two only. 



It should be remarked that, if the distribution of de- 

 viation was constant throughout any large class of facul- 

 ties, though the O might differ in different sub-classes of it, 

 then, even though the distribution of that faculty was very 

 far indeed from being normal, an appropriate table of dis- 

 tribution could be drawn up to solve such problems as 

 those mentioned above. I have as yet no accurate data 

 to put this idea to a practical test. 



There are three convenient stages of expressing the 

 variety of the various measures in a series, each reaching 

 considerably nearer to precision than the one before. 

 The first is to give only Q and M ; the second is to 

 record the measures at the grades id", 25", 50"", 75°, and 

 90' ; the third is the more minute method, adopted m 

 the table of percentiles— viz. to give the measures at 5°, 

 10^, 20°, &c., 80°, 90°, and 95°. It may in some cases be 

 found worth while to go further, say to 1° and 99°, or 

 even also to 0° and 99°'9. So much for the expression 

 of variety. 



The use of O is by no means limited to the objects 

 just named. It is a necessary datum wherever the 

 law of frequency of error has ti be applied, and 

 the properties of this law are applicable to a very 

 large number of anthropological problems, with more 

 accuracy of result than might have been anticipated 

 when the series are only approximatively normal. This 

 has been practically shown by the agreement among 

 themselves of several inquiries to which I will shortly 

 allude, and it is theoretically defensible by two con- 

 siderations. The ore is that the law of frequency 

 supposes the amount of error or of deviation to be the 

 same in symmetrically disposed grades on either side of 

 50', their signs being alone different, niiniis on the one 

 side of 50 and plus on the other. Now, in an observed 

 series there may be, and often is, a want of symmetry, 

 but if the deviate, say at 70% is as much greater than the 

 normal as the deviate at 30' is less than the normal, then 

 the effects of these two upon the final result will be much 

 the same as if there had been exact symmetry at those 

 points. The other consideration is that any nonconformity 

 of the observed deviates with the theoretical ones towards 

 the end of the series has but a small and perhaps in- 

 sensible effect on the broad general conclusions We need 

 care little for any vagaries outside of the grades 5° and 

 95, if the intervening portion gives fairly good results. 

 The latter portion forms nine-tenths of the whole series, 

 and even considerable irregularities in the remaining 

 tenth are of small relative importance. 



One great use of Q 's to enable us to estimate the 

 trustworthmess of our average results. We require to 

 know both O and the number of observations before we 

 can estimate the degree of dependence to be placed on 

 M. If there was only one observation, then the degree 

 of dependence would be equal to Q ; in other words, the 

 error of M would be just as likely as not to exceed O. If 

 there were two, two hundred, two thousand, or any other 

 number of observations, the error of M would then be j 

 reduced, but not in simple proportion. It would be as j 

 likely as not to exceed a value equal to O divided by the 

 square roots of those numbers. When we desire to ascer- 

 tain the trustworthiness of the difference between the ] 

 M values of two series, as between the mean statures of, | 

 the professional and artisan class as derived from certain j 

 observations, the properties of the law of frequency of 

 error must again be appealed to. Anthr pologists are I 



much engaged in studying such differences as these ; bet 

 from their disregard of the simple datum O, and from not 

 being familiar with its employment, there is usually a 

 lamentable and quite unnecessary vagueness in the value 

 to be attached to their results. Thii is especially th^ 

 case in comparisons between the average dimensions of 

 the skulls of various races, which often depend upon the 

 measurement of only a few specimens. An almost solitary 

 exception to this needless laxity will be found in a brief 

 but admirably-expressed memoir by Dr. Venn, the weU- 

 known author of the "Logic of Chance.' It is upon 

 Cambridge anthropometry, and was published in the last 

 number of the Journal of this Institute. It deserves to be 

 a model to those who are engaged in similar inquiries.- 



Another class of investigations in which a knowledge of 

 Q is essential was spoken of some time back— namely.- 

 questions of correlation jn the widest sense of the word. 

 These problems have nothing to do with the relations of the 

 M value, but are solely concerned with those of the devia- 

 tions from M at the various grades. It is true that af 

 knowledge of M is requisite in order to subtract it from the 

 measures, and so to get at the deviations. But after this 

 is done, M is put aside. It has no part in the work of the 

 problem ; it is only after the results have been arrived at 

 without its use that it is again brought forward and added 

 to them. N umerous properties of the law of frequency of 

 error in which Q is the datum were utilized in my in- 

 quiries into family likeness in stature, and in all cases 

 they brought out consistent results. An excellent example 

 of this was seen in the success of the methods employed 

 to determine the variety in families of brothers. Foar 

 different properties of the law had to be applied to partly 

 different samples of the same group in order to determine 

 thevalue of the Q of stature in fraternities, and they respec- 

 tively gave io7,o'98, rio,and no inch, which, statistically 

 speaking, are much alike. Certain properties of the law 

 of frequency of error were also applied to family like- 

 ness in eye colour, with results that gave by calculation 

 the total number of light-eyed children in families 

 differently grouped according to their parentage .tnd 

 grandparentage, and according to three different sets of 

 data, as 623, 601, and 614 respectively, the observed 

 number being 629. Other properties of the same law 

 have been applied to determine the ratio of artistic to non- 

 artistic children in families whose parentages were known 

 to be either both artistic, one artistic, one not, or neither 

 artistic. They gave to 1507 children the ratios of 64, 39, 

 and 21, respectively, as against the observed values of 60, 

 39, and 17. 



Lastly, as regards the correlation of lengths of the 

 different limbs. It has already been shown that the cor- 

 relation lies between the deviations, and has nothing t<> do 

 with the values of M. Now, to express this relation truly, 

 so that it shall be reciprocal, the scale of deviation of the 

 correlated limbs, say, for example, of the cubit and of 

 the St iture, must be reduced to a common standard. We 

 therefore reduce them severally to scales in each of which 

 their own O is the unit. The (2 of the cubit is o" 56 inch, 

 therefore we divide each of its deviations by o 56. The Q 

 of the stature is 175 inch, so we divide each of its devialirns 

 by 175. When this is done the correlation is perfect. 

 The value of regression is found to be 08, whether the 

 cubit be taken as the " subject " and the mean of the 

 corresponding statures as the " relative," or vice versA. 



The value of the regression has been ascertained for 

 each of many pairs of the following elements, and a com- 

 parison was made in each case between the correlated 

 values as observed and those calculated from the ratio of 

 regression. The coincidence was close throughout, quite 

 as much so as the small number of cases under examina- 

 tion, 350 in all, could lead us to hope. The elements were 

 nine in all, viz. head-length, breadth of head, length 

 of right leg below the knee, of left cubit, of left 

 middle finger, of the height sitting above the chair, of 



