112 Prof. K. Pearson on some Applications of the 



the frequency distribution itself suggests the limits between 

 which the required root must lie. The determination of the 

 moments, on which the constants of the equation depend, is 

 such an e very-day task in statistical investigations that the 

 trained statistician soon performs it with the greatest rapidity- 

 In laboratories where such work is regularly done, the 

 Brunsviga and the Comptometer are always at the disposal 

 of the calculators, and reduce this labour to the lightest of 

 tasks. During the past five or six years I have had a number 

 of workers training for statistical research, and have made a 

 point of their resolving at least one heterogeneous frequency 

 distribution. The result is that we possess a considerable 

 number of cases relating partly to the animal and vegetable 

 kingdoms, partly to economic statistics, in which it has been 

 possible to differentiate heterogeneous material. We have 

 dealt with flowers, insects, fish, human characters (eyesight,, 

 judgment, skull-measurements, &c), price of investments, 

 and other topics. The object of the present paper is to> 

 illustrate the method by a selection of this material from one 

 field — that of skull-measurements. We hope thereby to 

 emphasize the type of problems for which the method offers 

 some solution, and further to indicate that it is not so 

 laborious that it cannot be used whenever it seems likely to 

 be of service. 



For our present purposes I have selected data bearing on 

 one character only of the human skull, namely, the cephalic 

 index. 



(2) Illustration I. — One of the most difficult and yet 

 important problems in craniology is the identification of sex. 

 There is a considerable element of speculation in the sexing 

 of skulls, and a control-investigation would be of great 

 value. Now it is clear that any skull-material before sexing, 

 even if from a pure race, is heterogeneous, consisting of two 

 groups, male and female. Hence it occurred to me that my 

 method of differentiation might be used as a control ex- 

 periment to test the sexing of the expert craniologists. I 

 chose the cephalic index for two reasons : (i.) because the 

 absolute dimensions of the skull change considerably and the 

 cephalic index hardly at all with age ; there would thus be 

 no danger due to a casting-out of juvenile skulls, which 

 might to some extent be really female, (ii.) From rather 

 extended observations on the cephalic index, I had noticed 

 that it approximated more to normal distribution than other 

 measurements. Against these advantages must be set the 

 disadvantage that the cephalic index is not a very markedly 

 sexual character. I accordingly chose the French race, in 



