116 Prof. K. Pearson on some Applications of the 



It may be that there are other sources of heterogeneity 

 than sex in the skulls of the Paris Catacombs ; the skulk 

 were collected about the year 1800 from graveyards which 

 had been used for several centuries. Or, it may be that the 

 cephalic index for the Parisians has in itself a distribution 

 so markedly skew that the skewness due to sex-difference 

 is completely masked, 



(3) With a view of considering this point I suggested to 

 Miss Fawcett that she should endeavour to fit the material 

 with a skew frequency-curve. She found the following 

 equation to the curve: — 



( 7 , \ 13-88905 / ™ x 479372 



i-nsm) (^.irfess) • 



where y8x is the number of skulls with indices between arand 

 ,t, + S#. The modal value is 78*0988, or 78*1 say, and this is 

 the origin of or. The total range of cephalic index is from 

 66*2891 to 1123155, corresponding to an observed ran^e 

 from 68 to 97. The observed and calculated frequencies are 

 given in the table on p. 117, e being their difference. 



Forming the value of ^ 2 = S(<? 2 /y), where?/ is the theoretical 

 frequency, we find : 



% 2 = 23'0121. 



Hence following the method described in the Phil. Mag. 

 June 1900, pp. 163-164, we deduce * 



P = -8148. 



That is to say, in upwards of 81 cases out of a hundred random 

 samplings w T e should have obtained a system of frequencies 

 for our cephalic indices, differing as much or more from our 

 theoretical curve as the observed system. In other words, 

 the skew-curve is a remarkably good fit. Fig. 1 shows 

 this graphically. We thus conclude that in this case the 

 skew-curve achieves what neither one nor two normal curves 

 can do. If it be supposed that possibly the Paris skulls are 

 a triple racial mixture, then the diagram suggests that one of 

 these components will have a mean cephalic index as high as 

 87 to 88, i. e. a value 2 to 3 units above that of the medieval 

 Jews, and corresponding, as far as I am aware, to no race 

 likely to be found in Paris from 1600 to 1800. I see no hope 



* It has "been suggested that in the paper cited I have used the normal 

 distribution to prove that the normal distribution is of no special value. 

 The suggestion, however, overlooks the point, which I readily admit, that 

 the normal curve is a good fit to a binomial distribution, and that it is 

 precisely such binomial distributions to which the errors of random 

 sampling belong. 



