Craniologlcal Notes M'l 



mixturo of craiiia bolongod or mit to the naiiio catogory as tlie .series dcalt witli (jm \k \i\ uf 

 Miss l''a\vcett'.s |iapei'. Haviiif; showii tliat tlio alisimlity of coiiipariiig it witli tlicso serics was 

 nieasui'od by ockls of Ai,i>')i to 1, 1 .said that tlic wliole prolilcni dcscrvcd coiisideratimi from a 

 wider standpoint. In othor words I toolc up de noeo tho quo.stioii of liioiiictrically detenaiiiing 

 the degrec of lioniogeiicity in cranial seriea on the liasis of longth and breadth mea-siireinentB, 

 and obtiiined rcsults which I liojicd might be hclpful in future. 1 did this because mere contro- 

 versy is very (hstastoful to nie, even whcn it is striving to [lull a groat branch of seience oiit of 

 the discreihtable rut it has lioen brought into by the u.se of liopclessly inisiiontiiic Statistical 

 niethcids. 1 fail to find any changc of front in tliis, for tlic catacondi data had alread)' been 

 dealt witli as suspeot on account of the reniarkablo pn>portioii of the two sexcs. Dr Myors' iiiixod 

 series was dismissed ou the odds of 42,552 to 1 ; it is hopelessly niore heterogeneoiis thaii the 

 worst of the four series quoted by Miss Fawcett. 1 tlion procoeded to got a nwre rej!ni'.<l tcst, liy 

 replacing the Frciich data, — which 1 huld not to lie a race niixture, but a niixtnre of individual.s 

 (and possibly sexes) fi-om the same race at diflurcnt periods, — by a series of Parisian crania froru 

 oiie period, provided for nie by the doyen of Kreuch craniologists as a honiogeneous series. 

 There is no change of front, niorcly an endeavour to reacli a still better critcrion*. 



(iii) Lastly, I turn to the saddost part of Dr Mycrs' letter, for it shows liow little hope there 

 is that he caii ever be converted to the inore modern view of statistics as an exact science. Instcad 

 of in tlie least replying to my chief contention that no numbcr is absohitely great or snidll, he 

 writes as follows about my calciilation of the odds against hi.s mixture of 42,552 to 1 : 



"Suiely commou sense lepudiates the legitimacy of such a calculation. But if mathematics allowa 

 it, what must also be the odds against the appearance of the figure 7'202 in the same series ! Yet this 

 fi^ure aetually oecurs there, being the measure of variability of tlie KCiO Fiench skuUs from the Paris 

 catacombs." 



All niatheinatics are inerely common sense in synibols and the divergenco betwoeii the two 

 can oiily arise from a slip in the mathematics, or because a writer labels as "common sense'' what 

 is not sen.se but what he wants to believe. Let nie suggest to Dr Myers the following experi- 

 ment ; Let him take 12 pennies and toss them 100,000 times. He will lind that only about 

 24 times in 100,000 trials, or once in 4167 trials, he will obtain all twelve heads. He would 

 obtain nine heads in about 5371 cases out of his 100,000 trials, or about once in 18 trials. 

 This is mathematical, but I know from experience that it is very closely indeed " common 

 sense." Now would Dr Myers be justified in saying that 12 — 9 = 3 is very small and arguing 

 as follows ? 



"Surely common sense repudiates the legitimacy of a calculation giving 41(5(1 to 1 against 

 12 heads. But if mathematics allows it what inu.st also be the odds against the appearance of 

 the tiguro i) in the same series of pciiiiy tossings ! Yet this figure aetually oecurs in penny 

 tossings." 



I hope Dr Myei'S will not think nie tlippant, but the arguments are really quite parallel, 

 except that his 8'389 is more than ten times as improbable as the ttcelve heads occwring. The 

 Standard deviation in this oase of penny tossings is 1-732 and the mean number of heads of 

 course, six. Thus, nine heads difler by not quite twice, and 12 heads by not quite four times, 

 the Standard deviation from tho mean. Now in the casc of tho crania the mean was 6'279, 

 the Standard deviation '51 85, and the French skulls' variability 7'2()2, or it difi'ers r780 times 

 the Standard deviation from the mean. A deviation greater than this will occiir once in 25 trials, 



* Dr Myers' mixture is absurd iu compaiison with the French cataeouib crania. Why is it a 

 change of front to show it is infinitely more absurd wlieu we use still more honiogeneous material ? His 

 proper reply would be to show tbat as we improve our material the Naqada series becomes very 

 improbable, but this is not the case. 



