Cratdologieal Notes 



:U3 



Table II. givcs the obscrvcil /iinl u.ilciil.itod values. A|)plying Uic inctlioJ for tcMtiiig 

 goodiicss of fit givcn in Biomtilrika, W>\. I. p. ITif), \vc timl in thc lirst ca.se: ;^- = 40'429 and 

 P='37, and in thc .sccond caso, ;(^ = 33'362 atid l'=i)'.). Or, .supixwing the Hungarian data 

 dealt with by I'rofes.sc n- v. Turök to actually follow tlio norni.il ov (lau.sMian curve, in evcry tlirce 

 saniples of 2000 .skull.s Im woidd on the avcnige havo found one fitting theory worse for 

 forehcad lireadtli than his .s,inii)le actually docs ; and for .skull brcadth overy two out of thrce 

 sauiples of 2000 skulLs would on tlie a\ei'age givc a worsc rcsult. In the face of such probabilities 

 ;is thi.s any sound statistician would not bcsitate to say that for skull broadtlis l'rofcs.s()r 

 V. Török's Hungarian data obcy tho nonual law of frequency, o.\actly as \ve have proved i.s 

 the casc for uiany scrics of otlier i-rani;d uieasurenicnt.s. The acconipanying diagr'iin.i .show the 

 observation.s fitted with thc theoretii'.U curvcs and denioiistrato at a glance how idlu i.s any 

 argument based ou Hcheingipfdu. 



TABLE II. 



Had thc mean diftcred from the mode, modern .Statistical theory could have deduced the 

 modal value without appealing to Sckeingipfeln. As it turns out, however, for these skull.s' 

 breadths Gaussian theory is pcrfectly applicable : mean, mode aml Diediaii ne/tsihl// foim-ide, and 

 Professor v. Törok's attack on thc arithnietical mean falls absolutely, and this bccause he 

 is attempting, as .so niany other craniologists do, to advance without any knowledgc of Statistical 

 theory. 



If the median wore to coincidc with the mean Professor v. Török teils us that thcre would Ije 

 a fixed point to start thc " Problem einer Typusbcstimumng " from. Well, it actually does 



