342 Miscellanca 



We find : 



Greatest Forehead Breadth. 

 Mean= 1200475 mm. 

 S.D. = 5-8124 mm. 

 ^, = 33-783,910 

 ,1^ = 11-370,311 

 p =1192-0125 

 y = 5-9425 

 a = 200-5918 

 j/o = 137-273 

 Modc= 119-8792 mm. Mode = 142-4482 mm. 



The modo beiiig tlio oi-igin tlie eqiiations to the frequency distributions are resiiectively : 



/ ,. \ 1192 0125 



y= 137-273 (l+_:3^^) e— - 



and : 



i 40r-9270 



^='-^*'-^''*H'+T3(H279) «"""'•''• 



The high vaUics of p and a show how vcry close the distributions arc to thc Gaussian curve 

 in which mean, mode, and median coincide. Let us consider this matter a little more at length 

 by finding thc pro})ablc errors of ^jj and </, the di.stance bctweon mean and mode. The vahies of 

 these probable en-oi-s are to l« found in the piper on the Probable Errors of Frequency 

 Constants*, .and wo thus find for : 



Greatest Sl-till Breadth. Greatest Forehead Breadth. 



;i3 = 11-3703 ± 17-3500 ^3= 17-8107 ±7-5783 



d = -1683± 1081 d = -2163± -1088. 



Thus ^»3 is in ono case slightly more than double the probable error and may, perhaps, be 

 signiticant. In both cases d is lo.ss than twico its probable error and is thercfore very likcly to 

 be insigniticant. Actually this is the slendor sort of biusis lipon which Professor von Türok 

 would discanl the aritiimetic mean from craniology ! Lot>king at the rcsiilt a little more closely, 

 we See that Professor v. Török liaving no conception of the probable error of any sub-gi-oup 

 of frequency places his mode at the Scheingipfel of 121 mm. for the forehead breadth. The 

 actual mode of the graduated observations is at 119-88 min., only -17 mm. from the mean 

 at 120-05 mm. ; and thc diftcronco bctwccn the two i.s really withiii thc liniits of tlie errors 

 i)f random sampling. Tuniing to thc maxiniuni skull breadth, the aotual mode is at 142-45 mm. 

 and the mean at 14271 mm., wliile l'rofessor v. Türok points with triiiniph to Seheingipfeln at 

 141 and 144 as if they subverted the whole theory of the arithmetic mean ! Again we see that 

 the true mode as obtained from a proper .system of graduation is quitc clo.se to the mean, 

 i.e. within a sinall fraction of a millimetrc, and the difference is qiiite po.s.sibly diie to the errors 

 of random sampling. 



In Order to illustrate thcse points still furthor tlic normal ciirves for thc two series 

 were calculated ; they are : 



Greatest Forehead Breadth. Greatest Shdl Breadth. 



y= 137-273 e"Kö?iii' y = 136-684 «"'(s^stri' . 



Origin at mean: 120-0475 mm. Origin at mean: ]42-7095mra. 



* niomilriku. Vol. 11. p. '273. See also I'liil. Traii.'<. A., Vol. 1!)1, p. 275. 



