Mykr M. Orensteen 
5. Frequency Distributions. The total number of individuals measured being 
802, the measurements were grouped so as to save time and labour. 
The grouping of the data has been arranged so as to give approximately eleven 
sub-ranges of grouping in each case with limits as follows : 
Length ^ the head 
Breadth of the head 
Left middle finger 
Left foot ... 
Left cubit 
Stature 
every 3 inillimetres 
3 
7 
15 
4 centimetres 
Table III gives the mean, the standard deviation corrected by Sheppard's 
adjustments, and the corresponding probable error for each character observed. 
The cephalic index is computed as usual from the ratio of the breadth to the 
length multiplied by a hundred ; this gives a measure as to whether the head is 
relatively long or broad. 
The frequency distributions were further analysed. Sheppard's corrections 
were applied to all the moments, and the criterion k for the classification of the 
curves determined, k in all cases was less than unity ; jSj and ^.^ computed from 
|Lt2, f-sj j(^4 (<^he second, third and fourth moments) were practically 0 and 3 re- 
spectively for all distributions. 
The normal curve of the form Y = y^e'^l'^'^' was therefore chosen. 
The theoretical frequencies for every sub-group of each distribution of measure- 
ments have been computed from the following formulae: 
Length of the head: /= 162-73e-^-=/«''«i32, 
Breadth of the head: /= 205-50e-^'/*3«i94^ 
Left middle finger: /= 208-43e-^V75-3952^ 
Left foot: /= 181-20e-^V305-5i5o^ 
Left cubit: / = 226-30e-^'/8'"'-325'', 
Stature: /= 215-77 e-^'/'o-^o's, 
where / is the frequency for every unit of grouping, 
X is the distance from the origin (in these cases the origin and the mean 
coincide) to the foot of the ordinate to be calculated, 
e is the base of the Napierian system of logarithms. 
The comparison between the observed frequencies and the theoretical fre- 
quencies computed from the above formulae is given in Table IV. is calculated 
from the sum of the squares of the departures of the theoretical values from the 
observed ones, divided by the corresponding theoretical values. This is shown 
by the formula 
= 2 [{ijo - Vcflyc^ 
P is the probability that, in another random sample of the same size, a worse 
fit would be obtained. 
