454 Vai'iation and Correlation of the Human Skull 
Calculated 
Calculated 
mm. 
Observed 
Skew 
Normal 
mm. 
Observed 
Skew 
Normal 
119 & under 
0-5 
0-2 
0'7 
1 132 
11-5 
14"8 
15'8 
120 
1-0 
0-5 
0-6 
1 133 
15-0 
14-3 
15-0 
121 
0-5 
ro 
1-1 
134 
15-5 
13-3 
13-8 
122 
2-0 
1-9 
1-7 
135 
8-0 
11-8 
11-9 
123 
2-5 
31 
2-7 
136 
10-5 
10-1 
9-9 
124 
3'5 
4-6 
4-0 
137 
8'5 
8-3 
7-8 
125 
3-0 
6-3 
5-6 
138 
8-5 
6-4 
5-9 
126 
13-5 
8-2 
7-5 
i 139 
2-5 
4-7 
4-2 
127 
iro 
10-1 
9-6 
140 
3-5 
3-2 
2-9 
128 
11-5 
11-8 
11-7 
141 
4-0 
2-0 
1-9 
129 
13-5 
13-2 
13-5 
142 
0-0 
1-1 
1-2 
130 
10-5 
14-3 
14-9 
143 & over 
10 
1-0 
1-1 
131 
19-5 
14-8 
13-7 
Totals 
181 
181 
180-7 
Applying the test for goodness of fit*, we find for sl^evv curve: = 15"65 and 
P = '90, and for normal curve: ;i^'- = l7'46 and P = '83. In other words, if the 
skulls, of which the Naqadas are a sample, obeyed the former frequency distribu- 
tion we should get a more peaked polygon in 90 out of every 100 trial samples of 
181 crania ; and if they obeyed the latter, or normal distribution, in 83 out of 
every 100 trials. We see therefore quite clearly in a special case : first, that the 
multimodal appearance of short series of crania such as those represented in our 
diagrams may be wholly due to random sampling and be no sign of racial hetero- 
geneity ; secondly, that for many cranial series the normal curve, if it presents a 
worse fit than the skew curve, still gives a distribution quite good enough for 
most craniometric purposes. 
This is not to be interpreted as meaning that skewness and the distinction of 
mode and mean are not to be regarded, when enough material is available. They 
may ultimately lead the craniologist to important conclusions such as that sug- 
gested above on p. 443, but the data provided by most craniological series is not 
sufficient to determine them with significance. In other words, our present more 
elaborate investigations justify for short series of craniological characters the use 
of the normal curve. It has already been shown that such use is justified as 
a first approximation in the case of many other characters in manf. 
(10) On. the Correlation of Cranial Characters. 
Attention has already been drawn to the fact that the correlation of cranial 
characters in man is remarkably low and is also very irregular from race to racej. 
Unfortunately cranial series are always relatively small, and a small series com- 
bined with low correlation means a high probable error. Further, if we add possible 
* See Biometrika, Vol. i. p. 155. 
+ It is widely recognised now that the normal curve is not a general description of frequency. 
Hence its use for each class of cases must be independently demonstrated. Its simplicity and easy 
theoretical handling make any justifieation of its use such as the above of great importance. 
X Pearson: Phil. Trans. Vol. 187, A, 1896, pp. 279—281; Boas: American Anthropologist, Vol. i. 
1899, pp. 448 et seq.; Lee : Phil. Trans. Vol. 19G, A, pp. 228, 229. 
