414 LEO MM KYLE [DECEMBER 
heading are arranged the frequencies of the variations, under the third 
the average or standard deviation of the variations for each successive 
group of five characters, and under the fourth the probable error of 
each group. 








Average 
| Deviations? expressed in terms of the “Errors of Mean | Devia- | Prob- 
Character. | Squares.” tion of | able 
Varia- |Error. 
tions. 
SUZ emo oo -O0O+ "5 1 sby (2) 2sbeeee 
Intermaxilla (2) 2 9 20 14 7 2 
Intermaxilla (a) | 3 5 14 24 4 + 
Tail (2) . : 1 10 15 14 9 5 114| -96 
Eye (/) . ie Ps a 816] | 5 | 
Mandible i 4/6) 4] 4 9 512] 2 
Head (d) 3 5| | 10 6} {9 4 
Head (2) P38 017 | 011 8] 01 iOS 1 | 
Head (/) . 6 16 21 ql + 1-42 |1-29 
Body-Height .|2/0|/4/ 5] 7] 8 4} 5/ 0/8] 5 | 
C. Vertebrae . | 4 hy 
A. Vertebrae. | 5 2 
Fin-rays P(r). | 2 24 26 2 
1 PEN: 1a) 8 13 2 1:33 1-12 
reenter) Ze oh S| 6 4) 6.3, Ses 
i ll 2/3| 7] 3/12 81 610 131 








Average deviation for all the characters 1:29 
Probable error A Py = 109: 
The conclusion one would draw from these two examples is that the 
assumption which permits theory and observation to agree so closely 
cannot be far from correct. The characters were not taken in any 
fixed order, and the results show how the observed values fluctuate 
about the theoretical, the first group having an average deviation 
lower than the theoretical value, whilst the second and third have higher. 
This fluctuation gives point to an important corollary founded by Prof. 
Heincke on the theory shown above, namely, that if one or two char- 
acters only of all the individuals of a fixed group could be examined 
the variations would also agree with this law. The difficulty of ob- 
taining this fixed group of individuals similar in all respects as to age 
or size, place or living conditions, etc., is therefore the only drawback. 
A second corollary of even greater importance is also deduced 
from this theory. If this formula represents the “essence” of the 
variations of all the individuals, the conditions being the same, it must 
also do so for one. The variations of each individual are equally 
balanced on either side of the average, that is, if the variations of all 
the characters of each single individual could be tabulated they would 
be distributed about the mean according to the above law. 
1 The signs + and — are not used in obtaining the ‘‘ Average deviation.” 
