342 Trans. Acad. Sci. of St. Louis. 
The mean height, weight, girth of chest, etc., of each 
of these groups at any age will be the type of a certain 
degree of deviation from the mean of the age. That is to 
say, the weights, etc., of each group will be symmetrically 
distributed above and below the mean weight, etc., of the 
group in the manner already illustrated for the entire undivided 
number of observations, ¢. e., the entire thousand. Each 
group, therefore, will be characterized by a physical develop- 
ment definitely determined by the means of height, weight 
and other physical dimensions. These means, taken together 
form the type or norm of the group. The individual devia- 
tions from this norm follow the theory of probability, and 
the degree of abnormality presented by any individual devia- 
tion can be expressed in the terms of this theory. An exam- 
ple will illustrate this. A boy X shows a deviation in height 
of +1.5d from the mean height of hisage. He falls therefore 
in Group III. The boys in this group possess a mean weight 
of M kilogrammes, with a probable deviation of +d; that 
is, boys between d and 2d taller than the norm of their age 
should weigh M+d kilogrammes. In like manner, they 
should have a girth of chest of M+d centimetres, and a 
span of arms of M-+d centimetres, and so on. If the 
weight, etc., of the boy X coincide with the means of his 
group (Group III) his physique is normal, the accuracy of 
this conclusion being proportionate to the number of different 
dimensions on which it is based. If the boy X deviate more 
than +1d from the mean in ‘one or more dimensions, his 
development is abnormal, and the degree of abnormality is 
measured by the amount of his deviation. 
The necessity of choosing some one dimension as the basis 
of such a system is self-evident. There are good reasons, 
partly theoretical and partly practical, why height rather than 
weight should be taken as a basis. Height is more stable, 
less liable to irrelevant fluctuations, than weight. An excess 
in weight can be reduced ; a child whose weight is out of pro- 
portion to itsheight may be brought into proportion by suitable 
diet and exercise ; but height, once attained, cannot be reduced, 
nor can the growth in height be easily influenced: Practically, 
therefore, the physical trainer must be content to bring the 
