292 Trans. Acad. Sci. of St. Louis. 
from theaverage. It is seen in this table that the probable 
deviations are small, that is, one-half of the observations 
deviate but little from the middle value, which is one of the 
fundamental attributes of deviations due to accidental causes. 
But additional evidence is hardly required, and few critics will 
object to regarding the middle values in this investigation as 
types of their respective series. 
The objection sometimes made that the errors of observation 
materially affect the truth of the values obtained is of little 
weight, partly because such errors are ‘* accidental’’ and com- 
pensate each other as already explained, and partly because a 
deviation from the middle value due to an uncompensated 
error in measurement forms, as a rule, an inconsiderable part 
of that greater deviation which expresses the physiological 
difference between the individual and the type of his age and 
class. Accidental errors of observation need not give concern 
in measurements of great numbers of school children. Nor 
need there be much fear of constant errors of observation, 
provided the collection of material is made by many persons 
and with a good number of each sort of measuring instrument. 
If hundreds of teachers take part in the measurements, as in 
the present investigation, a constant cause of error due to a 
teacher’s unconscious bias or personal equation on one side of 
the middle value will very probably be compensated by the 
bias of another teacher on the opposite side, and, similarly, 
if a number of scales are used, the errors of those which weigh 
too lightly are likely to be compensated by the errors of those 
whose readings are too heavy. 
The trustworthiness of this material must be encouraging to 
those whose hold on the theory underlying these matters is 
not very strong, because it illustrates the truth that the types 
of physical development, and the laws of growth of the type 
may be induced from measurements made by comparatively 
unskilled hands and demonstrates that a system of anthropo- 
metrical measurements may be fruitfully employed in the 
public schools. 
he use to which middle values and the deviations from 
middle values shall be put is in part the subject of controversy, 
__ and it will be well to state here the manner in which they 
