from Science, N. S., Vol. V., No. 119, Pages 570-573, April 9, 1897. ] 
SCIENCE 
Editorial Committee : S. Newcomb, Mathematics; R. S. Woodward, Mechanics; E. C. Pickering 
Astronomy; T. C. Mendenhall, Physics; R. H. Thurston, Engineering; Ira Remsen, Chemistry; 
J. Le Conte, Geology; W. M. Davis, Physiography; O. C. Marsh, Paleontology; W. K. 
Brooks, C. Hart Merriam, Zoology; S. H. Scudder, Entomology; N. L. Britton, 
Botany; Henry F. Osborn, General Biology; H. P. Bowditch, Physiology; 
J. S. Billings, Hygiene ; J. McKeen Cattell, Psychology ; 
Daniel G. Brinton, J. W. Powell, Anthropology. 
THE GRO WTH OF CHILDREN. 
In the years 1891 and 1892 I collected 
statistics on the growth of children in Wor- 
cester, Mass., mainly with a view to investi- 
gating individual growth. Although it was 
not possible, as was my original intention, 
to continue the series through a number of 
years, some results of interest have been 
obtained. The measurements were taken 
partly by myself, partly by fellows and 
students of Clark University. I am in- 
debted to Dr. G. M. West for many of the 
measurements. 
The stature of the same children was 
measured in May, 1891, and in May, 1892. 
The average annual increases and the vari- 
ability of the amount of growth (the mean 
of the squares of individual variations) for 
these intervals were as follows: 
ing the years of adolescence. After this 
period it must necessarily fall. It disap- 
pears as soon as all the individuals have 
ceased growing. This increase in varia- 
bility must be considered due to the 
effect of retardation and acceleration of 
growth. During the period preceding pu- 
berty some individuals will have reached 
their full growth, while others are still 
growing at a very rapid rate. As 
the rate of growth during the early 
years of childhood does not vary very 
much, retardation and acceleration will 
not have any effect of this sort. For the 
same reason the distribution of amounts of 
growth during the years preceding puberty 
is very asymmetrical and must be more so 
for the years from 17 to 20, for which I have 
no observations. It will be noticed that 
Average Increases in Stature of Children Between the Following Years (cm.). 
5 and 6 
6 and 7 
7 and 8 
8 and 9 
9 and 10 
10 and 11 
11 and 12 
12 and 13 
13 and 14 
14 and 15 
15 and 16 
Boys 
6.55 
5.70 
5.37 
4.89 
5.10 
5.02 
4.99 
5.91 
7.88 
6.23 
5.64 
Girls 
5.75 
5.90 
5.70 
5.50 
5.97 
6.17 
6.98 
6.71 
5.44 
3.34 
- 
Variability of Annual Growth. 
Boys 
±0.68 
±0.86 ±0.96 
±1.03 
±0.88 
±1.26 
±1.86 
±2.39 
±2.91 ±3.46 
Girls 
±0.88 ±0.98 
±1.10 ±0.97 
±1.23 
±1.85 
±1.89 
±2.06 
±2.89 
±2.71 
This table shows that young children 
grow more uniformly than older children. 
The increasing variability is very great dur- 
the growth of girls is more variable than 
that of boys. 
I have furthermore divided the series for 2 
SCIENCE. 
each year in two equal halves, the one em- 
bracing the tall children, the other embra- 
cing the short children. The average an- 
nual increases of these two groups are as 
follows: 
L 2 - m2 1 
a = _1 - 1 
\ p3 
where p and the variabilities of stature 
at the periods t and and where m the 
Average Increase in Stature Between the Following Years : 
6 and 7 
7 and 8 
8 and 9 
9 and 10 
10 and 11 
11 and 12 
12 and 13 
13 and 14 
14 and 15 
15 and 16 
Short boys 
5.5 
5.2 
4.8 
4.8 
4.8 
4.8 
5.2 
7.3 
7.5 
6.8 
Tall boys 
5.9 
5.5 
5.0 
5.4 
5.3 
5.2 
6.6 
8.5 
5.0 
4.4 
Difference 
+0.4 
+0.3 
+0.2 
+0.6 
+0.5 
+0.4 
+0.6 
+1.2 
-2.5 
-2.4 
Short girls 
5.7 
5.5 
5.3 
5.5 
5.8 
7.0 
7.4 
6.5 
4.5 
Tall girls 
6.1 
5.9 
5.6 
6.4 
6.5 
7.0 
6.0 
4.4 
2.2 
Difference 
+0.4 
+0.4 
+0.3 
+0.9 
+0.7 
0.0 
-1.4 
-2.1 
-2.3 
It appears that during the early years of 
childhood short children grow more slowly 
than tall children, that is to say their gen- 
eral development continues to be slow. 
Later on, during the period of adolescence, 
they continue to grow while tall children 
have more nearly reached their full de- 
velopment. That is to say, small children 
are throughout their period of growth re- 
tarded in development, and smallness at 
any given period as compared to the aver- 
age must in most cases be interpreted as due 
to slowness of development. During early 
life slowness of development which has 
manifested itself is likely to continue, while 
some of the effects of retardation will be 
made good during the period of adolescence, 
which is liable to be longer than in children 
who develop rapidly in early life. 
We will call the average stature at the 
age t A(; the amount of growth of an indi- 
vidual whose stature at that period is At 4- x 
may be called dx. We assume that the re- 
lation between the actual size of an indi- 
vidual and the average amount of its annual 
growth be expressed by the simple relation 
dx - d + ax 
where a is a constant. 
Furthermore we will assume that the vari- 
ability of dx will be the same for all values 
of x. Then it can easily be proved that 
variability of the amount of growth during 
the period t-t. 
From these data the following values of 
a have been computed: 
Age. 
Boys. 
Girls. 
6 
0.05 
0.05 
7 
0.05 
0.05 
8 
0.01 
0.01 
9 
0.03 
0.03 
10 
0.06 
0.06 
11 
0.06 
0.07 
12 
0.10 
- 0.11 
13 
0.08 
- 0.17 
14 
- 0.03 
- 0.20 
15 
- 0.22 
The values cannot claim any great weight, 
since the series of observations is very 
small. Only about fifty individuals for 
each year and sex are available. They 
prove, however, that the values of a first 
decrease until about the eighth year. Then 
they increase and decrease again very 
rapidly after the thirteenth year in boys 
and after the eleventh year in girls. 
According to the assumptions made be- 
fore, the average individual which meas- 
ured A + x at the period t will measure 
(A + x) + (<Z + ax) = A + c? + a; -1 il---2 
P P \ 
at the period tv 
If it measured 
A + J + 
p SCIENCE. 
3 
it would remain in the same percentile grade 
while according to the above formula its 
percentile standing will be nearer the aver- 
age than at the initial period t. Only when 
all the children of the initial measurement 
A -f- x grow equally, i. e., if m = 0 could 
they remain in the same percentile grade. 
This conclusion agrees with Dr. Henry G. 
Beyer's observations.* 
The above approximation is fairly satis- 
factory during the early years of childhood. 
During the period of adolescence it is not 
satisfactory, because the values of a are too 
large. More extended observations will 
enable us to include terms of higher order 
in the considerations and to obtain more 
accurate knowledge of the laws of growth. 
The results of this investigation suggest 
that the differences of growth observed in 
children of different nationalities and of 
parents of different occupations may also 
be partly due to retardation or acceleration 
of growth, partly to differences of develop- 
ment in the adult stage. 
In order to decide this question we may 
assume that in the averages obtained for all 
the series representing various social groups 
accidental deviation from the general aver- 
age only occurred. Then it is possible to 
calculate the average deviation which would 
result under these conditions. When the 
actual differences that have been found by 
observation are taken into consideration 
another average deviation results. If the 
latter nearly equals the former, then the 
constant causes that affect each social group 
are few and of slight importance. If it is 
much larger than the former, then the 
causes are many and powerful. The pro- 
portion between the theoretical value of the 
deviation and the one obtained by observa- 
tion is therefore a measure of the number 
and value of the causes influencing each 
series. 
I have applied these considerations to the 
* Proc. U. S. Naval Institute, Vol. XXI., No. 2. 
measurements of Boston school children ob- 
tained by Dr. H. P. Bowditch. I have used 
thirteen different classes in my calculations, 
namely, five nationalities : American, Irish, 
American and Irish mixed, German and 
English ; and eight classes grouped accord- 
ing to nationalities and occupations: Amer- 
ican professional, mercantile, skilled labor 
and unskilled labor, and the same classes 
among the Irish. 
The results are as follows : 
Boys. 
Deviation. 
Girls. 
Deviation. 
Age. 
Theory. 
Obs. 
Ratio. 
Theory. 
Obs. 
Ratio. 
5 
0.34 
0.34 
1.0 
0.40 
0.58 
1.5 
6 
0.28 
0.46 
1.6 
0.34 
0.57 
1.8 
7 
0.29 
0.76 
2.6 
0.32 
0.81 
2.5 
8 
0.28 
0.54 
1.9 
0.30 
0.71 
2.4 
9 
0.32 
0.89 
2.7 
0.36 
0.40 
1.1 
10 
0.33 
0.76 
2.4 
0.38 
0.83 
2.1 
11 
0.35 
1.05 
3.0 
0.46 
1.04 
2.2 
12 
0.40 
1.18 
3.0 
0.52 
1.89 
3.6 
13 
0.46 
1.65 
3.6 
0.52 
1.44 
2.8 
14 
0.57 
2.69 
44 
0.53 
0.98 
1.9 
15 
0.67 
2.06 
2.9 
0.53 
1.02 
1.9 
16 
0.72 
1.50 
2.1 
0.54 
0.53 
1.0 
We see that the values obtained by actual 
observation are always greater than those 
obtained under the assumption that only 
accidental causes influence the averages for 
each class. We also see that these causes 
reach a maximum during the period of 
growth and decrease as the adult stage is 
reached. The maximum is found in the 
fourteenth year in the case of boys, in the 
twelfth year in the case of girls, i. e., in 
those years in which the effects of accelera- 
tion and retardation of growth are strongest. 
Although the values given here cannot 
claim any very great weight on account of 
the small number of classes, this phenom- 
enon is brought out most clearly. 
The figures prove, therefore, that the 
differences in development between various 
social classes are, to a great extent, results 
of acceleration and retardation of growth 
which act in such a way that the social 4 
SCIENCE. 
groups which show higher values of meas- 
urements do so on account of accelerated 
growth, and that they cease to grow earlier 
than those whose growth is in the beginning 
less rapid, so that there is a tendency to 
decreasing differences between these groups 
during the last years of growth. 
Franz Boas.