228 



SCIENCE. 



[N. S. Vol. I. No. 9. 



dren will be exactly on the stage of develop- 

 ment belonging to their age. 



At a ijeriod when the rate of growth is 

 decreasing rapidly, those children whose 

 growth is retarded will be further remote 

 from the value belonging to their age 

 than those whose growth is accelerated. As 

 the number of children above and below 

 the average of development are equal, those 

 with retarded growth will have a greater 

 influence upon the average measurement 

 than those whose growth is accelei-ated, 

 therefore the average value of the measure- 

 ment of all the children of a certain age 

 will be lower than the typical value, when 

 the rate of growth is decreasing ; higher 

 than the typical value when the rate of 

 growth is increasing. This shows that the 

 averages and means of such curves have no 

 meaning as types. I have shown in the 

 place quoted above, how the typical values 

 can be computed, and also that for stature 

 they differ from the average up to the 

 amount of 17 mm. 



These considerations also show clearly 

 that the curves must be asymmetrical. 

 Supposing we consider the weights of girls 

 of thirteen years of age, the individuals 

 composing this group will consist of the fol- 

 lowing elements : girls on their normal 

 stage whose weight is that of the group 

 considered, advanced girls, and retarded 

 girls. In each of these groups which are 

 represented in the total group in varying 

 numbers, the weights of the individuals are 

 probably distributed according to the laws of 

 chance, or according to the distribution of 

 weights in the adult population. What, 

 however, will be the general distribution '? 

 As the rate of increase of weight is de- 

 creasing, there -will be crowding in those 

 parts of the curves which represent the 

 girls in an advanced stage of development, 

 and this must cause an asymmetry of the 

 resultant general curve, which will depend 

 upon the composition of the series. This 



asymmetry does actually exist at the 

 period when the theory demands it, and 

 this coincidence of theorj^ and observation 

 is the best argument in favor of the opin- 

 ion that advance and retardation of devel- " 

 opment are general and do not refer to any 

 single measurement. 



Futhermore, the increase in variability 

 until the time when growth begins to de- 

 crease, and its subsequent decrease, are en- 

 tirely in accord with this theory. I have 

 given a mathematical proof of this phenom- 

 enon in the paper quoted above (Science, 

 May, 1892). Dr. Porter has called atten- 

 tion to the same phenomenon in his paper 

 of Ifovember, 1893, but I believe his formu- 

 lation is not sufficiently general, nor does he 

 give an interpretation of the phenomenon 

 which may be explained as follows : The 

 probability of a child not being in the stage of 

 development corresponding to its age fol- 

 lows the laws of chance. With increas- 

 ing age the mean deviation from the normal 

 type must increase. Assuming that at 

 the age of four years, .5 year represents the 

 mean deviation, then a certain number of 

 children will be in the stage of develop- 

 ment corresponding to 3.5 and 4.5 j^ears. 

 At the age of sixteen years the mean devia- 

 tion will probably be one year, and just as 

 many children would be on the stages of 

 fifteen and seventeen years as there were of 

 the four-year old childi-en on the stages of 

 3.5 and 4.5 years. The absolute amount of 

 gTOwth (iu girls) from fifteen to seventeen 

 j^ears is less than from 3.5 to 4.5, so that 

 for this reason a decrease in variability 

 must be found at the time when the rate of 

 growth begins to decrease. On the other 

 hand, the difference between individuals 

 which will finally become tall or short, in- 

 creases with the increase of growth, so that 

 the combined effect of these counteracting 

 causes will be a maximum of variability at 

 the period preceding puberty. Dr. Porter's 

 formulation of the phenomenon (No. 2, p. 



