Increase in stature of girls. Number of girls whose increase in stature was observed between the ages of 9 and 10 and 11 and 12 and 13 and 14 and 15 and 10. 1 Increase in centimeters. 5 and 6. 6 and 7. 7 and 8. 8 and 9.) 13. 1848 10.0 10.4. 10.5- 10.9. 11.0- 11.4. 11.5 11.9. 1 1 1 I next divided the series into two equal parts, the first embracing the short, the second the tall, individuals. The following amounts of growth were found for these two groups: Average annual increase (d+4) in stature of short and tall children between the following years: Difference (2 A). +0.37 +0.37 +0.17 +0.62 +0.51 +0.41 +1.31 Difference (24). +0.31 | +0.41 +0.33 +0.89, +0.71 -0.06 -1.35 -2.17 -2.22 Short Tall. That there must be an interdependence between the rate of growth and the actual size attained at a certain period can be shown to be a theoretical necessity. If the variability of a series at the age t is, and if the variability of the annual increment d is m, then, according to the theory of probabilities, the variability at the age t+1 must be Vu+ m2 if the amount of annual growth does not depend upon the size attained at the period t. Observations show that m is small as compared to μ. Observations also show that u first increases quite rapidly from year to year, and that at the period of adolescence it suddenly decreases very rapidly. It is clear that these phenomena do not agree with the assumption made. We must conclude, therefore, that the amount of annual growth depends upon the size attained at a certain period. It is possible to give an approximate value of this relation. If the average of all measurements for the period t is A, that for the period t, is 4+d, where d is the average amount of growth for the period t1-t. We will consider in what manner a value +d+v in the series of the period t, develops from the series of the period t. We will suppose that the relation between the actual size of an individual and the average amount of his annual growth is expressed by the simple relation ddax, where a is a constant. Ꮧ Furthermore, we will assume that the variability of d is the same for all values of r. The annual growth of a single individual of the size A+ will be, according to these assumptions, d+ax+y, where y expresses the accidental variation of the annual increment. The size of the individual at the period t, will therefore be By observation we find the variability at the period t- that is, that of r-equals M1. Therefore As a must be a small value, the positive root only is available, and we have It follows from this equation that as long as 4, is considerably larger than u, a positive; when, is smaller than , it is always negative. As during the early years increases with age, among young children the small ones are in a period of retarded growth, while the tall ones are in a period of accelerated growth, while among older children when begins to decrease again the tall ones cease growing, while the smaller ones grow rapidly. The values given on page 1549 for the amount of growth of short and tall children may be considered as equaling It is therefore possible to calculate a from the data contained in the table on page 1549. The two series of values show a fairly close agreement, considering the small number of repeated measurements. is a very rough approximation to actual conditions, and that, particularly during the period preceding puberty, the distribution of annual increase will differ considerably from this law. Dr. H. P. Bowditch, in a paper published in the Twenty-second Annual Report of the State Board of Health of Massachusetts, assumes that the growth of children is such that they always remain in the same percentile grade-that is to say, if the variability at the period t is μ, and at the period t, is u,, then the average child which has at the period t the measurement A+x=A+μ will have at the period t, the measurement A,+. Its growth during the intervening period will therefore be x μ The assumption is therefore narrower than the one made above, as a, which we tried to determine by means of the various data, is here given the arbitrary value M. It will be noticed that for The data given on pages 1546 and 1547 show that μ m is so large that it can not be neglected. can not be true, and we conclude that the average percentile grade of growing individuals is constantly changing. The average individual of the measurement A + x at the period t will be at the period t1 A+ x + d + ax = A + d + x (1 + a) If the individual remained on the same percentile grade, his measurement would It will be seen that the deviation (1) is smaller than (2). It follows, therefore, that the average of all growing individuals who in one year have a certain percentile grade will be nearer the general average the following year. This agrees with the results found by Dr. Henry G. Beyer. These facts and considerations have an important bearing upon the theory of the statistics of growth. When we consider children of a certain age, we find that they are not all in the same stage of development. Some have reached a point just corresponding to their age, while others are a little behind, and still others a little in advance, of their age. Consequently the values of their measurements will not exactly correspond to those of their age. We may assume that the difference between their stage of development and that belonging to their exact age is due to accidental causes, so that the number less developed than the average of a particular age will be the same as the number of those more developed: or there will be as many children in a stage of development corresponding to that of their age plus a certain length of time as in a stage corresponding to that of their age minus a certain length of time. The number of children who have a certain amount of deviation may be assumed to be arranged according to the laws of probability, so that the average of all the children will be exactly in the stage of development belonging to their age. Observations have shown that growth during childhood is quite regular, and that it decreases rapidly during the period of adolescence. At this period, when the rate of growth is decreasing, those children whose growth is retarded will be more remote from the value belonging to their age than those whose growth is accelerated. As the numbers above and below the average are equal, those with retarded growth will have a greater influence upon the average than those whose growth is accelerated; therefore the average of all values of the measurement of all the children of a certain age will be too low when the rate of growth is decreasing and too high when it is increasing. These considerations may be expressed in mathematical form as follows: In the adult the relative frequency of the variation a from the average value of the measurement s will generally be expressed by the formula where is the measure of the variability of the series. 1The Growth of United States Naval Cadets" (Proc. U. S. Naval Institute, Vol. XXI, No. 2, whole No. 74). 2The following theory was first published in "Science," Vol. XIX, 1892, May 6, p. 256; May 20, p. 281. The value of the measurement belonging to the average of all those individuals who will finally reach the value s is, at any given period, a function of that period, and may be called s. The value of the measurement at the period t of all those individuals who will finally reach the stature s+a is a function of s, and x, and may be expressed by f(s; ). The individuals constituting the adult series will not develop quite regularly, but some will be in advance of others. We assume that at any given time these variations in period will be distributed according to the law of probabilities. The relative frequency of the variation y from the period under consideration, t, will be The probability, therefore, of finding an individual who will finally have the statures+x, standing at the period of development t+y, and whose measure ment is therefore f(8+y; x) is equal to P1+z. P1+y; or, Pƒ (8 + y ; x): 1 μ.μ.2π e x2 y2 (3) The individuals who will finally have the measurement s+x, will have at a period ty, the same measurement that other individuals who will finally be 8+, have at the period t+y. Consequently there will be an infinitely large number of combinations of x and y, which will result in the same values + v. This will be the case whenever ƒ (81+y; x)=8+ V y = p(s; + v; x). By substituting this value of y in (3), and taking the integral for all values of x, The distribution of probabilities about the type will then be asymmetrical. It is possible to compute from these data the typical values for each year, and at the place quoted above I have given a method of approximation. The latter is, however, not sufficient. I have disregarded values of the order ab and b2 in arriving at the results given. This is, however, not sufficient. By including terms of higher order it is possible to compute the series more accurately, but the calculation is so exceedingly long and entails so much labor that I have given it up, particularly as it must be verified by actual observation. It seems more economical to wait until a satisfactory series of measurements, taken at annual intervals, is available. Dr. H. P. Bowditch' has called attention to the asymmetry of the curves, which he expressed by the difference between the probable and average values. His observations were corroborated by the study of material collected in St. Louis, Mo., by Dr. W. T. Porter, who followed the method laid down by Dr. Bowditch. In order to gain a better insight into the character of the annual curves I have combined all the available American material. This computation was carried out for me by Dr. G. M. West, according to my instructions. The computations were made under his immediate supervision, and he is responsible for the preliminary interpolation, while I made the final combination myself. 1 Twenty-second Annual Report of the State Board of Health of Massachusetts, pp. 479 ff. 2 Transactions of the Academy of Science of St. Louis, Vol. VI, No. 12, 1894, pp. 350 ff. ED 97-98 |