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 a will assume the value "". The data given on pages 1546 and 1547 show that μ m is so large that it can not be neglected. Therefore the assumption a 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 t 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.1 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. 2, whole No. 74). "The Growth of United States Naval Cadets (Proc. U. S. Naval Institute, Vol. XXI, No. 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 8+ is a function of s, and a, and may be expressed by f(s; 2). 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 y2 22 dy. 1 Pt+y= e μην απ (2) The probability, therefore, of finding an individual who will finally have the statures+, standing at the period of development t+y, and whose measure ment is therefore ƒ (8+y; x) is equal to P1+2 · Pi + y; or, The individuals who will finally have the measurement s+x, will have at a period t+y, the same measurement that other individuals who will finally be + 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 (8+y; x)=&+V y = P(8 + 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. he Dr. H. P. Bowditch has called attention to the asymmetry of the curves, which 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. Twenty-second Annual Report of the State Board of Health of Massachusetts, pp. 479 ff. The method of procedure was the following. Observations are available from the following six cities: Boston, Milwaukee, St. Louis, Worcester, Toronto, Oakland. These represent a variety of conditions. We may assume that the variations represented by various cities are due to accidental causes, that is to say, that when the children in all the towns and cities of the country are measured we expect to find the results to vary around a certain average, according to the laws of probability. The type of the total population would embrace statistics of all the individuals of various ages. These are not available, and we must consider the cities in which the measurements were taken as representatives of the total population. In order to unite the material properly we ought to know how large a portion of the population is represented by each city. We can not obtain any satisfactory information on this point, and the only practicable way of uniting the material seems to be to add all the measured individuals, without regard to the varying numbers that were measured in each city. This has been done. It was necessary to reduce the observations that were recorded in inches to centimeters. Similar reductions were necessary in the tables of weights. This required a lengthy interpolation. The St. Louis measurements required an additional interpolation, as the age of the measured children was recorded at the nearest birthday, while all the other observers counted age from the last birthday. The results of this calculation are given on pages 1555 and 1556. It will be noticed that the distribution is rather unexpectedly irregular. I presume this is due to the fact that observers developed a tendency to round their observations, so that full inches and the centimeters ending with 0 or 5 (110, 115, 120, etc.) were given undue preference. It is likely that if this fact had been considered, the resulting curves would have been smoother. Frequencies of statures of American boys, in percentages. Height in centimeters. Ages, in years. 5.589 6.536 7.511 8.504 9.496 10.494 11.492 12.489 13.481 14.467 15.454 16.445 17.453 18.424 0.1 0.5 0.1 0.5 0.1 0.4 0.1 0.3 variation Mean vari ation Corrected average for half year Mean vari ation cor rected Mean vari ation at half 1,535 3,975 5,379 5, 633 5,531 5, 151 +3.83 +3.98 ± 4.174.35 4.56 ± 4.78 14.97 +5.35 +6.046.88 ± 7.3116.15 +5.7845.45 6.326.797.69 8.65 £8.92 47.777.256.76 14.81 +4.92 +5.22 -5.53 +5.66 5.90 105.90 111.58 116. 83 122. 04/123. 91 131. 78 136. 20, 140. 74 146.00 152. 89 159. 72164. 90 168.91 171.07 14.80 14.92 £5.22 ±5.53 £5, 66 ±5.90 ± 6.32 ± 6.80 ± 7.71 8.668.877.75 47.23 £6.74 [ year'± (4.40) 4.66 ±5.00 +5.34 ±5.48 £5.74 16.20 +6.62 +7.54 18.49 8.61 17.63 7. 15 Frequencies of statures of American girls, in percentages. Height in centi meters. Ages, in years. 5. 611 6.545 7.513 8.501 9. 497 10. 495 11. 494 12. 490 13. 479 14. 471 15. 466 16. 473 17.466 Cases 1,260 3,618, 4, 913 5,289 5, 132 4,827 4,507 4, 187 3,411 2,537 1,656 1,171 790 Average height 105. 45 110. 32 116. 16 121. 21 126. 13 131. 24 136.58 142. 46 148.58 153. 41 156. 45 158.00 159. 11 corrected Mean variation at half year +4.785.01 ±5. 46 ±5.54 ±6.00 ±6.63 +7.41 ± 7.20 +6.57 +5.88 +5.65 From the preceding facts and considerations we conclude that the averages and variabilities of growing children must not be considered more than indices of the typical conditions characteristic of a certain age. In order to determine these accurately, the asymmetry of the distributions must be taken into account. This, however, can not be done, except by the expenditure of a vast amount of labor, until a sufficient series of observations, taken according to the individualizing method, is available. GROWTH AS DETERMINED BY THE TOTAL SERIES OF TORONTO CHILDREN. I give first of all a table of statures grouped in periods of quarter years. In this tabulation all those individuals who did not expressly state that their age was so |