CHAPTER II. STATISTICS OF GROWTH. By FRANZ Boas and CLARK WISSLER. During the last thirty years a vast amount of anthropometric material relating to physical and mental growth has accumulated. Unfortunately, up to the present time there has been little agreement in regard to the methods of collecting and of treating such material. The following pages contain a discussion of some of the more important problems which have a bearing upon the methods of collecting and reducing observations on growth. In a paper published in the Report of the Commissioner of Education for 1896-97 (Chap. XXXIV, pp. 1541-1599), F. Boas discussed some of the data on the growth of American children that were available at the time, and, in a theoretical introduction, le treated some of the problems that confront us in the discussion of material of this character. In that paper the effect of retardation and acceleration of growth upon the distribution of measurements was somewhat fully discussed. It was shown that the assumption of a symmetrical distribution of variations in period-i. e., of accelerations and retardationsfollowing the laws of chance gives an adequate explanation of the characteristics of the observed curves of growth. (Ibid., p. 15.2.) The verification of this theory is of great importance, because, if it is correct, it follows that the developmental stage of a child at a certain period depends primarily on phenomena of retardation and acceleration, which influence the whole body at the same time, so that all measurements should show a tendency to vary in the same direction ; either all of them would tend to lag behind the normal average or all would be in advance of it. If this is so, tlien the assumption that is so often made, that during a period of energetic physical growth there is a rest of mental development and vice versa, would lose much of its probability. Assuming that during the period of growth deviations from the normal in the values of a certain measurement are partly due to variations in period, partly to hereditary and other causes of a permanent character, it would follow that other measurements of the same individual would be affected by the same groups of causes, particularly by the same variation in period. The more rapid the rate of growth, the greater will be the effect of variation in period upon all the different measurements. Retardation of developmental period, for instance, would considerably depress the values of all the measurements of the individual. Consequently, the correlations between different measurements ought to be closer during periods of rapid growth than at other periods. We hare investigated this problem by means of statistics collected in Worcester, Mass., and Toronto, Ontario, and by Peckham's measurements collected in Milwaukee, Wis., which the collector had the great kindness to place at our disposal. Unless stated otherwise, the theory and plan of investigation were worked out by F. Boas, while the calculations were made by Clark Wissler. The theory of correlations during the period of growth may be formulated as follows: We assume that the value of a certain measurement depends partly upon variation in period, partly upon other causes, and that the amount of growth is proportional to time. We call t the deviation in time and dr the amount of growth during the period 7, while the deviations due to other causes may be called r. The total deviation, 5, from the normal for any individual of a given age will then le For another measurement of the same individual the corresponding values may be called d., y, and n. Then m=dot+y. We will designate averages by brackets. [*] =0, [7"]+[...] [5]=d,1,[T"] +- [cy] H?=,26,2 + 6,2 N=0,6,2+6,7 (1) R4,11,=d,(1,6,2+r6,6. It appears from this that, if r6,6, remains fairly constant, R}/4, will be the greater, the greater d,(1,67. It does not seem probable that r6,6, should undergo very great changes during the later periods of growth. If, therefore, it can be shown that RM111, increases with the rapidity of growth--that is, with the value of d.d.-our theory would seem to be corroborated. In order to investigate this question, we have calculated the correlations between stature, weight, height sitting, length of head, and width of head from the measurements collected in Worcester, Mass. Unfortunately, the series is not long enough to give quite satisfactory results. We give first the general series of averages and variabilities for various measurements. TABLE Ia.-drerages and variabilities of measurements of boys, Worcester, Mass. Jeasurements [Figures printed in italics indicate the number of individuals measured. are in millimeters and in pounds aroirdupois. ] + sitting of head. of head. of head. of head of head. of head. of head. TABLE Ib.-Averages and variabilities of measurements of girls, Worcester, Jass. [Figures printed in italies indicate the number of individuals measured. Measurements Width of Length of Width of head. 1 head. forearm. hand. 6. 1 1,120 : 47 616_27 43.84.8 173.1 + 4.6. 138.0 3.5 304+ 18 58 +5 | 1,171 +56 63929 47.96.1 171.7 - 6.0 139. 1.4.1 315 + 24 1.28 101 8. 1,221 58 670 +31 51.9.6.6 175.0 - 5.9 110.3. 4.3 39818 60 +3 14,6 10.3 10.3 9. | 1,270 +34 680) 28 58.0 +8.4 176.3 6.3 140.2: 4.8 837 + 15 62 +3 147 119 65 +3 215 170 70.0 10.4 178. 2. 3. 4 142.1 1.9 370 + 23 10. 12 1, 443+72 78.31 81.0.13.1 180.0) +5.4 113.2- 4.7 399 19 70 + 4 1 788 + 42 89.7 15.3 181.7.-6.8 141.0 - 4.9 401 + 24 72 + 4 12 14.. 1,519 +12 815 +40 100.6.16.8 182.5 6.4 144.0.5.3 413.20 743 1.11 15. 1,569 +53 835 32 106.213.9 184.3 +5.2 105.4 4.5 427 +20 74+4 16. 1,572+60 840 +29 108.7 +12.1 183.7 +5.7 144.6 + 4.9 422 +26 101 17 19 17. 1, 591 +52 853 +31 114.6+13.6 184.8:5.5 145.2.4.3 In Table II are contained the coefficients of correlation for various ages and the number of observations from which the coefficients have been calculated. Table III contains the values for the product Rill, and the approximate values of bilo. Table IIa.-(orrelations for boys, Worcester, Jase. (n=number of cases; r =coefficient of correlation.] Stature Weight Stature Stature Height Height Weight Weight Length Stature and and and and sitting 'sitting and and and and and and height length width 1 width length length width width sitting Age. weight.' height 6 9 10 0.86 101 0.78 110' 0.701 1101 0.30 105 0.20. 10.1 0.34 99 0.24 970. 16 9 0.40 90.30 105 27 147 .81 20.3, .38 14.01 29 143 .18 143 .14 121 .30 110 4611 .25 121 41 107 12 23 1611 37 138 29 1.38 29 167 .39 1:34 in 157 .31 129 39 1.11 29 171 28 12: 26 1.1 12 151 .51 113 .30 177 117 19 156 45 149 123 23. 120 31 1301 .90 2!! 15 168 34 154 87 2013 03 14 1.34 39 131, 14 152 .48 1.20 96 17 02 . 14 .51 28 18 48 19 49 16 12 15 16 17 TABLE IIb.—Correlations for girls, Worcester, Afass. [n=number of cases; r=coefficient of correlation.] Stature and weight. Stature Weight Stature Stature Height Height Weight Weight Length and and and and sitting sitting and and and and length width of length width of width of of head. head. head. head. Age. 97 0.21 6 7 8 9 10 11 12 13 14 15 16 17 0.68 97 0.67 99 0.59 69 128 60 149 79 147 81 152 .76 147 .82 170 841 174 77 170 88 199 83 202 .84 199 74 901 81 208 85 201 .70 183 80 190 72 183 74 149 85 151 75 176 57 135 .81 138 .61 134 65 100 71 101 71 100 .59 64 69 . 47 64 84 0.34 79 0.23 77 0.44 77 0.30 84 .37 101 171 101 40 87 87 87 35 87 .09 101 32 104 12 105 .31 8.5 06 85 89 8.2 16 106 .38 118 .17 118 96 10 91 30 118 .36 160 .18 100 41 119 .28 119 .28 115 .31 115 233 160 .37 152 12 153 .14 123 41 120 19 1:20 29 130 38 145 . 44 145 .33 121 28 191 36 114 3.2 114 24 16.5 40 14 24 142 522 119 33 119 .51 113 43 113 3 14.2 .37 110 .42 115 40 102 .42 102 49 .39 115 . 14 109 09 109 14 98 17 98 .21 98 .26 109 .27 85 15 85 85 23 8.5 84 37 84 43 69 69 .06 69 .06 69 . 49 64 64 29 The preceding tables show at once that during the period of growth the correlation between the various measurements is greater than in the adult. This is particularly true in regard to those measurements which show the strongest increases, i. e., stature, weight, and height sitting. Table III shows that the greatest values of Rily My are found a little later than the greatest values of the products dél,, the difference in period being about one year in boys and two years in girls. This may be due to the increase of the values 60, 61, and 6, with increasing age (see p. 38). In order to obtain a better insight into the characteristics of the values of r during the period of growth, we have taken the averages of its values for three successive years and have given these averages to the middle years. The results are contained in the following table and are graphically represented in fig. 1 (p. 47). We have not plotted the variations of the correlations of head measurements, because these changes are too small and too uncertain. All the others show very clearly a rise of correlation during the period of rapid growth, followed by a decline during the period of decreasing growth. In accordance with the earlier periods of rapid growth and of decreasing growth the correlations of girls reach their maxima a year or two before those of boys, and it appears that the absolute maxima of correlations of boys are greater, corresponding to their absolutely greater rapidity of growth during the prepubertal period. As a consequence the values of the coefficients of correlation for girls are always a trifle lower than those of boys, except about the eighth, ninth, and tenth years. The correlations of the head may be considered as a whole, and we have taken the averages for all the years. TABLE V.-Correlations of head measurements with stature, height sitting, and uceight. It seems very doubtful that the slight excess in the correlations for boys is significant. The striking excess of the correlations between measurements indicative of bulk of body and length of head over those between measurements of bulk of body and width of head may be due in part to the variations in size of the frontal sinuses and of the occipital protuberance, which depend more closely upon the development of the skeleton than the transversal diameter. However, it might seem that, in young children at least, other causes must be looked for to account for the considerable difference between the coefficients. It is quite probable that it is mainly an expression of the closer correlation of antero-posterior measurements among themselves, the length of head being an axial measurement, the same as stature and height sitting. |