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 grow th.

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, he 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. 1552.)

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, then 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 have 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 the deviation in time and dr the amount of growth during the period r, while the deviations due to other causes may be called a. The total deviation, §, 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 d2, y, and 7. Then

(1)

If we call [2],["2], [§»],[t2],[x2],[y2],[xy], respectively,

μ12, M2, Rμ12, 672, 612, 6,2,ró12, we have

It appears from this that, if ró16, remains fairly constant, Ru, will be the greater, the greater d1d ̧67. It does not seem probable that ró,6, should undergo very great changes during the later periods of growth. If, therefore, it can be shown that Ru12 increases with the rapidity of growth-that is, with the value of d1d-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 Iа.-Averages and variabilities of measurements of boys, Worcester, Mass.

[Figures printed in italics indicate the number of individuals measured. are in millimeters and in pounds avoirdupois.]

Measurements

TABLE Ib.-Averages and variabilities of measurements of girls, Worcester, Mass. [Figures printed in italics indicate the number of individuals measured. Measurements are in millimeters and in pounds avoirdupois. ]

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 R12 and the approximate values of d¿d.

TABLE IIb.-Correlations for girls, Worcester, Mass.

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 Ru, u2 are found a little later than the greatest values of the products did, 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 6t, 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).

TABLE IV.-Correlations.

0.78 0.81 0.690.37 0.220.39 0.280.44 0.28 0.21 0.73 0.74 0.63 0.30 0.21 0.31 0.22 0.37 0.39 0.23 .73.81 67 40 23 40 29 42.28.20 .76 .79 .68 .36 .15 .35 .13 .47 .33 .23

39 .33.20.80.82.71 35 16.35 .15 .39 .32 .28 41 .33.19.83 .83 .79.37 16.35 17 40 25.28 43.28.20 .81.84.82 37 25 35.23.35 27.26 40 28.19 77 82.80.38 27 39 25 43.31.28 30.26.73 83.77 .38 .37 42 34 46.41.32 35.21.67 82.69 30 38.32

37 .36 .36

37 31 .33

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

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.