Читать книгу On Growth and Form - D'Arcy Wentworth Thompson - Страница 11

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The magnitudes and velocities which we are here dealing with are, of course, mean values derived from a certain number, sometimes a large number, of individual cases. But no statistical account of mean values is complete unless we also take account of the amount of variability among the individual cases from which the mean value is drawn. To do this throughout would lead us into detailed investigations which lie far beyond the scope of this elementary book; but we may very briefly illustrate the nature of the process, in connection with the phenomena of growth which we have just been studying.

It was in connection with these phenomena, in the case of man, that Quetelet first conceived the statistical study of variation, on lines which were afterwards expounded and developed by Galton, and which have grown, in the hands of Karl Pearson and others, into the modern science of Biometrics.

When Quetelet tells us, for instance, that the mean stature of the ten-year old boy is 1·273 metres, this implies, according to the law of error, or law of probabilities, that all the individual measurements of ten-year-old boys group themselves in an orderly way, that is to say according to a certain definite law, about this mean value of 1·273. When these individual measurements are grouped and plotted as a curve, so as to show the number of individual cases at each individual length, we obtain a char­ac­ter­is­tic curve of error or curve of frequency; and the “spread” of this curve is a measure of the amount of variability in this particular case. A certain math­e­mat­i­cal measure of this “spread,” as described in works upon statistics, is called the Index of Variability, or Standard Deviation, and is usually denominated by the letter σ. It is practically equivalent to a determination of the point upon the frequency curve where it changes its curvature on either side of the mean, and where, from being concave towards the middle line, it spreads out to be convex thereto. When we divide this {79} value by the mean, we get a figure which is independent of any particular units, and which is called the Coefficient of Variability. (It is usually multiplied by 100, to make it of a more convenient amount; and we may then define this coefficient, C, as = (σ ⁄ M) × 100.)

In regard to the growth of man, Pearson has determined this coefficient of variability as follows: in male new-born infants, the coefficient in regard to weight is 15·66, and in regard to stature, 6·50; in male adults, for weight 10·83, and for stature, 3·66. The amount of variability tends, therefore, to decrease with growth or age.

Similar determinations have been elaborated by Bowditch, by Boas and Wissler, and by other writers for intermediate ages, especially from about five years old to eighteen, so covering a great part of the whole period of growth in man108.

Coefficient of Variability (σ ⁄ M × 100) in Man, at various ages.
Age	5	6	7	8	9
Stature (Bowditch)	4·76	4·60	4·42	4·49	4·40
Stature (Boas and Wissler)	4·15	4·14	4·22	4·37	4·33
Weight (Bowditch)	11·56	10·28	11·08	9·92	11·04
Age	10	11	12	13	14
Stature (Bowditch)	4·55	4·70	4·90	5·47	5·79
Stature (Boas and Wissler)	4·36	4·54	4·73	5·16	5·57
Weight (Bowditch)	11·60	11·76	13·72	13·60	16·80
Age	15	16	17	18
Stature (Bowditch)	5·57	4·50	4·55	3·69
Stature (Boas and Wissler)	5·50	4·69	4·27	3·94
Weight (Bowditch)	15·32	13·28	12·96	10·40

The result is very curious indeed. We see, from Fig. 11, that the curve of variability is very similar to what we have called the acceleration-curve (Fig. 4): that is to say, it descends when the rate of growth diminishes, and rises very markedly again when, in late boyhood, the rate of growth is temporarily accelerated. We {80} see, in short, that the amount of variability in stature or in weight is a function of the rate of growth in these magnitudes, though we are not yet in a position to equate the terms precisely, one with another.

Fig. 11. Coefficients of variability of stature in Man (). from Boas and Wissler’s data.

If we take not merely the variability of stature or weight at a given age, but the variability of the actual successive increments in each yearly period, we see that this latter coefficient of variability tends to increase steadily, and more and more rapidly, within the limits of age for which we have information; and this phenomenon is, in the main, easy of explanation. For a great part of the difference, in regard to rate of growth, between one individual and another is a difference of phase—a difference in the epochs of acceleration and retardation, and finally in the epoch when growth comes to an end. And it follows that the variability of rate will be more and more marked, as we approach and reach the period when some individuals still continue, and others have already ceased, to grow. In the following epitomised table, {81} I have taken Boas’s determinations of variability (σ) (op. cit. p. 1548), converted them into the cor­re­spon­ding coefficients of variability (σ ⁄ M × 100), and then smoothed the resulting numbers.

Coefficients of Variability in Annual Increment of Stature.
Age	7	8	9	10	11	12	13	14	15
Boys	17·3	15·8	18·6	19·1	21·0	24·7	29·0	36·2	46·1
Girls	17·1	17·8	19·2	22·7	25·9	29·3	37·0	44·8	—

The greater variability of annual increment in the girls, as compared with the boys, is very marked, and is easily explained by the more rapid rate at which the girls run through the several phases of the phenomenon.

Just as there is a marked difference in “phase” between the growth-curves of the two sexes, that is to say a difference in the periods when growth is rapid or the reverse, so also, within each sex, will there be room for similar, but individual phase-differences. Thus we may have children of accelerated development, who at a given epoch after birth are both rapidly growing and already “big for their age”; and others of retarded development who are comparatively small and have not reached the period of acceleration which, in greater or less degree, will come to them in turn. In other words, there must under such circumstances be a strong positive “coefficient of correlation” between stature and rate of growth, and also between the rate of growth in one year and the next. But it does not by any means follow that a child who is precociously big will continue to grow rapidly, and become a man or woman of exceptional stature. On the contrary, when in the case of the precocious or “accelerated” children growth has begun to slow down, the backward ones may still be growing rapidly, and so making up (more or less completely) to the others. In other words, the period of high positive correlation between stature and increment will tend to be followed by one of negative correlation. This interesting and important point, due to Boas and Wissler109, is confirmed by the following table:—

Correlation of Stature and Increment in Boys and Girls. (From Boas and Wissler.)
Age		6	7	8	9	10	11	12	13	14	15
Stature	(B)	112·7	115·5	123·2	127·4	133·2	136·8	142·7	147·3	155·9	162·2
	(G)	111·4	117·7	121·4	127·9	131·8	136·7	144·6	149·7	153·8	157·2
Increment	(B)	5·7	5·3	4·9	5·1	5·0	4·7	5·9	7·5	6·2	5·2
	(G)	5·9	5·5	5·5	5·9	6·2	7·2	6·5	5·4	3·3	1·7
Correlation	(B)	·25	·11	·08	·25	·18	·18	·48	·29	− ·42	− ·44
	(G)	·44	·14	·24	·47	·18	− ·18	− ·42	− ·39	− ·63	·11

{82}

A minor, but very curious point brought out by the same investigators is that, if instead of stature we deal with height in the sitting posture (or, practically speaking, with length of trunk or back), then the correlations between this height and its annual increment are throughout negative. In other words, there would seem to be a general tendency for the long trunks to grow slowly throughout the whole period under in­ves­ti­ga­tion. It is a well-known anatomical fact that tallness is in the main due not to length of body but to length of limb.

The whole phenomenon of variability in regard to magnitude and to rate of increment is in the highest degree suggestive: inasmuch as it helps further to remind and to impress upon us that specific rate of growth is the real physiological factor which we want to get at, of which specific magnitude, dimensions and form, and all the variations of these, are merely the concrete and visible resultant. But the problems of variability, though they are intimately related to the general problem of growth, carry us very soon beyond our present limitations.