Variability and Correlation of Growth.
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 man[108].
| 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 (