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exemplified by A. L. Bowley's grouping of the wages paid to different classes.^[1]

119. The division of concrete errors which has been proposed is not to be confounded with another twofold classification, namely, A Variant Classification. observations which stand for a real objective thing, and such statistics as are not thus representative of something outside themselves, groups of which the mean is called “subjective.” This division would be neither clear nor useful. On the one hand so-called real means are often only approximately equal to objective quantities. Thus the proportional frequency with which one face of a die—the six suppose—turns up is only approximately given by the objective fact that the six is one face of a nearly perfect cube. For a set of dice with which Weldon experimented, the average frequency of a throw, presenting either five or six points, proved to be not .3$.3{\dot {3}}$, but 0.3377.^[2] The difference of this result from the regulation .3$.3{\dot {3}}$ is as unpredictable from objective data, prior to experiment, as any of the means called subjective or fictitious. So the mean of errors of observation often differs from the thing observed by a so-called “constant error.” So shots may be constantly deflected from the bull's-eye by a steady wind or “drift.”

120. On the other hand, statistics, not purporting to represent a real object, have more or less close relations to magnitudes which cannot be described as fictitious. Where the items averaged are ratios, e.g. the proportion of births or deaths to the total population in several districts or other sections, it sometimes happens that the distribution of the ratios exactly corresponds to that which is obtained in the simplest games of chance—“combinational” distribution in the phrase of Lexis.^[3] There is unmistakably suggested a sortition of the simplest type, with a real ascertainable relation between the number of “favourable cases” and the total number of cases. The most remarkable example of this property is presented by the proportion of male to female (or to total) births. Some other instances are given by Lexis^[4] and Westergaard.^[5] A similar correspondence between the actual and the “combinational” distribution has been found by Bortkevitch^[6] in the case of very small probabilities (in which case the law of error is no longer “normal”). And it is likely that some ratios—such as general death-rates—not presenting combinational distribution, might be broken up into subdivisions—such as death-rates for different occupations or age periods—each distributed in that simple fashion.

121. Another sort of averages which it is difficult to class as subjective rather than objective occurs in some social statistics, under the designation of index-numbers. The percentage which represents the change in the value of money between two epochs is seldom regarded as the mere average change in the price of several articles taken at random, but rather as the measure of something, e.g. the variation in the price of a given amount of commodities, or of a unit of commodity.^[7] So something substantive appears to be designated by the volume of trade, or that of the consumption of the working classes, of which the growth is measured by appropriate index-numbers,^[8] the former due to Bourne and Sir Robert Giffen,^[9] the latter to George Wood.^[10]

122. But apart from these peculiarities, any set of statistics may be related to a certain quaesitum, very much as measurements are related to the object measured. That quaesitum is the limiting or ultimate mean to which the series of statistics, if indefinitely prolonged, would converge, the mean of the complete group; this conception of a limit applying to any frequency-constant, to “c,” for instance, as well as “a” in the case of the normal curve.^[11] The given statistics may be treated as samples from which to reason up to the true constant by that principle of the calculus which determines the comparative probability of different causes from which an observed event may have emanated.^[12]

123. Thus it appears that there is a characteristic more essential to the statistician than the existence of an objective quaesitum, namely, the use of that method which is primarily, but not exclusively, proper to that sort of quaesitum—inverse probability.^[13] Without that delicate instrument the doctrine of error can seldom be fully utilized; but some of its uses may be indicated before the introduction of technical difficulties.

124. Having established the prevalence of the law of error,^[14] we go on to its applications. The mere presumption that wherever three or Applications of the Normal Law. four independent causes co-operate, the law of error tends to be set up, has a certain speculative interest.^[15] The assumption of the law as a hypothesis is legitimate. When the presumption is confirmed by specific experience this knowledge is apt to be turned to account. It is usefully applied to the practice of gunnery,^[16] to determine the proportion of shots which under assigned conditions may be expected to hit a zone of given size. The expenditure of ammunition required to hit an object can thence be inferred. Also the comparison between practice under different conditions is facilitated. In many kinds of examination it is found that the total marks given to different candidates for answers to the same set of questions range approximately in conformity with the law of error. It is understood that the civil service commissioners have founded on this fact some practical directions to examiners. Apart from such direct applications, it is a useful addition to our knowledge of a class that the measurable attributes of its members range in conformity with this general law. Something is added to the truth that “the days of a man are threescore and ten,” if we may regard that epoch, or more exactly for England, 72, as “Nature's aim, the length of life for which she builds a man, the dispersion on each side of this point being . . . nearly normal.”^[17] So Herschel says: “An [a mere] average gives us no assurance that the future will be like the past. A [normal] mean may be reckoned on with the most complete confidence.”^[18] The existence of independent causes,^[19] inferred from the fulfilment of the normal law, may be some guarantee of stability. In natural history especially have the conceptions supplied by the law of error been fruitful. Investigators are already on the track of this inquiry: if those members of a species whose size or other measurable attributes are above (or below) the average are preferred—by “natural” or some other kind of selection—as parents, how will the law of frequency as regards that attribute be modified in the next generation?

125. A particularly perfect application of the normal law of error in more than one dimension is afforded by the movements of Normal Distribution of Molecular Velocities. the molecules in a homogeneous gas. A general idea of the rôle played by probabilities in the explanation of these movements may be obtained without entering into the more complicated and controverted parts of the subject, without going beyond the initial very abstract supposition of perfectly elastic equal spheres. For convenience of enunciation we may confine ourselves to two dimensions. Let us imagine, then, an enormous billiard-table with perfectly elastic cushions and a frictionless cloth on which millions of perfectly elastic balls rush hither and thither at random—colliding with each other—a homogeneous chaos, with that sort of uniformity in the midst of diversity which is characteristic of probabilities. Upon this hypothesis, if we fix attention on any n balls taken at random—they need not be, according to some they ought not to be, contiguous—if n is very large, the average properties will be approximately the same as those of the total mixture. In particular the average energy of the n balls may be equated to the average energy of the total number of balls, say T/N, if T is the total energy and N the total number of the balls. Now if we watch any one of the n specimen balls long enough for it to undergo a great number of collisions, we observe that either of its velocity-components, say that in the direction of x, viz. u, receives accessions from an immense number of independent causes in random fashion. We may presume, therefore, that these will be distributed (among the n balls) according to the law of error. The law will not be of the type which was first supposed, where the “spread” continually increases as the number of the elements is increased.^[20] Nor will it be of the type which was afterwards mentioned^[21] where the spread diminishes as the number of the elements is increased. The linear function by which the elements are aggregated is here of an intermediate type; such that the mean square of deviation corresponding to the velocity remains constant. The method of composition might be illustrated by the process of taking r digits at random from mathematical tables adding the differences between each digit and 4.5 the mean value of digits, and dividing the sum by √r. Here are some figures obtained by taking at random batches of sixteen digits from the expansion of π, subtracting 16 × 4.5 from the sum of each batch, and dividing the remainder by √16:—

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1. Wages in the United Kingdom in the Nineteenth Century; and art. “Wages” in the Ency. Brit., 10th ed., vol. xxxiii.
2. Phil. Mag. (1900), p. 168.
3. Cf. Journ. Stat. Soc., Jubilee No., p. 192.
4. Massenerscheinungen.
5. Grundzüge der Statistik. Cf. Bowley, Elements of Statistics, p. 302.
6. Das Gesetz der kleinen Zahlen.
7. See for other definitions Report of the British Association (1889), pp. 136 and 161, and compare Walsh's exhaustive Measurement of General Exchange-Value.
8. Cf. Bowley, Elements of Statistics, ch. ix.
9. Journ. Stat. Soc. (1874 and later). Parly. Papers [C. 2247] and [C. 3079].
10. “Working-Class Progress since 1860,” Journ. Stat. Soc. (1899), p. 639.
11. On this conception compare Venn, Logic of Chance, chs. iii. and iv., and Sheppard, Proc. Lond. Math. Soc., p. 363 seq.
12. Laplace's 6th principle, Théorie analytique, intro. x.
13. See above, pars. 13 and 14.
14. Cf. above, par. 102.
15. Cf. Galton's enthusiasm, Natural Inheritance, p. 66.
16. A lucid statement of the methods and results of probabilities applied to gunnery is given in the Official Text-book of Gunnery (1902).
17. Venn, Journ. Stat. Soc. (1891), p. 443.
18. Ed. Rev. (1850), xcii. 23.
19. Cf. Galton, Phil. Mag. (1875), xlix. 44.
20. Above, par. 112.
21. Ibid.