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~~, but 0.3377.^{[2]} The
difference of this result from the regulation ~~.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:—

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