The maximum frequency is, as it ought to be, at the point *x* = 6*i*,
*y* = 6*i*′. The density is particularly great along a line through that
point, making 45° with the axis of *x*; particularly small in the
complementary direction. This also is as it ought to be. For if
the centre is made the origin by substituting *x* for (*x* − *a*) and *y* for
(*y* − *b*), and then new co-ordinates X and Y are taken, making an
angle θ with *x* and *y* respectively, the curve which is traced on the
plane of *z*X by its intersection with the surface is of the form

*z* = J exp −X^{2}[*k* sin^{2} θ − 2*l* cos θ sin θ + *m* cos^{2} θ]/2(*km* − *l*^{2}),

a probability-curve which will be more or less spread out according
as the factor *k* sin^{2} θ − 2*l* cos θ sin θ + *m* cos^{2} θ is less or greater.
Now this expression has a minimum or maximum when
(*k* − *m*) sin θ−2*l* cos 2θ = 0; a minimum when (*k* − *m*) cos 2θ+2 *l*sin 2θ is
positive, and a maximum when that criterion is negative; that
is, in the present case, where *k* = *m*, a minimum when θ = ¼π and a
maximum when θ = ¾π.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |

12 | |||||||||||||

11 | 1 | 1 | 5 | 1 | 1 | ||||||||

10 | 2 | 6 | 28 | 27 | 19 | 2 | |||||||

9 | 1 | 2 | 11 | 43 | 76 | 57 | 54 | 15 | 4 | ||||

8 | 6 | 18 | 49 | 116 | 138 | 118 | 59 | 25 | 5 | ||||

7 | 12 | 47 | 109 | 208 | 213 | 118 | 71 | 23 | 1 | ||||

6 | 9 | 29 | 77 | 199 | 244 | 198 | 121 | 32 | 3 | ||||

5 | 3 | 12 | 51 | 119 | 181 | 200 | 129 | 69 | 18 | 3 | |||

4 | 2 | 16 | 55 | 100 | 117 | 91 | 46 | 19 | 3 | ||||

3 | 2 | 14 | 28 | 53 | 43 | 34 | 17 | 1 | |||||

2 | 7 | 12 | 13 | 18 | 4 | 1 | 1 | ||||||

1 | 2 | 4 | 1 | 2 | 1 | ||||||||

0 |

116. *Characteristics of the Law of Error*^{[1]}—As may be presumed
from the examples just given, in order that there should be some
approximation to the normal law the number of elements need not
be very great. A very tolerable imitation of the probability-curve
has been obtained by superposing *three* elements, each obeying a
law of frequency quite different from the normal one,^{[2]} namely, that
simple law according to which one value of a variable occurs as
frequently as another between the limits within which the variation
is confined (*y* = 1/2*a*, between limits *x* = +*a*, *x* = −*a*). If the
component elements obey unsymmetrical laws of frequency, the
compound will indeed be to some extent unsymmetrical, unlike the
“normal” probability-curve. But, as the number of the elements is
increased, the portion of the compound curve in the neighbourhood
of its centre of gravity tends to be rounded off into the normal shape.
The portion of the compound curve which is sensibly identical with
a curve of the “normal” family becomes greater the greater the
number of independent elements; *caeteris paribus*, and granted certain
conditions as to the equality and the range of the elements. It
will readily be granted that if one component predominates, it
may unduly impress its own character on the compound. But it
should be pointed out that the characteristic with which we are
now concerned is not average magnitude, but deviation from the
average. The component elements may be very unequal in their
contributions to the average magnitude of the compound without
prejudice to its “normal” character, provided that the fluctuation
of all or many of the elements is of one and the same order. The
proof of the law requires that the contribution made by each element
to the mean square of deviation for the compound, *k*, should be
small, capable of being treated as differential with respect to *k*.
It is not necessary that all these small quantities should be of the
same order, but only that they should admit of being rearranged,
by massing together those of a smaller order, as a numerous set of
independent elements in which no two or three stand out as *sui*
*generis* in respect of the magnitude of their fluctuation. For example,
if one element consist of the number of points on a domino (the sum
of two digits taken at random), and other elements, each of either
1 or 0 according as heads or tails turn up when a coin is cast, the
first element, having a mean square of deviation 16.5, will not be
of the same order as the others, each having 0.25 for its mean square
of deviation. But *sixty-six* of the latter taken together would constitute
an independent element of the same order as the first one;
and accordingly if there are several times sixty-six elements of the
latter sort, along with one or two of the former sort, the conditions
for the generation of the normal distribution will be satisfied. These
propositions would evidently be unaffected by altering the average
magnitude, without altering the deviation from the average, for any
element, that is, by adding a greater or less *fixed* magnitude to each
element. The propositions are adapted to the case in which the
elements fluctuate according to a law of frequency other than the
normal. For if they are already normal, the aforesaid conditions
are unnecessary. The normal law will be obeyed, by the sum of
elements which each obey it, even though they are not numerous and
not independent and not of the same order in respect of the extent
of fluctuation. A similar distinction is to be drawn with respect
to some further conditions which the reasoning requires. A limitation
as to the range of the elements is not necessary when they are
already normal, or even have a certain affinity to the normal curve.
Very large values of the element are not excluded, provided they are
sufficiently rare. What has been said of curves with special
reference to one dimension is of course to be extended to the case
of surfaces and many dimensions. In all cases the theorem that
under the conditions stated the normal law of error will be generated
is to be distinguished from the hypothesis that the conditions are
fairly well fulfilled in ordinary experience.

117. Having deduced the genesis of the law of error from ideal (B) Verification of the Normal Law. conditions such as are attributed to perfectly fair games of chance, we have next to inquire how far these conditions are realized and the law fulfilled in common experience.

118. Among important concrete cases errors of observation
occupy a leading place. The theory is brought to bear on this case
Errors proper.
by the hypothesis that an error is the algebraic sum of
numerous elements, each varying according to a law
of frequency special to itself. This hypothesis involves
two assumptions: (1) that an error is dependent on numerous
independent causes; (2) that the function expressing that dependence
can be treated as a linear function, by expanding in terms of ascending
powers (of the elements) according to Taylor's theorem and
neglecting higher powers, or otherwise. The first assumption seems,
in Dr Glaisher's words, “most natural and true. In any observation
where great care is taken, so that no large error can occur, we can
see that its accuracy is influenced by a great number of circumstances
which ultimately depend on independent causes: the state of the
observer's eye and his physiological condition in general, the state
of the atmosphere, of the different arts of the instrument, &c.,
evidently depend on a great number of causes, while each contributes
to the actual error.”^{[3]} The second assumption seems to be frequently
realized in nature. But the assumption is not always safe. For
example, where the velocities of molecules are distributed according
to the normal law of error, with zero as centre, the *energies* must be
distributed according to a quite different law. This rationale is
applicable not only to the fallible perceptions of the senses, but also
to impressions into which a large ingredient of inference enters,
such as estimates of a man's height or weight from his appearance,^{[4]}
and even higher acts of judgments.^{[5]} Aiming at an object is an act
similar to measuring an object, misses are produced by much the
same variety of causes as mistakes; and, accordingly, it is found
that shots aimed at the same bull's-eye are apt to be distributed
according to the normal law, whether in two dimensions on a target
or according to their horizontal deviations, as exhibited below
(par. 156). A residual class comprises miscellaneous statistics,
physical as well as social, in which the normal law of error makes
Miscellaneous Statistics.
its appearance, presumably in consequence of the action
of numerous independent influences. Well-known
instances are afforded by human heights and other
bodily measurements, as tabulated by Quetelet^{[6]} and
others.^{[7]} Professor Pearson has found that “the normal curve
suffices to describe within the limits of random sampling the distribution
of the chief characters in man.”^{[8]} The tendency of social
phenomena to conform to the normal law of frequency is well

- ↑ Experiments
*in pari materia*performed by A. D. Darbishire afford additional illustrations. See “Some Tables for illustrating Statistical Correlation,”*Mem.*and*Proc. Man. Lit., and Phil. Soc.*, vol. li. pt. iii. - ↑
*Journ. Stat. Soc.*(March 1900), p. 73, referring to Burton,*Phil. Mag.*(1883), xvi. 301. - ↑
*Memoirs of Astronomical Society*(1878), p. 105. - ↑
*Journ. Stat. Soc.*(1890), p. 462 seq. - ↑
*E.g.*the marking of the same work by different examiners. Ibid. - ↑
*Lettres sur la théorie des probabilités*and*Physique sociale*. - ↑
*E.g.*the measurements of Italian recruits, adduced in the*Atlante**statistico*, published under the direction of the Ministero de Agricultura (Rome, 1882); and Weldon's measurements of crabs,*Proc.**Roy. Soc.*liv. 321; discussed by Pearson in the*Trans. Roy. Soc.*(1894), vol. clxxxv. A. - ↑
*Biometrika*, iii. 395. Cf. ibid. p. 141.