Section I.—*The Law of Error*.

98. (1) *The Normal Law of Error*.—The simplest and best recognized
statement of the law of error, often called the “normal
law,” is the equation

,

more conveniently written , where *x*
is the magnitude of an observation or “statistic,” *z* is the proportional
frequency of observations measuring *x*, *a* is the arithmetic
mean of the group (supposed indefinitely^{[1]} multiplied) of similar
statistics: *c* is a constant sometimes called the “modulus”^{[2]}
proper to the group; and the equation signifies that if any large
number N of such a group is taken at random, the number of
observations between *x* and *x* + ∆*x* is (approximately) equal to
the right-hand side of the equation multiplied by N∆*x*. A
graphical representation of the corresponding curve—sometimes
called the “probability-curve”—is here given (fig. 10), showing
the general shape of the curve, and how its dimensions vary with
the magnitude of the modulus *c*. The area being constant (viz.
unity), the curve is furled up when *c* is small, spread out when *c*
is large. There is added a table of integrals, corresponding to
areas subtended by the curve; in a form suited for calculations
of probability, the variable, τ, being the length of the abscissa
referred to (divided by) the modulus.^{[3]} It may be noted that the
points of inflexion in the figure are each at a distance from the
origin of 1/√2 modulus, a distance equal to the square foot of
the mean square of error—often called the “standard deviation.”
Another notable value of the abscissa is that which divides the area
on either side of the origin into two equal parts; commonly called
the “probable error.” The value of τ which corresponds to this
point is 0.4769. . . .

Fig. 10.

99. An a priori proof of this law was given by Herschel^{[4]} as
A priori proof.
follows: “The probability of an error depends solely on its magnitude
and not on its direction;” positive and negative
errors are equally probable. “Suppose a ball dropped
from a given height with the intention that it should
fall on a given mark,” errors in all directions are equally probable,
and errors in perpendicular directions are independent. Accordingly
the required law, “*which must necessarily be general and*
*apply alike in all cases, since the causes of error are supposed alike*
*unknown*,”^{[5]} is for one dimension of the form φ(*x*^{2}), for two
dimensions
φ(*x*^{2} + *y*^{2}); and φ(*x*^{2} + *y*^{2}) ≡ φ(*x*^{2}) × φ(*y*^{2}); a functional
equation of which the solution is the function above written.
A reason which satisfied Herschel is entitled to attention, especially
if it is endorsed by Thomson and Tait.^{[6]} But it must be confessed
that the claim to universality is not, without some strain of
interpretation,^{[7]} to be reconciled with common experience.

*Table of the Values of the Integral* I = .

τ | I |

0.00 | 0.00000 |

.01 | .01128 |

.02 | .02256 |

.03 | .03384 |

.04 | .04511 |

.05 | .05637 |

.06 | .06762 |

.07 | .07886 |

.08 | .09008 |

.09 | .10128 |

.1 | .11246 |

.2 | .22270 |

.3 | .32863 |

.4 | .42839 |

.5 | .52050 |

.6 | .60386 |

.7 | .67780 |

.8 | .74210 |

.9 | .79691 |

1.0 | .84270 |

1.1 | .88020 |

1.2 | .91031 |

1.3 | .93401 |

1.4 | .95229 |

1.5 | .96611 |

1.6 | .97635 |

1.7 | .98379 |

1.8 | .98909 |

1.9 | .99279 |

2.0 | .99532 |

2.1 | .99702 |

2.2 | .99814 |

2.3 | .99886 |

2.4 | .99931 |

2.5 | .99959 |

2.6 | .99976 |

2.7 | .99986 |

2.8 | .99992 |

2.9 | .99996 |

3.0 | .99998 |

∞ | 1.00000 |

100. There is, however, one class of phenomena to which Herschel's
reasoning applies without reservation. In a “molecular chaos,” such
as the received kinetic theory of gases postulates, if a molecule be
placed at rest at a given point and the distance which it travels
from that point in a given time, driven hither and thither by colliding
molecules, is regarded as an “error,” it may be presumed that
errors in all directions are equally probable and errors in perpendicular
directions are independent. It is remarkable that a similar
presumption with respect to the *velocities* of the molecules was
employed by Clerk Maxwell, in his first approach to the theory
of molecular motion, to establish the law of error in that region.

101. *The Laplace-Quetelet Hypothesis*.—That presumption has,
indeed, not received general assent; and the law of error appears to
be better rested on a proof which was originated by Laplace. According
to this view, the normal law of error is a first approximation
to the frequency with which different values are apt to be assumed
by a variable magnitude dependent on a great number of independent
variables, each of which assumes different values in random
fashion over a limited range, according to a law of error, not in
general the law, nor in general the same for each variable. The
normal law prevails in nature because it often happens—in the
world of atoms, in organic and in social life—that things depend
on a number of independent agencies. Laplace, indeed appears
to have applied the mathematical principle on which this explanation
depends only to examples (of the law of error) artificially
generated by the process of taking averages. The merit of accounting
for the prevalence of the law *in rerum natura* belongs rather
to Quetelet. He, however, employed too simple a formula^{[8]} for
the action of the causes. The hypothesis seems first to have been
stated in all its generality both of mathematical theory and statistical
exemplification by Glaisher.^{[9]}

102. The validity of the explanation may best be tested by first
(A) deducing the law of error from the condition of numerous
(A) Deduction from Hypothetical Conditions.
independent causes; and (B) showing that the law is
adequately fulfilled in a variety of concrete cases, in
which the condition is probably present. The
condition may be supposed to be perfectly fulfilled in *games*
*of chance*, or, more generally, *sortitions*, characterized by
the circumstance that we have a knowledge prior to
specific experience of the proportion of what Laplace
calls favourable cases^{[10]} to all cases—a category which includes,
for instance, the distribution of digits obtained by random extracts
from mathematical tables, as well as the distribution of the numbers
of points on dominoes.

103. The genesis of the law of error is most clearly illustrated by
the simplest sort of “game,” that in which the sortition is between
two alternatives, heads or tails, hearts or not-hearts, or, generally,
success or failure, the probability of a success being *p* and
Games of Chance.
that of a failure *q*, where *p* + *q* = 1. The number of
such successes in the course of *n* trials may be
considered as an aggregate made up of *n* independently
varying elements, each of which assumes the values 0 or 1 with
respective frequency *q* and *p*. The frequency of each value of the

- ↑ On this conception see below, par. 122.
- ↑
*E.g.*in the article on “Probability” in the 9th ed. of the*Ency.**Brit.*; also by Airy and other authorities. Bravais, in his article*Sur la probabilité des erreurs*. . . . “Mémoires présentés par divers savants” (1846), p. 257, takes as the “modulus or parameter” the inverse square of our*c*. Doubtless different parameters are suited to different purposes and contexts;*c*when we consult the common tables, and in connexion with the operator, as below, par. 160;*k*( = ½*c*^{2}) when we investigate the formation of the probability-curve out of independent elements (below, par. 104);*h*( = 1/*c*^{2}) when we are concerned with weights or precisions (below, par. 134). If one form of the coefficient must be uniformly adhered to, probably, σ( =*c*/√2), for which Professor Pearson expresses a preference, appears the best. It is called by him the “standard deviation.” - ↑ Fuller tables are to be found in many accessible treatises.
Burgess's tables in the
*Trans. of the Edin. Roy. Soc.*for 1900 are carried to a high degree of accuracy. Thorndike, in his*Mental**and Social Measurements*, gives, among other useful tables, one referred to the standard deviation as the argument. New tables of the probability integral are given by W. F. Sheppard,*Biometrics*, ii. 174 seq. - ↑
*Edinburgh Review*(1850), xcii. 19. - ↑ The italics are in the original. The passage continues: “And it is on this ignorance, and not on any peculiarity in cases, that the idea of probability in the abstract is formed.” Cf. above, par. 6.
- ↑
*Natural Philosophy*, pt. i. art. 391. For other a priori proofs see Czuber,*Theorie der Beobachtungsfehler*, th. i. - ↑ Cf. note to par. 127.
- ↑ He considered the effect as the sum of causes each of which obeys the simplest law of frequency, the symmetrical binomial.
- ↑
*Memoirs of Astronomical Society*(1878), p. 105. Cf. Morgan Crofton, “On the Law of Errors of Observation,”*Trans. Roy. Soc.*(1870), vol. clx. pt. i. p. 178. - ↑ Above, par. 2.