108. (3) De Forest has given a proof unencumbered by imaginaries of what is the fundamental proposition in Laplace's theory that, if a polynomial of the form
A0 + A1z + A2z2 + . . . + Amzm
be raised to the nth power and expanded in the form
B0 + B1z + B2z2 + . . . + Bmnzmn
then the magnitudes of the B's in the neighbourhood of their maximum (say B1) will be disposed in accordance with a “probability-curve,” or normal law of error.
109. (4) Professor Morgan Crofton's original proof of the law of error is based on a datum obtained by observing the effect which the introduction of a new element produces on the frequency-locus for the aggregate of elements. It seems to be assumed, very properly, that the sought function involves as constants some at least of the mean powers of the aggregate, in particular the mean second power, say k. We may without loss of generality refer each of the elements (and accordingly the aggregate) to its respective centre of gravity. Then if y, = f(x), is the ordinate of the frequency-locus for the aggregate before taking in a new element, and y = ∂y the ordinate after that operation, by a well-known principle, y + ∂y = [Sφm(ξ)f(x − ξ)∆ξ], where η, = φm(ξ), is the frequency-locus for the new element, and the square brackets indicate that the summation is to extend over the whole range of values assumed by that element. Expanding in ascending powers of (each value of) ξ and neglecting powers above the second, as is found to be legitimate under the conditions specified, we have (since the first mean power of the element vanishes)
∂y = ½[Sξ2φm(ξ)∆ξ]d2f.
From the fundamental proposition that the mean square for the aggregate equals the sum of mean squares for the elements it follows that [Sξ2φm(ξ)∆ξ] the mean second power of deviation for the mth element is equal to ∂k, the addition to k the mean second power of deviation for the aggregate. There is thus obtained a partial differential equation of the second order
A subsidiary equation is (in effect) obtained by Professor Crofton from the property that if the unit according to which the axis of x is graduated is altered in any assigned ratio, there must be a corresponding alteration both of the ordinate expressing the frequency: of the aggregate and of the mean square of deviation for the aggregation. By supposing the alteration indefinitely small he obtains a second partial differential equation, viz. (in the notation here adopted)
From these two equations, regard being had to certi
an other conditions
of the problem, it is deducible that y = Ce−x2/2k, where C is a
constant of which the value is determined by the condition that
110. (5) The condition on which Professor Crofton's proof is based may be called differential, as obtained from the introduction of a single new element. There is also an integral condition obtained from the introduction of a whole set of new elements. For let A be the sum of m1 elements, fluctuating according to the sought law of error. Let B be the sum of another set of elements m2 in number (m1 and m2 both large). Then Q a quantity formed by adding together each pair of concurrent values presented by A and B must also conform to the law of error, since Q is the sum of m1 + m2 elements. The general form which satisfies this condition of reproductivity is limited by other conditions to the normal law of error.
111. The list of variant proofs is not yet exhausted, but enough has been said to establish the proposition that a sum of numerous elements of the kind described will fluctuate approximately according to the normal law of error.
112. As the number of elements is increased, the constant above designated k continually increases; so that the curve representing Varieties of Linear Function. the frequency of the compound magnitude spreads out from its centre. It is otherwise if instead of the simple sum we consider the linear function formed by adding the m elements each multiplied by 1/m. The “spread” of the average thus constituted will continually diminish as the number of the elements is increased; the sides closing in as the vertex rises up. The change in “spread” produced by the accession of new elements is illustrated by the transition from the high to the low curve, in fig. 10, in the case of a sum; in the case of an average (arithmetic mean) by the reverse relation.
113. The proposition which has been proved for linear functions may be extended to any other function of numerous variables, Extension to Non-linear Functions. each representing the value assumed by an independently fluctuating element; if the function may be expanded in ascending powers of the variables, according to Taylor's theorem, and all the powers after the first may be neglected. The matter is not so simple as it is often represented, when the variable elements may assume large, perhaps infinite, values; but with the aid of the postulate above enunciated the difficulty can be overcome.
114. All the proofs which have been noticed have been extended to errors in two (or more) dimensions. Let Q be the sum of a Extension to two or more Dimensions. number of elements, each of which, being a function of two variables, x and y, assumes different pairs of values according to a law of frequency zr = fr(x, y), the functions being in general different for different elements. The frequency with which Q assumes values of the variables between x and +∆x and between y and y + ∆y is z∆x∆y, if
where, as in the simpler case, a = ∑ar, ar being the arithmetic mean of the values of x assumed in the long run by one of the elements, b is the corresponding sum for values of y, and
k = ∑[∬(x − ar)2fr(x, y)dxdy]
m = ∑[∬(y − br)2fr(x, y)dxdy]
l = ∑[∬(x − ar)(y − br)fr(x, y)dxdy];
the summation extending over all the elements, and the integration between the extreme limits of each; supposing that the law of frequency for each element is continuous, otherwise summation is to be substituted for integration. For example, let each element be constituted as follows: Three coins having been tossed, the number of heads presented by the first and second coins together is put for x, the number of heads presented by the second and third coins together is put for y. The law of frequency for the element is represented in fig. 11, the integers outside denoting the values of x or y, the fractions inside probabilities of particular values of x and y concurring.
If i is the distance from 0 to 1 and from 1 to 2 on the abscissa, and i′ the corresponding distance on the ordinate, the mean of the values of x for the element—∆a, as we may say,—is i, and the corresponding mean square of horizontal deviations is ½i2. Likewise ∆b = i′; ∆m = ½i′2; and ∆l = ⅛(+i × +i′ − i × −i′) = ¼ii′. Accordingly, if n such elements are put together (if n steps of the kind which the diagram represents are taken), the frequency with which a particular pair of aggregates x and y will concur, with which a particular point on the plane of xy, namely, x = ri and y = ri, will be reached, is given by the equation
115. A verification is afforded by a set of statistics obtained with dice by Weldon, and here reproduced by his permission. A success is in this experiment defined, not by obtaining a head when a coin is tossed, but by obtaining a face with more than three points on it when a die is tossed; the probabilities of the two events are the same, or rather would be if coins and dice were perfectly symmetrical. Professor Weldon virtually took six steps of the sort above described when, six painted dice having been thrown, he added the number of successes in that painted batch to the number of successes in another batch of six to form his x, and to the number of successes in a third batch of six to form his y. The result is represented in the annexed table, where each degree on the axis of x and y respectively corresponds to the i and i′ of the preceding paragraphs, and i = i′. The observed frequencies being represented by numerals, a general correspondence between the facts and the formula is apparent.
- The Analyst (Iowa), vols. v., vi., vii. passim; and especially vi. 142 seq., vii. 172 seq.
- Morgan Crofton, loc. cit. p. 781, col. a. The principle has been used by the present writer in the Phil. Mag. (1883). xvi. 301.
- For a criticism and extension of Crofton's proof see the already cited paper on “The Law of Error,” Camb. Phil. Trans. (1905), pt. i. § 2. Space does not permit the reproduction of Crofton's proof as given in the 9th ed. of the Ency. Brit. (art. “Probability,” § 48).
- Loc. cit. pt. I. § 4; and app. 6.
- Loc. cit. p. 122 seq.
- Loc. cit. pt. ii. § 7.
- The second by Burbury, in Phil. Mag. (1894), xxxvii. 145; the third by its author in the Analyst for 1881; and the remainder by the present writer in Phil. Mag. (1896), xii. 247; and Camb. Phil. Trans. (1905), loc. cit.
- Compare the formula for the simple case above, § 4.
- On the irregularity of the dice with which Weldon experimented, see Pearson, Phil. Mag. (1900), p. 167.