The maximum frequency is, as it ought to be, at the point x = 6i, y = 6i′. 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 zX by its intersection with the surface is of the form
z = J exp −X2[k sin2 θ − 2l cos θ sin θ + m cos2 θ]/2(km − l2),
a probability-curve which will be more or less spread out according as the factor k sin2 θ − 2l cos θ sin θ + m cos2 θ is less or greater. Now this expression has a minimum or maximum when (k − m) sin θ−2l cos 2θ = 0; a minimum when (k − m) cos 2θ+2 lsin 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.
116. Characteristics of the Law of Error—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, 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 = 1/2a, 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.” 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, and even higher acts of judgments. 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 and others. 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.” 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.