These values being substituted for the coefficients in the general formula, there results an expression which may be obtained directly by continuing to expand the expression for a term of the binomial. In virtue of the second approximation a set of observations is not to be excluded from the affinity to the normal curve because, like the curve of barometric heights, it is slightly asymmetrical. In virtue of the third approximation it is not excluded because, like the group of shot-marks above examined, it is, though almost perfectly symmetrical, in other respects apparently somewhat abnormal.

160. If the third approximation is not satisfactory there is still available a fourth, or a still higher degree of approximation.10 The general expression for y which (multiplied by Ax) represents the probability that an error will occur at a Higher particular point (within a particular small interval) Approxima may be written

с

16

Pearsonian criterion-and in other calculations involving the number of observations, in particular the determinations of probable error a good margin is to be left for the possibility that the n observations are not perfectly independent: e.g. the accidents of wind or nerve which affected one shot may have affected other shots immediately before or after.

158. (2) The Generalized Law of Error.-That the normal law of error should not be exactly fulfilled is not disconcerting to those who ground the law upon the plurality of independent causes. On that view the normal law would only be exact when the numbers of elements from which it is generated is very great. In general, when that number is large, but not indefinitely great, there is required a correction owing to one or other of the following imperfections: that the elements do not fluctuate according to the normal law of frequency; that their fluctuations are not independent of each other; that the function whereby they are aggregated is not linear. The correction is formed by a series of terms descending in the order of magnitude.

159. The first term of this series may be written -2(k/c3)[x/c−2x3/3c3];

where /2 is the mean square of deviation for the compound and also the sum of the mean squares of deviations for the component elements, k is the mean cube of deviations for the Second compound and the sum of the mean cubes for the comand Third ponents, and the elements are supposed to be such and Approxima so numerous that ki/c is of the order 1/n. This second tions. approximation, first given by Poisson, was rediscovered by De Forest. The present writer has obtained it by a variety of methods. By a further extension of these methods a third and further approximations may be found. The corrected normal law is then of the form

-- (exp-2) [1-2k (-3)+k3 (−3+10

2=

The Analyst (Iowa), vol. ix.

Phil. Mag (Feb. 1896) and Camb. Phil. Trans. (1905). The part of the third approximation affected with k2 may be found by proceeding to another step in the method described (Phil. Mag., 1896, p. 96). The remaining part of the third approximation is found by the same method (or the variant on p. 97) from the dy I d'y new partial differential equation=24 where k1, is the difference between the actual mean fourth power of deviation and what it would be if the normal law held good. Further approximations may be obtained on the same principle.

H4-342 in the notation which Professor Pearson has made familiar.

7 Cf. Pearson, Trans. Roy. Soc. (1895), A, clxxxvi. 347.

XXII 7*

I -x2/2k, k is the

where % is (the norinal error-function)√(27)

mean square of deviation; ki, k,..., &c., are coefficients formed from the mean powers of deviation according to the rule that k, is the difference between the ith mean power as it actually is and what it would be if the (-1)th approximation were perfectly correct. Thus ki is the difference between the actual mean third power and what the third power would be if the first approximation, the normal law, were perfectly correct, that is, the difference between the actual mean third power, often written us, and zero, that is us. Similarly k is the difference between the actual mean fourth power of deviation, say 4, and what that mean power would be if the second approxima ki, ks. ks, &c., k, kz, k4, &c., form each a succession of terms descendtion were perfectly correct, viz. 3k. Thus k1 =μ-3k. The series ing in the order of magnitude, when each k, e.g. k, has been divided by the corresponding power, i.e. the power (+2) of the parameter or modulus c=√(2k), which division is secured by the successive differentiations of yo, with which each k is associated, e.g. k, with ( Moreover, the first term of the odd series of k's when divided by the proper power of the parameter, viz. c3 is small in comparison with the first term of the even series, viz. k, properly referred→→ divided by c2 (= 2k).

161. Whatever the degree of approximation employed, it is to be remembered that the law in general is only applicable to a certain range of the compound magnitude here represented by the abscissa x." The curve of error, even when general- Character ized as here proposed, coincides only with the central of the portion-the body, as distinguished from the extremities Approxima of the actual locus; a greater or less proportion. tion. 162. The law thus generalized may be extended, with similar reservations, to two or more dimensions. For example, the second approximation in two dimensions may be written d'zo d'zo 3,0 k 2x + 32,1 kg +31,2k dxdys dx dy where zo is (the normal error-function)

x and y are (as before) co-ordinates measured from the centre of gravity of the group as origin, each referred to (divided by) its proper modulus; r is the ordinary coefficient of regression; ok is the mean value of the cubes x3, 2k is the mean value of the products x2y, and so on; all these k's being quantities of an order less than unity. This form lends itself readily to the determination of a second approxi mation to the regression-curve, which is the locus of that y, which is the most probable value of the ordinate corresponding to an assigned value of x. Form the logarithm of the above-written expression (for the frequency-surface); and differentiate that logarithm with respect to x. The required locus is given by equating this

Above, § 103, referring to Todhunter, History, art. 993. The third (or second additional term of) approximation for the binomial, given explicitly by Professor Pearson, Trans. Roy. Soc. (1895), A, footnote of p. 347, will be found to agree with the general formula above given, when it is observed that the correction affecting the absolute term, his yo, disappears in his formula by division. Soc. (1898), A.

Journ. Stat. Soc. (1899), p. 550, referring to Pearson, Trans. Roy.

19 Practically no doubt the law is not available beyond the third or fourth approximation, for a reason given by Pearson, with reference to his generalized probability-curve, that the probable error incident to the determination of the higher moments becomes very great.

This consideration does not present the determination of the true moments from the complete set of observations if homogeneous, according as the system of elements fulfils more or less perfectly certain conditions.

differential to zero (the second differential being always negative). The resulting equation is of the form

y-rx-T-ax2 - 2ẞxy — vy2 = 0,

where T. a. B. are all small, linear functions of the k's. As y is nearly equal to r x, it is legitimate to substitute r x for y, when y is multiplied by a small coefficient. The curve of regression thus reduces to a parabola with equation of the form

y-T-rx-qx2;

where q is a linear function of the third mean powers and moments of the given group.

163. Dissection of certain Heterogeneous Groups.-Under the head of law of error may be placed the case in which statistics relating to two (or more) different types, each separately conforming to the normal law, are mixed together; for instance, the measurements of human heights in a country comprising two distinct races.

In this case the quaesita are the constants in a curve of the form: y = a(1/√πc1) exp−(x−a)2/c,2+B(1/√πC2) exp — (x — b)2/cz2, where a and B are the proportionate sizes of the two groups (a+8=1); a and b are the respective centres of gravity; and c, c the respective moduli. The data are measurements each of which relates to one or other of these component curves. A splendid solution of this difficult problem has been given by Professor Pearson. The five unknown quantities are connected by him with the centre of gravity of the given observations, and the mean second, third, fourth and fifth powers of their deviations from that centre of gravity. by certain rational algebraic equations, which reduce to an equation in one variable of the ninth dimension. In an example worked by Professor Pearson this fundamental equation had three possible roots, two of which gave very fair solutions of the problem, while the third suggested that there might be a negative solution, importing that the given system would be obtained by subtracting one of the normal groups from the other; but the coefficients for the negative solution proved to be imaginary. "In the case of crabs' foreheads, therefore, we cannot represent the frequency curve for their forehead length as the difference of two normal curves." In another case, which primâ facie seemed normal, Professor Pearson found that "all nine roots of the fundamental nonic lead to imaginary solutions of the problem. The best and most accurate representation is the normal curve." 164. This laborious method of separation seems best suited to cases in which it is known beforehand that the statistics are a mixture of two normal groups, or at least this is strongly suggested by the two-headed character of the given group. Otherwise the less troublesome generalized law of error may be preferable, as it is appropriate both to the mixture of two-not very widely different-normal | groups, and also the other cases of composition. Even when a group of statistics can be broken up into two or three frequency curves of the normal-or not very abnormal-type, the group may yet be adequately represented by a single curve of the "generalized type, provided that the heterogeneity is not very great, not great enough to prevent the constants ki, ka, ka, &c., from being small. Thus, suppose the given group to consist of two normal curves each having the same modulus c, and that the distance between the centres is considerable, so considerable as just to cause the central portion of the total group to become saddle-backed. This phenomenon sets in when the distance between the centre of gravity of the system and the centre of either component=c. Even in this case ka is only -0.125; k, is 0.25 (the odd k's are zero).

Section II.-Laws of Frequency.

Finally, the slope of the curve correlated to the hypergeometrical series (which expresses a probability distribution in which the contributory causes are not independent, and not equally likely to give equal deviations in excess and defect), as the above curves to their respective binomials, is given by a relation of the form

1 Cf. Journ. Stat. Soc. (1899), Ixii. 131. A similar substitution of the generalized law of error may be recommended in preference to the method of translating a normal law of error (putting x=f(x), where x obeys the normal law of error) suggested by the present writer (Journ. Stat. Soc., 1898), and independently by Professor J. C. Kapteyn (Skew Frequency Curves, 1903).

This latter curve comprises the two others as special cases, and, so far as my investigations have yet gone, practically covers all homogeneous statistics that I have had to deal with. Something still more general may be conceivable, but I have found no necessity for it." The "hypergeometrical series," it should be explained, had appeared as representative of the distribution of black balls, in the following case. "Take n balls in a bag, of which pn are black and qn are white, and let r balls be drawn and the number of black be recorded. If r>pn, the range of black balls will lie between o and pn; the resulting frequency-polygon is given by a hypergeometrical series."

Further reasons in favour of his construction are given by Professor Pearson in a later paper. "The immense majority, if not the totality, of frequency distributions in homogeneous material show, when the frequency is indefinitely increased, a tendency to give a smooth curve characterized by the following properties. (i.) The frequency falls again to zero-probably at a quite different rate-as the characstarts from zero, increases slowly or rapidly to a maximum and then ter for which the frequency is measured is steadily increased. This is the almost universal unimodal distribution of the frequency of homogeneous series .. (i.) In the next place there is generally contact of the frequency-curve at the extremities of the range. These characteristics at once suggest the following of frequency curve, if yox measure the frequency falling between x and x+dx: — dy_y(x+a) dx F(x) Now let us assume that F(x) can be expanded by Maclaurin's theorem.. Then our differential equation to the frequency will be I dy

...

Experience shows that the form (x) [" keeping ba, b, b1, only " suffices for certainly the great bulk of frequency distributions." 166. The " generalized probability-curve" presents two main forms y=yo(1+x/a1)) I −x/ɑ?) v31⁄2, - tan-1x/a. and y-yo = Yo (1+x3 / a2) me

When a1, a2,

are all finite and positive, the first form represents, in general, a skew curve, with limited range in both directions; in the particular case, when a, a, a symmetrical curve, with range limited in both directions. If a, the curve reduces to y=yo (2+x/a11le); representing an asymmetrical binomial with 2/3, and 21 = 2μ22/μ-Qμμ2, 2 and 3. being respectively the mean second and mean third power of deviation measured from the centre of gravity. In the particular case, when us is small, this form reduces to what is above called the "quasi-normal" curve; and when μ is zero, a becoming infinite, to the simple normal curve. The pregnant general form yields two less familiar shapes apt to represent curves of the character shown in figs. 14 and 15-the one occurring in a

good number of instances, such as infant deaths, the values of houses, the number of petals in certain flowers; the other less familiarily illustrated by Consumptivity and Cloudiness. The second solution represents a skew curve with unlimited range in both directions. Professor Pearson has successfully applied these formulae to a number of beautiful specimens culled in the most diverse fields of statis. tics. The flexibility with which the generalized probability-curve adapts itself to every variety of existing groups no doubt gives it a great advantage over the normal curve, even in its extended form. It is only in respect of a priori evidence that the latter can claim precedence.'

167. Skew Correlation.-Professor Pearson has extended his Trans. Roy. Soc. (1895), A. p. 381. Ibid. p. 360. "Mathematical Contributions to the Theory of Evolution" (Drapers' Company Research Memoirs, Biometric Series II.), xiv. 4. P. 7. loc. cit. • Ibid. p. 367.

7 Pearson, loc. cit., p. 364, and Proc. Roy. Soc.

A lucid exposition of Professor Pearson's various methods is given by W. Palin Elderton in Frequency-curves and Correlation (1906).

Journ. Stat. Soc. (1895), p. 506.

method to frequency-loci of two uimensions; constructing for the curve of regression (as a substitute for the normal right line), in the case of "skew correlation," a parabola, with constants based on the higher moments of the given group.

168. In this connexion reference may again be made to Mr Yule's method of treating skew surfaces as if they were normal. It is certainly remarkable that the correlation should be so well represented by a line the property of a normal surface-in cases of which normality cannot be predicated: for instance, the statistics of the number of husbands (or wives) living at each age who have wives (or husbands) living at different ages. It may be suggested that though in this case there is one dominant cause, the continual decrease of the population, inconsistent with the plurality of causes postulated for the law of error, yet there is a sufficient degree of accidental variation to realize one property at least of the normal locus.

between

bility.

169. There is possibly an extensive class of phenomena of which frequency depends largely on fortuitous causes, yet not Relations so completely as to present the genuine law of error. This mixed class of phenomena might be amenable Frequency to a kind of law of frequency that would be different and Proba from, yet have some affinity to, the law of error. The double character may be taken as the definition of the laws proper to the present section. The definition of the class is more distinct than its extent. Consider for example the statistics which represent the numbers out of a million born that die in each year of age after thirty of forty-the latter part of the column in a life-table. These are well represented by a species of Professor Pearson's "generalized probability-curve," 175 his type iii. of the form

Generating Functions." Not all parts of the book are as rewarding probabilités) and the fourth and subsequent chapters of the second as the Introduction (published separately as Essai philosophique des book. Among numerous general treatises E. Czuber's Wahrscheinlichkeitstheorie (1899) may be noticed as terse, lucid and abounding in references. Other authorities may be mentioned in relation to the different parts of the subject as above divided. First principles are discussed with remarkable acumen by J. Venn in Logic of Chance (1st ed., 1876, 3rd ed., 1888) and by J. v. Kries in Principien der Wahrscheinlichkeitsrechnung (1886). As a repertory of neat problems involving the calculation of probability and expectation W. A. Whitworth's Choice and Chance (5th ed., 1901), and DCC. Exercises... in Choice and Chance (1897) deserve mention. But this advantage is afforded in nearly as great perfection by more comprehensive works. Bertrand's Calcul des probabilités (1889) abounds in choice examples, while it excels in almost every other branch of the subject. Special mention is also deserved by H. Poincaré's geometrical probability Professor Morgan Crofton is one of the Calcul des probabilités (leçons professes, 1893-1894). On local or highest authorities. His paper on "Local Probability" in Phil. Trans. (1868), and on Geometrical Theorems," Proc. Lond. Math. Soc. (1887), viii., should be read in connexion with the section on "Local Probability in his article on "Probability in the 9th edition of the Ency. Brit., from which section several paragraphs have been transferred en bloc to the section on Geometrical Applications in the present article. The topic is treated exhaustively by Czuber in Geometrische Wahrscheinlichkeiten und of Theorie der Beobachtungsfehler, in which he has reproduced, often Mittelworten (1884). Czuber is also to be mentioned as the author with improvement, or referred to, almost everything of importance in the work of his predecessors. A. L. Bowley's Elements of Statistics, pt. 2 (2nd ed., 1902), forms an introduction to the law of error which given in Section I. of Part II. above. A list of writings on the cognate leads the beginner easily, yet far. References to other writers are topic, the method of least squares, has been given by Merriman (Connecticut Trans. vol. iv.). On laws of frequency, as above defined, Professor Karl Pearson is the highest authority. His "Contributions to the Mathematical Theory of Evolution," of which twelve have appeared in the Trans. Roy. Soc. (1894-1903) and others are being published by the Drapers' Company, teem with new theories in (F. Y. E.3)

The former law, the simplest species of the "generalized probability-curve," may well be attributed in part to the operation of a plexus of causes such as that which is apt to generate the law of error. In fact, a high authority, Professor Lexis, has seen in these statistics-or continental statistics in pari materia-a fulfilment of the normal law of error. They at least fulfil tolerably the generalized law of error above described. But the Gompertz-Makeham formula is not thus to be accounted for; at least it is not thus that it was regarded by its discoverers. Gompertz justifies his law' by a "hypothetical deduction congruous with many natural effects," such as the exhaustion of air by a pump; and Makcham follows in the same track of explanation by way of natural laws. Of course it is not denied that mortality is subject to accident. But the Gompertz-Makeham law purports to be fulfilled in spite of, not by reason of, fortuitous agencies. The formula is accounted for not by the interaction of fleeting causes which is characteristic of probability, but by causes of that ordinary kind of which the investigation constitutes the greater part of natural science. Laws of frequency thus conceived do not belong to the theory of Probabilities.

AUTHORITIES.-As a comprehensive and masterly treatment of the subject as a whole, in its philosophical as well as mathematical character, there is nothing similar or second to Laplace's Théorie analytique des probabilités. But this "ne plus ultra of mathematical skill and power as it is called by Herschel (Edinburgh Review, 1850) is not easy reading. Much of its difficulty is connected with the use of a mathematical method which is now almost superseded, 1 " Contributions," No xiv. (above cited). Not the same parabola as that proposed at par. 162. 3 Census of England and Wales General Report (cod. 2174), p. 226. Cf. p. 70, as to the rationale of the phenomenon. A good example of the suggested blend between law and chance is presented by an hypothesis which Benine (in a passage referred to above, par. 97) has proposed to account for Pareto's income-curve. "Contributions," No. ii., Phil. Trans. (1895), vol. 186, A. Lexis, Massenerscheinungen, § 46. Cf. Venn, cited above, par. 124. Phil. Trans. (1-25).

Assurance Magazine (1866), xi. 315.

"

PROBATE, in English law, the "proving" (Lat. probatio) of a will. The early jurisdiction of the English ecclesiastical courts over the probate of wills of personality is discussed under WILL. The Court of Probate Act 1857 transferred the jurisdiction both voluntary and contentious of all ecclesiastical, royal peculiar, peculiar and manorial courts to the court of probate thereby constituted, created a judge and registrars of that court, abolished the old exclusive rights in testamentary matters of the advocates of Doctors' Commons, and laid down rules of procedure. Contentious jurisdiction was given to county courts when the personal estate of the deceased was under £200 in value. The Judicature Act 1873 merged the old court of probate in the probate divorce and admiralty division of the High Court of Justice. The division now consists of the president and one other judge. The practice of the division is mainly regulated by the rules of the Supreme Court 1883. Appeals lie to the court of appeal and thence to the House of Lords, Probate may be taken out either in common or solemn form. In the former case, which is adopted when there is no dispute as to the validity of the will, the court simply recognizes the will propounded as the last will of the deceased. This formality is necessary to enable the executor to administer the estate of his testator. Probate in this form is granted simply as a ministerial act if the attestation clause declares that the formalities of the Wills Act have been complied with, or if other evidence to that effect is produced. Such grant is liable to revocation, but it is provided that any person dealing with an executor on the faith of a grant of probate in common form, shall not be prejudiced by its revocation. The executor may within thirty years be called upon to prove in solemn form, or a person who doubts the validity of the will propounded may enter a caveat which prevents the executor proving for six months and the caveat may be renewed each six months. The executor may however take out a summons to get the caveat "subducted" or withdrawn, but if an appearance to the summons is entered

These initials do not apply to certain passages in the above article, namely, the greater part of paragraphs 41, 52, 62 and 72, and almost the whole of the 4th section of Part. I. (pars. 76-93), which have been adopted from the article" Probability" in the 9th edition of the Ency. Bril., written by Professor Morgan Crofton.