96. The Law of Error

Of the propositions respecting average with which Probabilities is concerned the most important are those which deal with the relation of the average to its constituents, and are commonly called " laws of error." Error is defined in popular dictionaries as " deviation from truth "; and since truth commonly lies in a mean, while measurements are some too large and some too small, the term in scientific diction is extended to deviations of statistics from their average, even when that average - like the mean of human or barometric heights - does not stand for any real objective thing. A " law of error" is a relation between the extent of a deviation and the frequency with which it occurs: for instance, the proposition that if a digit is taken at random from mathematical tables, the difference between that figure and the mean of the whole series (indefinitely prolonged) of figures so obtained, namely, 4.5, will in the long run prove to be equally of ten E 0.5, d I. 5, ± 2 ' 5.3.5, X 4.5 .6 The assignment of frequency to discrete values - as o, I, 2, &c., in the preceding example - is often replaced by a continuous curve with a corresponding equation. The distinction of being the law of error is bestowed on a function which is applicable not merely to one sort of statistics - such as the digits above instanced - but to the great variety of miscellaneous groups, generally at least, if not universally. What form is most deserving of this distinction is not decided by uniform usage; different authorities do not attach the same weight to the different grounds on which the claim is based, namely the extent of cases to which the law may be applicable, the closeness of the application, and the presumption prior to specific experience in favour of the law. The term " the law of error " is here employed to denote (I) a species to which the title belongs by universal usage, (2) a wider class in favour of which there is the same sort of a priori presumption as that which is held to justify the more familiar species. The law of error thus understood forms the subject of the first section below.

97. Laws of Frequency

What , other laws of error may require notice are included in the wider genus " laws of frequency," which forms the subject of the second section. Laws of frequency, so far as they belong to the domain of Probabilities, relate much to the same sort of grouped statistics as laws of error, but do not, like them, connote an explicit reference to an average. Thus the sequence of ralidom digits above instanced as affording a law of error, considered without reference to the mean value, presents the law of frequency that one digit occurs as often as another (in the long run). Every law of error is a law of frequency; but the converse is not true. For example, it is a law of frequency - discovered by Professor Pareto 7 - that the number of incomes of different size (above a certain size) is approximately represented by the equation y = A/x a , where x denotes the size of an income, y the number of incomes of than size. But whether this generalization can be construed as a law of error(in the sense here defined) depends on the nice inquiry whether the point from which the frequency diminishes as the income x increases can be regarded as a " mode," y diminishing as x decreases from that point.

5 See above, pt. i., pars? 3 and 4. Accordingly the expected value of the sum of n (similar) constituents (x,--x2-l-... ± xn) may be regarded as an average, the average value of nx r where xr is any one of the constituents.

See as to the fact and the evidence for it, Venn, Logic of Chance, 3rd ed., pp. III, 114. Cf. Ency. Brit., 8th ed., art " Probability," p. 592; Bertrand, op. cit., preface § ii.; above, par. 59.

See his Cours d'economie politique, ii. 306. Cf. Bowley, Evidence before the Select Committee on Income Tax (1906, No. 365, Question 1163 seq.); Benini, Metodologica statistica, p. 324, referred to in the Journ. Stat. Soc. (March, 1909).

? a x 0 0 Section I.-The Law of Error. 98. (I) The Normal Law of Error.-The simplest and best recognized statement of the law of error, often called the " normal law," is the equation (x-(x z = -e -, lJ7rc more conveniently written (I /- rc) exp - (x -a) 2 /c 2, 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' 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+ A x is (approximately) equal to the right-hand side of the equation multiplied by NAx. A graphical representation of the corresponding curve-sometimes called the " probability-curve "-is here given (fig. io), 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, T, 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 IN 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 T which corresponds to this point is 0 4769... .

Y 0 FIG. 10.

99. An a priori proof of this law was given by Herschel' as follows: " The probability of an error depends solely on its magni priori tude and not on its direction;" positive and negative A. 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 O(x 2), for two dimen ' On this conception see below, par. 122.

2 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 probabilite des erreurs.. .. " Memoires presentes 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(= lc') when we investigate the formation of the probability-curve out of independent elements (below, par. 104); h(= I/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, 0-(=c11,12), 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.

5 The italics are in the original. The passage continues: " And sions 41(x 2 + 31 2); and 4, (x 2 + y2) - ¢ (x2) X 49(y2); 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.' But it must be confessed that the claim to universality is not, without some strain of interpretation,' to be reconciled_ with common experience.

Table of the Values of the Integral I =T-7,2fore-r2dx.

ioo. 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.

Ioi. 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' 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.° 102. The validity of the explanation may best be tested by first (A) deducing the law of error from the condition of numerous 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 that of a failure q, where p + q = I. 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 o or i with respective frequency q and p. The frequency of each value of the 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 Cf. note to par. 127.

3 He considered the effect as the sum of causes each of which obeys the simplest law of frequency, the symmetrical binomial.

0 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.

' 0 Above, par. 2.

-X +X (A) aggregate is given by a corresponding term in the expansion of (q+p), and by a well-known theorem 1 this term is approximately zqual to I - v2/2npq r2npg e; where v is the number of integers by which the term is distant from np (or an integer close to np); provided that v is of (or <) the order ,l n. Graphically, let the sortition made for each element be represented by the taking or not taking with respective frequency p and q a step of length i. If a body starting from zero takes successively n such steps, the point at which it will most probably come to a stop is at npi (measured from zero); the probability of its stopping at any neighbouring point within a range oft ni is given by the abovewritten law of frequency, vi being the distance of the stoppingpoint from npi. Put vi=x and 2npgi 2 =c; then the probability may be written (I // 7r c) exp - x2/c2.

104. It is a short step, but a difficult one, from this case, in which the element is binomial - heads or tails - to the general case, in which the element has several values, according to the law of frequency - consists, for instance, of the number of points presented by a randomly-thrown die. According to the general theorem, if Q is the sum 2 of numerous elements, each of which assumes different magnitudes according to a law of frequency, 7=f r (x), the function f being in general different for different elements, the number of times that Q assumes magnitudes between x and x+Ox in the course of N trials is NzOx, if z = (I/ A i 27rk) exp - (x - a) 2 /2k; where a is the sum of the arithmetic means of all the elements, any one of which a, = [fxf 7 (x)dx], the square brackets denoting that the integrations extend between the extreme limits of the element's range, if the frequency-locus for each element is continuous, it being understood that [ff(X) dX ] = I; and k is the sum of the mean squares of error for each element, = [ f f2 fr(ar+E)dE ], if the frequency-locus for each element is con tinuous, where a r is the arithmetic mean of one of the elements, and E the deviation of any value assumed by that element from ar, denoting summation over all the elements. When the frequencylocus for the element is not continuous, the integrations which give the arithmetic mean and mean square of error for the element must be replaced by summations. For example, in the case of the dice above instanced, the law of frequency for each element is that it assumes equally often each of the values I, 2, 3, 4, 5, 6. Thus the arithmetic mean for each element is 3.5, and the mean square of error ((3.5 - 1) 2 + (3.5 - 2) 2 + &c.} /6 = 2.916. Accordingly, the sum of the points obtained by tossing a large number, n, of dice at random will assume a particular value x with a frequency which is approximately assigned by the equation z=(I /,v 7r5.8 3 n) exp - (x-3-5)2/5.83n.

The rule equally applies to the case in which the elements are not similar; one might be the number of points on a die, another the number of points on a domino, and so ron. Graphically, each element is no longer represented by a step which is either null or i, but by a step which may be, with an assigned probability, one or other of several degrees between those limits, the law of frequency and the range of i being different for the different elements.

105. Variant Proofs

The evidence of these statements can only be indicated here. All the proofs which have been offered involve some postulate as to the deviation of the elements from their respective centres of gravity, their " errors." If these errors extended to infinity, it might well happen that the law of error would not be fulfilled by a sum of such elements. 3 The necessary and sufficient postulate appears to be that the mean powers of deviation for the elements, the second (above written) and the similarly formed third, fourth, &c., powers (up to some assigned power), should be finite.4 106. (i) The proof which seems to flow most directly from this postulate proceeds thus. It is deduced that the mean powers of deviation for the proposed representative curve, the law of error (up to a certain power), differ from the corresponding powers of the actual locus by quantities which are negligible when the number of the elements is large.' But loci which have their, mean powers of deviation (up to some certain power) approximately equal may be considered as approximately coincident.' 107. (2) The earliest and best-known proof is that which was ' By the use of Stirling's and Bernoulli's theorems, Todhunter, History... of Probability. 2 The statement includes the case of a linear function, since an element multiplied by a constant is still an element.

3 E.g. if the frequency-locus of each element were 1/7r(I +x2), extending to infinity in both directions. But extension to infinity would not be fatal, if the form of the element's locus were normal. 4 For a fuller exposition and a justification of many of the statements which follow, see the writer's paper on " The Law of Error " in the Camb. Phil. Trans. (1905).

Loc. cit. pt. i. § I.

6 On this criterion of coincidence see Karl Pearson's paper "On the Systematic Fitting of Curves," Biometrika, vols. i. and ii.

originated by Laplace and generalized by Poisson. Some idea of this celebrated theory may be obtained from the following free version, applied to a simple case. The case is that in which all the elements have one and the same locus of frequency, and that locus is symmetrical about the centre of gravity. Let the locus be represented by the equation r) =4)(i), where the centre of gravity is the origin, and 0(+E) =4(-); the construction signifying that the probability of the element having a value (between say E - Ig and + 1A0 is ct,(),U. Square brackets denoting summation between extreme limits, put x (a) for [Scp()e" 1a ,g] where is an integer multiple of ,g (or Ox) =Ox, say. Form the mth power of x(a).

The coefficient of e ll - iarLx in (x(a)) m is the probability that the sum of the values of the m elements should be equal to rzx; a probability which is equal to Oxy r ., where y is the ordinate of the locus representing the frequency of the compound quantity (formed by the sum of the elements). Owing to the symmetry of the function 4 the value of Yr, will not be altered if we substitute _ larax _ _ +v _larAx for e e nor if we substitute 2(e, e _ _ larOx) that is cos Thus (x(a)) m becomes a sum of terms of the form Oxy 7 cos ar6a, where y_ T =y +7. Now multiply (x(a)) m thus expressed by cos tzxa, where, t being an integer, tzx = x, the abscissa of the " error " the probability of whose occurrence is to be determined. The product will consist of a sum of terms of the form Oxy,. 2(cos a(r+t)Ax+cos a(r - t)6.x). As every value of k r - t (except zero) is matched by a value' equal in absolute magnitude, - r+t, and likewise every value of r+t is matched by value - r - t, the series takes the form Oxy,Z cos gaOx+Oxy t , where q has all possible integer values from 1 to the largest value of Id' increased by Ii; and the term free front circular functions is the equivalent of ixy 7 cos a(r+t)Ox, when r= - t, together with Oxy,. cos a(r - t)%x, when r= +t. Now substitute for a0x a new symbol 0'; and integrate with respect to /3, the thus transformed (x(a)) m cos tOxa between the limits /3=o and /3=7r. The integrals of all the terms which are of the form Oxy r cos q0 will vanish, and there will be left surviving only 7rOxyt. We thus obtain, as equal to 7rzxy t ,.. 07 { x(a/Ox) } "'cos t,6d/3. Now change the independent variable to a; then as da=dazx, 1 (7/Ox Oxy t =fixda(x(a))m cos tOxa.

7r Replacing tOx by x, and dividing both sides by Ox, we have y 2 = f o 7 ' 1 ' 6x S a (x(a)) m cos ax. Now expanding the cos which enters into the expression for x(a), we obtain x(a) = [ 5 4, (a)1 - 2 [ S O(a) a2]x2+-. j [Sch(a)a 4 ]x 4.. 4 Performing the summations indicated, we express x(a) in terms of the mean powers of deviation for an element. Whence x(a) m is expressible in terms of the mean powers of the compound locus. First and chief is the mean second power of deviation for the compound, which is the sum of the mean second powers of deviation for the elements, say k. It is found that the sought probability may be equated to 2 cos ax+ I k 2 f 7r 0 4! cos x-. ., where k 2 is the coefficient defined below. 9 Here 7r/Ax may be replaced by oo, since the finite difference Lx is small with respect to unity when the number of the elements is large;l0 and thus the integrals involved become equateable to known definite integrals. If it were allowable to neglect all the terme of the series but the first the 2 expression would reduce to a7rk) e_u , the normal law of error. A / (But it is allowable to neglect the terms after the first, in a first approximation, for values of x not exceeding a certain range, the number of the elements being large, and if the postulate above enunciated is satisfied. 11 With these reservations it is proved that the sum of a number of similar and symmetrical elements conforms to the normal law of error. The proof is by parity extended to the case in which the elements have different but still symmetrical frequency functions; and, by a bolder use of imaginary quantities, to the case of unsymmetrical functions.

7 Laplace, Theorie analytique des probabilites, bk. ii. ch. iv.; Poisson, Recherches sur la probabilite des jugements. Good restatements of this proof are given by Todhunter, History. .. of Probability art. 1004, and by Czuber, Theorie der Beobachtungsfehler,art. 38 and Th. 2, § 4.

3 The symbol II is used to denote absolute magnitude, abstraction being made of sign. 9 Below, pars. 159, 160.10 Loc. cit. app. I. 11 Loc. cit. p. 53 and context.

108. (3) De Forest 1 has given a proof unencumbered by imaginaries of what is the fundamental proposition in Laplace's theory that, if a polynomial of the form Ao+A,z+A 2 z'+ ... +Amzm be raised to the nth power and expanded in the form Bo+B i z+B 2 z 2 +. .. +Bmnznm, then the magnitudes of the B's in the neighbourhood of their maximum (say B,) will be disposed in accordance with a " probabilitycurve," or normal law of errpr.

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 =ay the ordinate after that operation, by a well-known principle,' y + ay = [So,"(e)f(x - 06,6 where i, =0 m (), 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) 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) a y = 2[S O"'(),E]dx From the fundamental proposition that the mean square for the aggregate equals the sum of mean squares for the elements it follows that [S '4m(E)L E] the mean second power of deviation for the mth element is equal to 8k, 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 dy_,d2y (1) dk - 'dk`' 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) dy Y+xdx+ 2kd k =o. (2) From these two equations, regard being had to certian other conditions of the problem,' it is deducible that y -x21 2k , where C is a constant of which the value is determined by the condition that ydx = 1. Ito. (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 m 1 elements, fluctuating according to the sought law of error. Let B be the sum of another set of elements m 2 in number (m, and m 2 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 m,+m 2 elements. The general form which satisfies this condition of reproductivity is limited by other conditions to the normal law of error.' III. 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 the frequency of the compound magnitude spreads out Var i ties from its centre. It is otherwise if instead of the simple sum we consider the linear function formed by adding the Function m elements each multiplied by I /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 ' 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.

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, 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 Functions. 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 number of elements, each of which, being a functionE xtension of two variables, x and y, assumes different pairs of w values according to a law of frequency rz r =f r (x, y), themore functions being in general different for different elements. Dimensions. The frequency with which Q assumes values of the variables between x and +6,x and between y and yd - Ay is zOxzy, if 1 a)'-2l(x - a)(9 - b) + k(y - b)' z - 2,r,lkm - l exp 2(km - l2) where, as in the simpler case, a = Z.a r, a r 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 [jfx= - a r)'fr(x, y)dxdy] m = [ff(y - br)'fr(x, y)dxdy] = [j/(x - ar) (y - b r).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 continz uous, 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. II, the integers outside denoting the values of x or y, the fractions inside probabilites of particular values of x and y concurring. FIG. I I. If i is the distance from o to I 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-6,a, as we may say, - is i, and the corresponding mean square of horizontal deviations is Zi t. Likewise ob =i'; Om=21; and zl=8(+iX +i' - i X - i') = 41i'. 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 z = i i 6 exp - - 8 [(r - n)'i' - (r - n)(r' - n)ii' + (r' - u)i'-]8 27r N 3n 3n 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.9 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.

s Loc. cit. pt. ii. § 7.

The second by Burbury, in Phil. Mag. (1894), xxxvii. 1 45; 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. s Compare the formula for the simple case above, § 4..

9 On the irregularity of the dice with which Weldon experimented, see Pearson, Phil. Mag. (1900), p. 167.

xxii. 13 a  ?

The maximum frequency is, as it ought to be, at the point x=61, y=61'. The densit y 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 0 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 - X 2 [k sin 2 0-2l cos 0 sin 8+m cos t 0]/2(km-12), a probability-curve which will be more or less spread out according as the factor k sin 2 8-21 cos 0 sin 0+m cos t 0 is less or greater. Now this expression has a minimum or maximum when (k - m) sin 0-21 cos 20=o; a minimum when (k - m) cos 20+2 'sin 20 is positive, and a maximum when that criterion is negative; that is, in the present case, where k =in, a minimum when 0 =1.71- and a maximum when 0=717r. 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, 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 /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 1 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. iji.

2 Journ. Stat. Soc. (March 1900), p. 73, referring to Burton, Phil. Mag. (1883), xvi. 301.

independent elements in which no two or three stand out as sui genesis 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 o 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 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 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: (I) 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 parts 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,' 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 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. ? 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 Memoirs of Astronomical Society (1878), p. 105.

' Journ. Stat. Soc. (2890), p. 462 seq.

5 E.g. the marking of the same work by different examiners. Ibid.

6 Lettres sur la theorie des probabilites and Physique sociale. 7 E.g. the measurements of Italian recruits, adduced in the Atlante statistico, published under the direction of the Ministero de Agricultura (Rome, 2882); and Weldon's measurements of crabs, Proc. Roy. Soc. liv. 322; discussed by Pearson in the Trans. Roy. Soc. (1894), vol. clxxxv. A.

8 Biometrika, iii. 395. Cf. ibid. p. 242.

exemplified by A. L. Bowley's grouping of the wages paid to different classes.' 119. The division of concrete errors which has been proposed is not to be confounded with another twofold classification, namely, A observations which stand for a real objective thing, and A such statistics as are not thus representative of something outside themselves, groups of which the mean is called subjective," This division would be neither clear nor useful. On the one hand so-called real means are often only approximately equal to objective quantities. Thus the proportional frequency with which one face of a die - the six suppose - turns up is only approximately given by the objective fact that the six is one face of a nearly perfect cube. For a set of dice with which Weldon experimented, the average frequency of a throw, presenting either five or six points, proved to be not. 3, but 0.3377.2 The difference of this result from the regulation 0.3 is as unpredictable from objective data, prior to experiment, as any of the means called subjective or fictitious. So the mean of errors of observation often differs from the thing observed by a so-called " constant error_" So shots may be constantly deflected from the bull's-eye by a steady wind or " drift." 120. On the other hand, statistics, not purporting to represent a real object, have more or less close relations to magnitudes which cannot be described as fictitious. Where the items averaged are ratios, e.g. the proportion of births or deaths to the total population in several districts or other sections, it sometimes happens that the distribution of the ratios exactly corresponds to that which is obtained in the simplest games of chance - " combinational " distribution in the phrase of Lexis. 3 There is unmistakably suggested a sortition of the simplest type, with a real ascertainable relation between the number of " favourable cases " and the total number of cases. The most remarkable example of this property is presented by the proportion of male to female (or to total) births. Some other instances are given by Lexis 4 and Westergaard. 5 A similar correspondence between the actual and the " combinational " distribution has been found by Bortkevitch 6 in the case of very small probabilities (in which case the law of error is no longer " normal "). And it is likely that some ratios - such as general death-rates - not presenting combinational distribution, might be broken up into subdivisions - such as death-rates for different occupations or ageperiods - each distributed in that simple fashion.

121. Another sort of averages which it is difficult to class as subjective rather than objective occurs in some social statistics, under the designation of index-numbers. The percentage which represents the change in the value of money between two epochs is seldom regarded as the mere average change in the price of several articles taken at random, but rather as the measure of something, e.g. the variation in the price of a given amount of commodities, or of a unit of commodity. ? So something substantive appears to be designated by the volume of trade, or that of the consumption of the working classes, of which the growth is measured by appropriate index-numbers, the former due to Bourne and Sir Robert Giffen,9 the latter to George Wood.'° 122. But apart from these peculiarities, any set of statistics may be -related to a certain quaesitum, very much as measurements are related to the object measured. That quaesitum is the limiting or ultimate mean to which the series of statistics, if indefinitely prolonged, would converge, the mean of the complete group; this conception of a limit applying to any frequency-constant, to ' ` c," for instance, as well as " a "' in the case of the normal curve." The given statistics may be treated as samples from which to reason up to the true constant by that principle of the calculus which determines the comparative probability of different causes from which an observed event may have emanated." 123. Thus it appears that there is a characteristic more essential to the statistician than the existence of an objective quaesitum, namely, the use of that method which is primarily, but not exclusively, proper to that sort of quaesitum - inverse probability.'3 1 Wages in the United Kingdom in the Nineteenth Century; and art. " Wages " in the Ency. Brit., 10th ed., vol. xxxiii.

2 Phil. Mag. (1900), p. 168.

3 Cf. Journ. Stat. Soc., Jubilee No., p. 192.

4 Massenerscheinungen. Grundziige der Statistik. Cf. Bowley, Elements of Statistics, p. 302.

6 Das Gesetz der kleinen Zahlen. 7 See for other definitions Report of the British Association (1889), pp. 136 and 161, and compare Walsh's exhaustive Measurement of General Exchange-Value. 3 Cf. Bowley, Elements of Statistics, ch. ix.

9 Journ. Stat. Soc. (1874 and later). Parly. Papers [C. 22471 and [C. 3070].

10 " Working-Class Progress since 1860," Journ. Stat. Soc. (1899), p. 639.

11 On this conception compare Venn, Logic of Chance, chs. iii. and iv., and Sheppard, Proc. Lond. Math. Soc., p. 363 seq.

12 Laplace's 6th principle, Theorie analytique, intro. x.

13 See above, pars. 13 and 14.

Without that delicate instrument the doctrine of error can seldom be fully utilized; but some of its uses may be indicated before the introduction of technical difficulties.

124. Having established the prevalence of the law of error, 14 we go on to its applications. The mere presumption that wherever three or four independent causes co-operate, the law of error tends to be set up, has a certain speculative interest.15 The assumption of the law as a hypothesis is legitimate. When the presumption is confirmed by specific experience this knowledge is apt to be turned to account. It is usefully applied to the practice of gunnery,' 0 to determine the proportion of shots which under assigned conditions may be expected to hit a zone of given size. The expenditure of ammunition required to hit an object can thence be inferred. Also the comparison between practice under different conditions is facilitated. In many kinds of examination it is found that the total marks given to different candidates for answers to the same set of questions range approximately in conformity with the law of error. It is understood that the civil service commissioners have founded on this fact some practical directions to examiners. Apart from such direct applications, it is a useful addition to our knowledge of a class that the measurable attributes of its members range in conformity with this general law. Something is added to the truth that " the days of a man are threescore and ten," if we may regard that epoch, or more exactly for England, 72, as " Nature's aim, the length of life for which she builds a man, the dispersion on each side of this point being. .. nearly normal." 17 So Herschel says: " An [a mere] average gives us no assurance that the future will be like the past. A [normal] mean may be reckoned on with the most complete confidence." The existence of independent causes, 19 inferred from the fulfilment of the normal law, may be some guarantee of stability. In natural history especially have the conceptions supplied by the law of error been fruitful. Investigators are already on the track of this inquiry: if those members of a species whose size or other measurable attributes are above (or below) the average are preferred - by " natural " or some other kind of selection - as parents, how will the law of frequency as regards that attribute be modified in the next generation?

125. A particularly perfect application of the normal law of error in more than one dimension is afforded by the movements of the molecules in a homogeneous gas. A general idea of the role played by probabilities in the explanation of these movements may be obtained without entering into the more complicated and controverted parts of the subject, without going beyond the initial very abstract supposition of perfectly elastic equal spheres. For convenience of enunciation we may confine ourselves to two dimensions. Let us imagine, then, an enormous billiard-table with perfectly elastic cushions and a frictionless cloth on which millions of perfectly elastic balls rush hither and thither at random - colliding with each other - a homogeneous chaos, with that sort of uniformity in the midst of diversity which is characteristic of probabilities. Upon this hypothesis, if we fix attention on any n balls taken at random - they need not be, according to some they ought not to be, contiguous - if n is very large, the average properties will be approximately the same as those of the total mixture. In particular the average energy of the n balls may be equated to the average energy of the total number of balls, say T/N, if T is the total energy and N the total number of the balls. Now if we watch any one of the n specimen balls long enough for it to undergo a great number of collisions, we observe that either of its velocity-components, say that in the direction of x, viz. u, receives accessions from an immense number of independent causes in random fashion. We may presume, therefore, that these will be distributed (among the n balls) according to the law of error. The law will not be of the type which was first supposed, where the " spread " continually increases as the number of the elements is increased. 20 Nor will it be of the type which was afterwards mentioned" where the spread diminishes as the number of the elements is increased. The linear function by which the elements are aggregated is here of an intermediate type; such that the mean square of deviation corresponding to the velocity remains constant. The method of composition might be illustrated by the process of taking r digits at random from mathematical tables adding the differences between each digit and 4.5 the mean value of digits, and dividing the sum by 1 r. Here are some figures obtained by taking at random batches of sixteen digits from the expansion of x, subtracting r6 X 4.5 from the sum of each batch, and dividing the remainder by 16: - 14 Cf. above, par. 102.

15 Cf. Galton's enthusiasm, Natural Inheritance, p. 66.

16 A lucid statement of the methods and results of probabilities applied to gunnery is given in the Official Text-book of Gunnery (1902).

17 Venn, Journ. Stat. Soc. (1891), p. 443.

Ed. Rev. (1850), xcii. 23.

19 Cf. Galton, Phil. Mag. (1875), xlix. 44.

10 Above, par. 112.

21 Ibid.

+ I.2 5, + 0.75, - I, - I, +5.5, - 2.75, + 0.75, - 2, + 1.75, +3.2 5, +0.25, - 2.75, - 2.25, - 0.5, +4.75, +0.25.

If, instead of sixteen, a million digits went to each batch, the general character of the series would be much the same; the aggregate figures would continue to hover about zero with a standard deviation of 8.25, a probable error of nearly 2. Here for instance are seven aggregates formed by recombining 252 out of the 256 digits above utilized into batches of 36 according to the prescribed rule: viz. subtracting 36X4.5 from the sum of each batch of 36 and dividing the remainder by .J 36: -0.5, +3.3, +2.6, - o 6, +I. 5, - 2, +I.

The illustration brings into view the circumstance that though the system of molecules may start with a distribution of velocities other than the normal, yet by repeated collisions the normal distribution will be superinduced. If both the velocities u and v are distributed according to the law of error for one dimension, we may presume that the joint values of u and v conform to the normal surface. Or we may reason directly that as the pair of velocities u and v is made up of a great number of elementary pairs (the co-ordinates in each of which need not, initially at least, be supposed uncorrelated) the law of frequency for concurrent values of u and v must be of the normal form which ma y be written 1 z = 2 (km l. 1 - r2) exp - [k2 - 2r, km + m2] /2 (I - r2). It may be presumed that r, the coefficient of correlation, is zero, for, owing to the symmetry of the influences by which the molecular chaos is brought about, it is not to be supposed that there is any connexion or repugnance between one direction of u, say south to north, and one direction of v, say west to east. For a like reason k must be supposed equal to m. Thus the average velocity=2k; which multiplied by m, the mass of a sphere, is to be equated to the average energy T/N. The reasoning may be extended with confidence to three dimensions, and with caution to contiguous molecules.

126. Correlation cannot be ignored in another application of the many-dimensioned law of error, its use in biological inquiries to investigate the relations between different generations.

It was found by Galton that the heights and other measurable attributes of children of the same parents range about a mean which is not that of the parental heights, but nearer the average of the general population. The amount of this " regression " is simply proportional to the distance of the " mid-parent's " height from the general average. This is a case of very general law which governs the relations not only between members of the same family, but also between members of the same organism, and generally between two (or more) coexistent or in any way co-ordinated observations, each belonging to a normal group. Let x and y be the measurements of a pair thus constituted. Then 2 it may be expected that the conjunction of particular values for x and y will approximately obey the two-dimensioned normal law which'has been already exhibited (see par. 114).

127. Regression-lines. - In the expression above given, put l/ J kin =r, and the equation for the frequency of pairs having values of the attribute under measurement becomes z - leXp [(x - a) 2 2r (x - a)(y - b) + (y - b)2] /2(I - r2).

27r?/ km1 I - r' k -1 k - N l m m This formula is of very general application.' If two sets of measurements were made on the height, or other measurable feature, of the proverbial " Goodwin Sands " and " Tenterden Steeple," and the first measurement of one set was coupled with the first of the other set, the second with the second, and so on, the pairs of magnitudes thus presented would doubtless vary according to the above-written law, only in that case r would presumably be zero; the expression for z would reduce to the product of the two independent probabilities that particular values of x and y should concur. But slight interdependences between things supposed to be totally unconnected would often be discovered by this law of error in two or more dimensions.' It may be put in a more convenient form by substituting E for (x - a)/ A t k and n for (y - b)/-1 m. The equation of the surface then becomes z = (I /27rA/ I - r 2) exp - [E 2 - 2 r17 + 7 7 2 ] /2 (I - r2). If the frequency of observations in the vicinity of a point is represented by the number of dots in a small increment of area, when r= o the dots will be distributed uniformly about the origin, the curves of equal probability will be circles. When r is different from zero 1 Above, par. 114, and below, par. 127.

2 Some plurality of independent causes is presumable.

3 Herschel's a priori proposition concerning the law of error in two dimensions (above, par. 99) might still be defended either as generally true, so many phenomena showing no trace of interdependence, or on the principle which justifies our putting z for a probability that is unknown (above, par. 6), or 5 for a decimal place that is neglected; correlation being equally likely to be positive or negative. The latter sort of explanation may be offered for the less serious contrast between the a priori and the empirical proof of the law of error in one dimension (below, par. 158).

4 Cf. above, par. 115.

the dots will be distributed so that the majority will be massed in two quadrants: in those for which and n are both positive or both negative when r is positive, in those for which and n have opposite signs when r is negative. In the limiting case, when r = I the whole host will be massed along the line n =i, every deviation E being attended with an equal deviation n. In general, to any deviation of one of the variables ' there corresponds a set or " array " (Pearson) of values of the other variable; for which the frequency is given by substituting ' for in the general equation. The section thus obtained proves to be a normal probability-curve with standard deviation A / (I - r 2). The most probable value of n corresponding to the assigned value of i is W. The equation n--- r, or rather what it becomes when translated back to our original co-ordinates (y - b)/o = r(x - a)oi, where o, 02 are our 1/ k, m respectively,' is often called a regression-equation. A verification is to hand in the abovecited statistics, which Weldon obtained by casting batches of dice. If the dice were perfect, r (=l/ A ' km) would equal 2, and as the dice proved not to be very imperfect, the coefficient is doubtless approximately =1. Accordingly, we may expect that, if axes x and y are drawn through the point of maximum-frequency at the centre of the compartment containing 244 observations, corresponding to any value of x, say 2vi (where i is the side of each square compartment), the most probable value of y should be vi, and corresponding to y = 2vi the most probable value of x should be vi. And in fact these regression-equations are fairly well fulfilled for the integer values of v (more than which could not be expected from discrete observations): e.g. when x = +41, the value of y, for which the frequency (25) is a maximum, is as it ought to be + 21; when x= - 21 the maximum (119) is at y= - i; when x= - 41 the maximum (16) is at y= - 21; when y is + 21 the maximum (138) is at x= +i; when y is-21 the maximum (117) at x= - i, and in the two cases (x= +21 and y= +41), where the fulfilment is not exact, the failure is not very serious.

128. Analogous statements hold good for the case of three or more dimensions of errors The normal law of error for any number of variables, x i x 2 x3, may be put in the form z = (I (27r)n/2 1/ A) exp - [Rnx1 2 + R22x2 2 + &c. + 2R12x1x2 + &c.]/20 where A is the determinant: - r23 ' I

each r, e.g. r23 (= r32), is the coefficient ' of correlation between two of the variables, e.g. x 2, x3; Rll is the first minor of the determinant formed by omitting the first row and first column; is the first minor formed by omitting the second row and the second column, and so on; R21) is the first minor formed by omitting the first column and second row (or vice versa). The principle of correlation plays an important role in natural history. It has replaced the notion that there is a simple proportion between the size of organs by the appropriate conception that there are simple proportions existing between the deviation from the average of one organ and the most probable value for the coexistent deviation of the other organ from its average. ? Attributes favoured by " natural " or other selection are found to be correlated with other attributes which are not directly selected. The extent to which the attributes of an individual depend upon those of his ancestors as measured by correlation. 8 The principle is instrumental to most of the important " mathematical contributions " which Professor Pearson has made to the theory of evolution. 9 In social inquiries, also, the principle promises a rich harvest. Where numerous fluctuating causes go to produce a result like pauperism or immunity from small-pox, the ideal method of eliminating chance would be to construct " regression-equations " of the following type: " Change % in pauperism [in the decade 1871-1881] in rural districts= - 2 7.0 7%, +0.299 (change % out-relief ratio), +0.271 (change % on proportion of old), + 064 (change % in population)."10 129. In order to determine the best values of the coefficients involved in the law of error, and to test the worth of the results obtained by using any values, recourse must be had to inverse probability. 130. The simplest problem under this head is where the quaesitum is a single real object and the data consist of a large number of observations, x i, x 2,. n , such that if the number were indefinitely increased, the completed series would form a normal probability-curve with the true point as its centre, and having a given modulus c. It is as if we had observed the position of the dints made by the fragments 5 Cf. note to par. 98, above.

Phil. Mag. (1892), p. 200 seq.; 1896, p. 211; Pearson, Trans. Roy. Soc. (1896), 187, p. 302; Burbury, Phil. Mag. (1894), p. 145.

7 Pearson, " On the Reconstruction of Prehistoric Races," Trans. Roy. Soc. (1898), A, p. 174 seq.; Proc. Roy. Soc. (1898), P. 418.

8 Pearson, " The Law of Ancestral Heredity," Trans. Roy. Soc.; Proc. Roy. Soc. (1898).

9 Papers in the Royal Society since 1895.

io An example instructively discussed by Yule, Journ. Stat. Soc. (1899).

of an exploding shell so far as to know the distance of each mark measured (from an origin) along a right line, say the line of an extended fortification, and it was known that the shell was fired perpendicular to the fortification from a distant ridge parallel to the fortification, and that the shell was of a kind of which the fragments are scattered according to a normal law 1 with a known coefficient of dispersion; the question is at what position on the distant ridge was the enemy's gun probably placed ? By received principles the probability, say P, that the given set of observations should have resulted from measuring (or aiming at) an object of which the real position was between x and x +Ax' is Ax J exp - [(x - x1) 2 -{- (x - x 2) 2 + &c.]/c2; where J is a constant obtained by equating to unity r Pdx (since the given set of observations must have resulted from some position on the axis of x). The value of x, from which the given set of observations most probably resulted, is obtained by making P a maximum. Putting dP/dx = o, we have for the maximum (d 2 P/dx 2 being negative for this value) the arithmetic mean of the given observations. The accuracy of the determination is measured by a probability-curve with modulus chi n. This in the course of a very long siege if every case in which the given group of shell-marks x 1, x2,.. .' n was presented could be investigated, it would be found that the enemy's cannon was fired from the position x', the (point right opposite to the) arithmetic mean of x1, x2, &c., xn, with a frequency assigned by the equation z = (.l n/,/ 7rc) exp - n(x - x')2/c2. The reasoning is applicable without material modification to the case in which the data and the quaesitum are not absolute quantities, but proportions; for instance, given the percentage of white balls in several large batches drawn at random from an immense urn containing black and white balls, to find the percentage of white balls in the urn - the inverse problem associated with the name of Bayes.

131. Simple as this solution is, it is not the one which has most recommended itself to Laplace. He envisages the quaesitum not so much as that point which is most probably the real one, as that point which may most advantageously be put for the real one. In our illustration it is as if it were required to discover from a number of shot-marks not the point 2 which in the course of a long siege would be most frequently the position of the cannon which had scattered the observed fragments but the point which it would be best to treat as that position - to fire at, say, with a view of silencing the enemy's gun - having regard not so much to the frequency with which the direction adopted is right, as to the extent to which it is wrong in the long run. As the measure of the detriment of error, Laplace 3 takes " la valeur moyenne de l'erreur a craindre," the mean first power of the errors taken positively on each side of the real point. The mean spare of errors is proposed by Gauss as the criterion. 4 Any mean power indeed, the integral of any function which increases in absolute magnitude with the increase of its variable, taken as the measure of the detriment, will lead to the same conclusion, if the normal law prevails.' 132. Yet another speculative difficulty occurs in the simplest, and recurs in the more complicated inverse problem. In putting P as the probability, deduced from the observations that the real point for which they stand is x (between x and x +ox), it is tacitly assumed that prior to observation one value of x is as probable as another. In our illustration it must be assumed that the enemy's gun was as likely to be at one point as another of (a certain tract of) the ridge from which it was fired. If, apart from the evidence of the shell-marks, there was any reason for thinking that the gun was situated at one point rather than another, the formula would require to be modified. This a priori probability is sometimes grounded on our ignorance; according to another view, the procedure is justified by a rough general knowledge that over a tract of x for which P is sensible one value of x occurs about as often as another.' 1 If normally in any direction indifferently according to the twoor three-dimensioned law of error, then normally in one dimension when collected and distributed in belts perpendicular to a horizontal right line, as in the example cited below, par. 155.

2 Or small interval (cf. preceding section).

" Toute erreur soit positive soit negative doit titre consideree comme un desavantage ou une perte reelle a un jeu quelconque," Theorie analytique, art. 20 seq., especially art. 25. As to which it is acutely remarked by Bravais (op. cit. p. 258), " Cette regle simple laisse a desirer une demonstration rigoureuse, car l'analogue du cas actuel avec celui des jeux de hasard est loin d'être complete." 4 Theoria combinationis, pt. i. § 6. Simon Newcomb is conspicuous by walking in the way of Laplace and Gauss in his preference of the most advantageous to the most probable determinations. With Gauss he postulates that " the evil of an error is proportioned to the square of its magnitude " (American Journal of Mathematics, vol. viii. No. 4).

As argued by the present writer, Camb. Phil. Trans. (1885), vol. xiv. pt. ii. p. 161. Cf. Glaisher, Mem. Astronom. Soc. xxxix. 108.

' The view taken by the present writer on the " Philosophy of Chance," in Mind (1880; approved by Professor Pearson, Grammar 133. Subject to similar speculative difficulties, the solution which has been obtained may be extended to the analogous problem in which the quaesitum is not the real value of an observed magnitude, but the mean to which a series of statistics indefinitely prolonged converges.' 134. Next, let the modulus, still supposed given, not be the same for all the observations, but c 1 for x1, c 2 for x2, &c. Then P becomes proportional to exp - [(x - x 1) 2 / c 1 2 -{- (x - x 2) 2 c 2 And the value of x which is both the most probable and the " most advantageous" is (xi/c12+x2/c22+&c.)/(I/c12+I/c22+&c.); each observation being weighted with the inverse mean square of observations made under similar conditions.' This is the rule prescribed by the " method squaresof least squares "; but as the rule in this case has been deduced by genuine inverse probability, the problem does not exemplify what is most characteristic in that method, namely, that a rule deducible from the hypothesis that the errors of observations obey the normal law of error is employed in cases where the normal law is not known, or even is known not, to hold good. For example, let the curve of error for each observation be of the form of z = [1/ A ! (xc)] X exp [ - x 2 /c 2 - 2j (x/c - 2x3/3c3)], where j is a small fraction, so that z may equally well be equated to 7rc)[i - 2j(x/c - 3 /3c 3)] exp - x 2 /c 2 , a law which is actually very prevalent. Then, according to the genuine inverse method, the most probable value of x is given by the quadratic equation dx log P = o, where log P = const. - E (x - x,.) 2 /c,. 2 - E2j [ (x - x,.) 3 /c r 3 - 2(x - x r) 3 /3c r 3 ], denoting summation over all the observations. According to the " method of least squares," the solution is the weighted arithmetic mean of the observations, the weight of any observation being inversely proportional to the corresponding mean square, i.e. 6.2 /2 (the terms of the integral which involve j vanishing), which would be the solution if the j's are all zero. We put for the solution of the given case what is known to be the solution of an essentially different case. How can this paradox be justified?

135. Many of the answers which have been given to this question seem to come to this. When the data are unmanageable, it is legitimate to attend to a part thereof, and to determine the most probable (or the " most advantageous ") value of the quaesitum, and the degree of its accuracy, from the selected portion of the data as if it formed the whole. This throwing overboard of part of the data in order to utilize the remainder has often to be resorted to in the rough course of applied probabilities. Thus an insurance office only takes account of the age and some other simple attributes of its customers, though a better bargain might be made in particular cases by taking into account all available details. The nature of the method is particularly clear in the case where the given set of observations consists of several batches, the observations in any batch ranging under the same law of frequency with mean x', and mean square of error k r , the function and the constants different for different batches; then if we confine our attention to those parts of the data which are of the type x', and kr ignoring what else may be given as to the laws of error - we may treat the x' r .'s as so many observations, each ranging under the normal law of error with its coefficient of dispersion; and apply the rules proper to the normal law. Those rules applied to the data, considered as a set of derivative observations each formed by a batch of the original observations) averaged, give as the most probable (and also the most advantageous combination of the observations the arithmetic mean weighted according to the inverse mean square pertaining to each observation, and for the law of the error to which the determination is liable the normal law with standard deviation 9 A / (Ek/n) - the very rules that are prescribed by the method of least squares.

136. The principle involved might be illustrated by the proposal to make the economy of datum a little less rigid: to utilize, not indeed all, but a little more of our materials - not only the mean square of error for each batch, but also the mean cube of error. To begin with the simple case of a single homogenous batch: suppose that in our example the fragments of the shell are no longer scattered according to the normal law. By the method of least squares it would still be proper to put the arithmetic mean to the given observations for the true point required, and to measure the accuracy of that determination by a probability-curve of which the modulus is -V (2k), where k is the mean square of deviation (of fragments from their mean). If it is thought desirable to utilize more of the data there is available, the proposition that the arithmetic mean of a of Science, 2nd ed. p. 146). See also " A priori Probabilities," Phil. Mag. (Sept. 1884), and Camb. Phil. Trans. (1885), vol. xiv. pt. ii. p. 1 47 Seq.

7 Above, pars. 6, 7.

8 The mean square J +00 (x Ire) exp - x 2 /c 2 dx = c2/2.

9 The standard deviation pertaining to a set of (n/r) composite observations, each derived from the original n observations by averaging a batch thereof numbering r, is -J (k/r)/ 1 1 (n/r) = -j (k/n), when the given observations are all of the same weight; mutatis mutandis when the weights differ.

numerous set of observations, say x, (taken as a sample from an indefinitely large group obeying any the same law of frequency) varies from set to set approximately according to the following law (to be established later) z= ? c exp - [ c 2 2 + 2 nj c - 3 / = f(x), say; 3c where c 2 /2 the mean square of deviation, and j = the mean cube of deviation, and j/c 3 , say j, is small. Then, by abstraction analogous to that which has just been attributed to the method of least squares, we may regard the datum as a single observation, the arithmetic mean (of a sample batch of observations) subject to the law of error z = f (x). The most probable value of the quaesitum is therefore given by the equation f'(x - x') = o, where x' is the arithmetic mean of the given observations. From the resulting quadratic equation, putting x = x' + e, and recollecting that e is small we have e = jc. That is the correction due to the utilization of the mean cube of error. The most advantageous solution cannot now be determined,' f(x) being unsymmetrical, without assuming a particular form for the function of detriment. This method of least squares plus cubes may easily be extended to the case of several batches.

137. This application of probabilities not to the actual data but to a selected part thereof, this economy of the inverse method, is widely practised in miscellaneous statistics, where the object is to determine whether the discrepancy between two sets of observation is accidental or significant of a real difference. 2 For instance, let the data be ages at death of individuals of two classes (e.g. temperate or not so, urban or rural, &c.) who have been under observation, since the age of, say, 20. Granted that the ages at death conform to Gompertz's law; the determination of the modal age at death, that age at which the proportion of the total observed dying (per unit of time) is a maximum for each class, would most perfectly be effected by the genuine inverse method. That method will also enable us to determine the probability that the two modes should have differed to the observed extent by mere accident.' According to the abridged method it suffices to proceed as if our data consisted of two observations x' and y', the average ages at death of the two classes, each average obeying the normal law of error, with respective moduli c, = [(x' - x 1)' + (x' - x2) 2 + &c.12/n, c2 = [(y ' - y i) 2 + (y' - y2)' + &c.]2/n, where x,, x2,' &c., y l y2, &c., are the respective sets of observed ages at death; as follows from the law of error, whatever the law of distribution of the given observations. According to a well-known property of the normal law, the difference between the averages of n and n observations respectively will range under a probability-curve with modulus c 1 2 + c 2 2, say c. Whence for the probability that a difference as great as the observed one, say e, should have occurred by chance we have 2[1-0(r)], where T =e/c, and 0(x) is the integral 21) ir fo(exp = x 2)dx, given in many treatises.

138. This sort of abridgment may be extended to other kinds of average besides the arithmetic, in particular the median (that point d which has as many of the given observations above as below it). By simple induction we know that the median of a large sample of observations is a probable value for the true median; how probable is determined as follows from a selection of our data. First suppose that all the observations are of the same weight. If x were the true median, the probability that as many as jn + r of the observations should fall on either side of that point is given by the normal law for which the exponent is - 2r 2 /n. 4 This probability that the observed median will differ from the true one by a certain number of observations is connected with the probability that they will differ by a certain extent of the abscissa, by the proposition that the number of observations contained between the true and apparent median is equal to the small difference between them multiplied by the density of observations at the median - in the case of normal and generally symmetrical curves the greatest ordinate. This is the second datum we require to select. In the case of the normal curve it may be calculated from the modulus itself, determined by induction from a selection of data. If the observations are not all of the same worth, weight may be assigned by counting one observation as if it occurred oftener than another. This is the essence of Laplace's Method of Situation.5 1 The use of the cubes is also contrasted with that of the squares (only) in this respect: that it is no longer a matter of indifference how many of the original observations we assign to the batch of which the mean constitutes the single (compound) observation.

The object of the writer's paper on " Methods of Statistics " in the Jubilee number of the Journ. Stat. Soc. (1885).

3 See on the use of the inverse method to determine the mode of a group, the present writer's paper on " Probable Errors " in the Journ. Stat. Soc. (Sept. 1908).

Above, par. 103.

Theorie analytique, 2nd supp. p. 164. Mecanique celeste, bk. iii. art. 40; on which see the note in Bowdich's translation. The method may be extended to other percentiles. See Czuber, Beobachtungsfehler, § 58. Cf. Phil. Mag. (1886), p. 375; and Sheppard, 139. In its simplest form, where all the given observations are of equal weight, this method is of wide applicability. Compared with the genuine inverse method, it is always more convenient, seldom much less accurate, sometimes even more accurate. If the given observations obey the normal law, the precision of the median is less than the precision of the arithmetic mean by only some 25 a discrepancy not very serious where only a rough estimate of the worth of an average is required. If the observations do not obey the normal law - especially if the extremities are abnormally divergent - the precision of the median may be greater than that of the arithmetic mean.' 140. Yet another instance of the contrast between genuine and abridged inversion is afforded by the problem to determine the modulus as well as the mean for a set of observations known to obey the normal law; what the first problem? a on o becomes when the coefficient of dispersion is not given. Frequency - . By inverse probability we ought in that case, in addition to the equation dP/dx = o, to put dP/dc = o. Whence c 2 = 2 [(x ' - x,) 2 + (x' - x2) 2 + &c. + (x' - x.)2] In, and x' (xi + x 2 + &c. + x")/n. This solution differs from that which is often given in the textbooks, in that there, in the expression for c 2, (n - i) occurs in the denominator instead of n. The difference is explained by the fact that the authorities referred to determine c, not by genuine inversion, but by ordinary induction, by a condition which certainly would be fulfilled in the long run, but does not express the whole of our data; a condition in this respect like the equation of c to 1r(Ee)/n, where e is the difference (taken positively, without regard to its sign) between any observation and the arithmetic mean of all the observations .9 141. Of course the determination of the most probable value is. subject to the speculative difficulties proper to a priori probability: which are particularly striking in this case, as it appears equally natural to take as that constant, of which the values are a priori equally probable, k(=c 2 /2), or even" h(= 1/c 2), the measure of weight, as in fact Laplace has done;" yet no two of these assumptions can be exactly true.12 142. A more convenient determination is obtained from simple induction by equating the modulus to some datum of the observed group to which it would be equal if the group were complete - in particular to the distance from the median of some percentile (or point which marks off a certain percentage, e.g 25 of the given observations) multiplied by a factor corresponding to the percentile obtainable from a familiar table. Mr Sheppard has given an interesting proof 13 that we cannot by way of percentiles obtain such good 14 results for the frequency-constants as by the use of " the average and average square " [the method prescribed by inverse probability].

143. The same philosophical subtleties, with greater mathematical complications, meet us when we pass on to the case of two or more quaesita. The problem under this head which mainly exercised the older writers was to determine a number of - unknown quantities, given a larger number, n, of equations involving them.

144. Supposing the true values approximately known, by substituting the approximate values in the given equations and expanding according to Taylor's theorem, there will be obtained for the corrections, say x, y..., 'n linear equations of the form a,x+b l y

=f, a2x+b2y

f2, where each a and b is a known coefficient, and each f is a fallible observation. Suppose that the error to which each is liable obeys the normal law, and that the modulus pertaining to each observation is the same - which latter condition can be secured by multiplying each equation by a proper factor - then if x' and y' are the true values of the quaesita, the frequency with which (a,x' + b l y' - f l ) assumes different values is given by the equation z

7 Above, par. 134.

8 E.g. Airy, Theory of Errors, art. 60.

9 It is a nice point that the expression for c', which has (n - I) instead of n for denominator, though not the more probable, may yet be the more advantageous (supposing that there were any sensible difference between the two). Cf. Camb. Phil. Trans. (1885), vol. xiv. pt. ii. p. 165; and " Probable Errors," Journ. Stat. Soc. (June 1908).

10 Above, par. 96, note.

Theorie analytique, 2nd supp. ed. 18 47, p. 578.

12 See the matter discussed in Camb. Phil. Trans., loc. cit. 13 Trans. Roy. Soc. (1899), A, cxcii. 135.

14 Good as tested by a comparison of the mean squares of errors in the frequency-constant determined by the compared methods.

if not known beforehand, may be inferred, as in the simpler case, from a set of observations. Similar statements holding for the other equations, the probability that the given set of observations f2, &c., should have resulted from a particular system of values for x, y. . is J exp [(a l x {-b'y - f')2/c,2-} (a2x +b2y - f2)2/c22-+-&c.], where J is a constant determined on the same principle as in the analogous simpler cases.' The condition that P should be a maximum gives as many linear equations for the determination of x' y' ... as there are unknown quantities.

145. The solution proper to the case where the observations are known to arrange according to the normal law may be extended to numerous observations ranging under any law, on the principles which justify the use of the Method of Least Squares in the case of a single quaesitum. 146. As in that simple case, the principle of economy will now justify the use of the median, e.g. in the case of two quaesita, putting for the true values of x and y that point for which the sum of the perpendiculars let fall from it on each of a set of lines representing the given equations (properly weighted) is a minimum.' 147. The older writers have expressed the error in the determination of one of the variables without reference to the error in the other. But the error of one variable may be regarded Corl as correlated with that of another; that is, if the system x,, y ... forms the solution of the given equations, while x' +E, y' +7 7 ... is the real system, the (small) values of r?.... which will concur in the long run of systems from which the given set of observations result are normally correlated. From this point of view Bravais, in 1846, was led to several theorems which are applicable to the now more important case of correlation in which t and i are given (not in general small) deviations from the means of two or more correlated members (organs or attributes) forming a normal group. more .

correlate To determine the frequency-constants of such a group it is proper to proceed on the analogy of the simple case of one-dimensioned error. In the case of two dimensions, for instance, the probability p i that a given pair of observations (x i, y,) should have resulted from a normal group of which the means are x' y' respectively, the standard deviations a i and Q2 and the coefficient of correlation r, may be written OxAyOQ i e0- 2 0r(1 /27r) U1Q2 (I - r 2) exp - 1E2 where E 2 = (x' - x,) 2 /Q, 2 - 2r(x' - x i)(y ' - y i)/o102 + (y ' - yi)2/0-22. A similar statement holds for each other pair of observations (x2y2), (x3y3)... with analogous expressions for p2, p3... Whence, as in the simpler case, we have p ' X p2 X &c. X pn/J (a constant) for P, the a posteriori probability that the given observations should have resulted from an assigned system of the frequency-constants. The most probable system is determined by making P a maximum, and accordingly equating to zero each of the following expressionsdP dP dP dP dP dx, dy, d01, do-2, dr.

The values of the arithmetic mean and of the standard deviation for each variable are what have been obtained in the simple case of one dimension. The value of r is E (x' - x,.) (y' - y r) /Q,Q 2.3 The probable error of the determination is assigned on the assumption that the errors to which it is liable are small.' Such coefficients have already been calculated for a great number of interesting cases. For instance, the coefficient of correlation between the human stature and femur is o8, between the right and left femur is 0.96, between the statures of husbands and wives is o28.5 149. This application of inverse probability to determine correlation-coefficients and the error to which the determination is liable has been largely employed by Professor Pearson' and other recent writers. The use of the normal formula to measure the probable - and improbable - errors incident to such determinations is justified by reasoning akin to that which has been employed in the general proof of the law of error. Professor Pearson has pointed out a circumstance which seems to be of great importance in the theory of evolution: that the errors incident to the determination of different frequency-coefficients are apt to be mutually correlated. Thus if a random selection be made from a certain population, the correlation-coefficient which fits the organs of that set is apt to differ from the coefficient proper to the complete group in the same sense as some other frequency-coefficients.

150. The last remark applies also to the determination of the coefficients, in particular those of correlation, by abridged methods, on principles explained with reference to the simple case; for instance by the formula r = EnÆE, where EE is the sum of (some or all) the Above, par. 130.

2 See Phil. Mag. (1888), " On a New Method of Reducing Observations "; where a comparison in respect of convenience and accuracy with the received method is attempted.

Corresponding to the k/-Ilm of pars. 14, 127 above.

Pearson, Trans. Roy. Soc., A, 1 9 1, p. 234.

5 Pearson, Grammar of Science, 2nd ed. p. 402, 431.

6 Trans. Roy. Soc. (1898), A, vol. 191; Biometrika, ii. 273.

Above, par. 107. Compare the proof of the " Subsidiary Law of Error," as the law in this connexion may be called, in the paper on " Probable Errors," Journ. Stat. Soc. (June 1908).

positive (or the negative) deviations of the values for one organ or attribute measured by the modulus pertaining to that member, and zr) is the sum of the values of the other member, which are associated with the constituents of fit. This variety of this method is certainly much less troublesome, and is perhaps not much less accurate, than the method prescribed by genuine inversion.

151. A method of rejecting data analogous to the use of percentiles in one dimension is practised when, given the frequency of observations for each increment of area, e.g. each Ox Ay, we utilize only the frequency for integral areas. Mr Sheppard has given an elegant solution of the problem: to find the correlation between two attributes, given the medians L, and M, of a normal group for each attribute and the distribution of the total group, as thus.' FIG. 12.

If cos D is put for r, the coefficient of correlation, it is found that D= 7rR/(P-FR). For example, let the group of statistics relating to dice already 'cited from Professor Weldon be arranged in four quadrants by a horizontal and a vertical line, each of which separates the total groups into two halves: lines of which equations prove to be respectively y=6. 11 and x=6.156. For R we have 1360.5, and for P 687.5 roughly. Whence D= 7rXo66; r= cos o66 X7r = - 1 nearly, as it ought; the negative sign being required by the circumstance that the lower part of Mr Sheppard's diagram shown in fig. 12 corresponds to the upper part of Professor Weldon's diagram shown in par. 115.

152. Necessity rather than convenience is sometimes the motive for resort to percentiles. Professor Pearson has applied the median method to determine the correlation between husbands and wives in respect of the darkness of eye-colour, a character which does not admit of exact graduation: " our numbers merely refer to certain groupings, arranged, it is true, in increasing darkness of colour, but in no way corresponding to equal increases in colour-intensity."'o From data of this sort, having ascertained the number of husbands with eye-colours above the median tint who marry wives with eyecolour above the median tint, Professor Pearson finds for r the coefficient of correlation -}-o 1. A general method for determining the frequency-constants when the data are, or are taken to be, of the integral sort has been given by Professor Pearson." Attention should also be called to Mr Yule's treatment of the problem by a sort of logical calculus on the lines of Boole and Jevons." 153. In the cases of correlation which have been so far considered, it has been presupposed that the things correlated range according to the normal law of error. But now, suppose the law A of distribution to be no longer normal: for instance, that the dots on the plane of 13 representing each a pair of members, are no longer grouped in elliptic (or circular) rings of equal frequency, that the locus of the maximum y deviation, corresponding to an assigned x deviation, is no longer a right line. How is the interdependence of these deviations to be formulated? It is submitted that such data may be treated as if they were normal: by an extension of the Method of Least Squares, in two or more dimensions. 14. Thus when the amount of pauperism together with the amount of outdoor relief is plotted in several unions there is obtained a distribution far from normal. Nevertheless if the average pauperism and average outdoor relief are taken for aggregates - say quintettes or decades - of unions taken at random, it may be expected that these means will conform to the normal law, with coefficients obtained from the original data, according to the rule which is proper to the case of the normal law. 15 By obtaining averages conforming to the normal law, as by the simple application of the method of least squares, we should not indeed have utilized the whole of our data, but we shall put a part of it in a very useful 8 Trans. Roy. Soc. (1899), A, 192, p. 141.

9 Above, par. 115.

"Grammar of Science, P. 432.

11 Trans. Roy. Soc., A, vol. 195. In this connexion reference should also be made to Pearson's theory of " Contingency " in his thirteenth contribution to the " Mathematical Theory of Evolution " (Drapers' Company Research Memoirs). '2 Trans. Roy. Soc. (190o), A, 1 94, p. 2 57; (1901), A, 197, p. 91.

13 Above, par. 127.

14 Above, par. 116.

" If from the given set of n observations (each corresponding to a point on the plane xy) there is derived a set of n/s observations each obtained by averaging a batch numbering s of the original observation; the coefficient of correlation for the derived system is the same as that which pertains to the original system. As to the standard deviation for the new system see note to par. 135.

shape. Although the regression-equations obtained would not accurately fit the original material, yet they would have a certain correspondence thereto. What sort of correspondence may be illustrated by an example in games of chance, which Professor Weldon kindly supplied. Three half-dozen of dice having been thrown, the number of dice with more than three points in that dozen which is made up of the first and the second half-dozen is taken for y, the number of sixes in the dozen made up to the first and the third half-dozen, is taken for x. Thus each twofold observation (xy) is the sum of six twofold elements, each of which is subject to a law of frequency represented in fig. 13; where 1 the figures outside denote the number of successes of each kind, for the ordinate the number of dice with n t more than three points (out of a cast FIG. 13. of two dice), for the co-ordinate the number of sixes (out of a cast of two dice, one of which is common to the aforesaid cast); and the figures inside denote the comparative probabilities of each twofold value (e.g. the probability of obtaining in the first two cast dice each with more than three points, and in the second cast two sixes, is 1/72). Treating this law of frequency according to the rule which is proper to the normal law, we have (for the element) if the sides of the compartments each =i a 1 =ix/s118; o.2=i/V2; r=i/-N120.

Whence for the regression-equation which gives the value of the ordinate most probably associated with an assigned value of the abscissa we have y=xXr6 2 /0 1 =o3x; and for the other regressionequation, x=y/6. Accordingly, in Professor Weldon's statistics, which are reproduced in the annexed diagram, when x =3 the 7 2 72 p most probable value of y ought to be 1. And in fact this expectation is verified, x and y being measured along lines drawn through the centre of the compartment, which ought to have the maximum of content, representing the concurrence of one dozen with two sixes and another dozen with six dice having each more than three points, the compartment which in fact contains 254 (almost the maximum content). In the absence of observations at x = - 31 or y= 61, the regression-equations cannot be further verified. At least they have begun to be verified by batches composed of six elements, whereas they are not verifiable at all for the simple elements. The normal formula describes the given statistics as they behave, not when by themselves, but when massed in crowds: the regressionequation does not tell us that if x' is the magnitude of one member the most probable magnitude of the other member associated therewith is rx', but that if x' is the average of several samples of the first member, then rx' is the most probable average for the specimens of the other member associated with those samples. Mr Yule's proposal to construct regression-equations according to the normal rule " without troubling to investigate the normality of the distribution " 2 admits of this among other explanations. 3 Mr Yule's own view of the subject is well worthy of attention.

1 Cf. above, par. 115.2 Proc. Roy. Soc., vol. 60, p. 477. Below, par. 168.

154. In the determination of the standard-deviation proper to the law of error (and other constants proper to other laws of frequency) it commonly happens that besides the inaccuracy, which has been estimated, due to the paucity of the data, there is an inaccuracy due to their discrete charac ter: the circumstance that measurement, e.g. of human heights, are given in comparatively large units, e.g. inches, while the real objects are more perfectly graduated. Mr Sheppard has prescribed a remedy for this imperfection. For the standard deviation let / 2 2 be the rough value obtained on the supposition that the observations are massed at intervals of unit length (not spread out continuously, as ideal measurements would be); then the proper value, the mean integral of deviation squared, say (/22) =µ2 - ,h 2, where h is the size of a unit, e.g. an inch. It is not to be objected to this correction that it becomes nugatory when it is less than the probable error to which the measurement is liable on account of the paucity of observations. For, as the correction is always in one direction, that of subtraction, it tends in the long run to be advantageous even though masked in particular instances by larger fluctuating errors.' 155. Professor Pearson has given a beautiful application of the theory of correlation to test the empirical evidence that a given group conforms to a proposed formula, e.g. the normal law of error.' Supposing the constants of the proposed function to Criterion of be known - in the case of the normal law the arithEmpirical metic mean and modulus - we could determine the verification. position of any percentile, e.g. the median, say a. Now the probability that if any sample numbering n were taken at random from the complete group, the median of the sample, a', would lie at such a distance from a that there should be r observations between 00 a and a' is A / 2/71-n exp f If, then, any observed set has an excess which makes the above written integral very small, the set has probably not been formed by a random selection from the supposed given complete group. To extend this method to the case of two, or generally n, percentiles, forming (n I) compartments, it must be observed that the excesses say e and e', are not independent but correlated. To measure the probability of obtaining a pair of excesses respectively as large as e and e', we have now (corresponding to the extremity of the probability-curve in the simple case) the solid content of a certain probability-surface outside the curve of equal probability which passes through the points on the plane xy assigned by e, e' (and the other data). This double, or in general multiple, integral, say P, is expressed by Professor Pearson with great elegance in terms of the quadratic factor, called by him x 2, which forms the exponent of the expression for the probability that a particular system of the values of the correlated e, e', &c., should concur - P=1/2/?r - 1x2d x I e - 'x2 + i 3 + I x n n - 2 J X L .3 () when n is odd; with an expression different in form, but nearly coincident in result, when n is even. The practical rule derived from this general theorem may thus be stated. Find from the given observations the probable values of the coefficients pertaining to the formula which is supposed to represent the observations. Calculate from the coefficients a certain number, say n, of percentiles; thereby dividing the given set into n -fi sections, any of which, according to calculation, ought to contain say m of the observations, while in fact it contains m'. Put e for m' - m; then x 2 =Ee l /m. Professor Pearson has given in an appended table the values of P corresponding to values of n -}- i up to 20, and values of x 2 up to 70. He does not conceal that there is some laxity involved in the circumstance that the coefficients employed are not known exactly, only inferred with probability.' 156. Here is one of Professor Pearson's illustrations. The table on next page gives the distribution of moo shots fired at a line in a target, the hits being arranged in belts drawn on the target parallel to the line. The " normal distribution " is obtained from a normal curve, of which the coefficients are determined from the observations. From the value of x 2, viz. 45.8, and of (n I), viz. II, we deduce, with sufficient accuracy from Professor Pearson's table, or more exactly from the formula on which the table is based, that P = 000,001,5 other words, if shots are distributed on a target according to the normal law, then such a distribution as that cited could only be expected to occur on an average some 15 or 16 times in io,000,000 times." 157. " Such a distribution " in this argument must be interpreted as a distribution for which it is claimed that the observations are all independent of each other. Suppose that there were only 500 independent observations, the Criticized. remainder being merely duplicates of these 500. Then in the above 4 Just as the removal of a tax tends to be in the long run beneficial to the consumer, though the benefit on any particular occasion may be masked by fluctuations of price due to other causes.

' Phil. Mag. (July, 1900). 6 As shown above, par. 103. Loc. cit. p. 166.

table the columns for the normal distribution and for the discrepancy e should each be halved; and accordingly the column for e 2 /m should be halved. Thus e 2 /m being reduced to 22.9, P as found from Professor Pearson's table is between 995 and 629. That is, such a distribution might be expected to occur once on an average some once or twice in a hundred times. If actual duplication of this sort is not common in statistics,' yet in all such applications of the 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/c 3)[x/c - 2x3/3c3] where c 2 /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 1 is the mean cube of deviations for the compound and the sum of the mean cubes for the components, and the elements are supposed to be such and so numerous that k l /c 3 is of the order i/- S l n. This second approximation, first given by Poisson, was rediscovered by De Forest. 3 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' r z 117rc (exp - C 2) [I - 2k (--3- p C) + k2 (-3+10 - 3?4-{-9? 6) +k2(2-2?2+3?4)' where k = k l /c 3, k 2 = k 2 /c 4, 1 and c are defined as above, k 2 is the sum of the respective differences for each element between its mean fourth power of error and thrice its mean square of error, 6 and also the corresponding difference for the compound. The formula may be verified by the case of the binomial, considered as a simple case of the law of great numbers. Here c 2 = 2npq, k, = npq (q - p), k2 = npq (1 - 6pq).7 It is frequent in the statistics of wages.

2 See on this subject, in addition to the paper on the " Law of Error " already cited (Camb. Phil. Trans., 1905), another paper by the present writer, on " The Generalized Law of Error," in the Journ. Stat. Soc. (September, 1906).

The Analyst (Iowa), vol. ix.

4 Phil. Mag (Feb.1896) and Camb. Phil. Trans. (1905).

The part of the third approximation affected with k 2 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 4 new partial differential equation = 2 4 d- - x4 , where k 2, is the differ ence 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.

6 A4-3/22 2 in the notation which Professor Pearson has made familiar.

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

These values being substituted for the coefficients in the general formula, there results an expression which may be obtained directly by continuing 8 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, 9 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.'° The general expression for y which (multiplied by o x) her represents the probability that an error will occur at a particular point (within a particular small interval) may be written l r l e - k1 3, (d) 3+k4, (') 4 -. .. +(- I)tk t (t+2), d ' / t +2. .. Ye, where yo is (the normal error function) 1 ex2/2k, k is the ? (2k7r) mean square of deviation; k 1, k 2, ... , &c., are coefficients formed from the mean powers of deviation according to the rule that t is the difference between the tth mean power as it actually is and what it would be if the (t - i)th approximation were perfectly correct. Thus kl 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 ,u3, and zero, that is Similarly k 2 is the difference between the actual mean fourth power of deviation, say and what that mean power would be if the second approximation were perfectly correct, viz. 3k 2. Thus k2=i-t3-3k2. The series k,, k 31 k5, &c., k, k 2, k 4, &c., form each a succession of terms descending in the order of magnitude, when each k, e.g. /e t has been divided by the corresponding power, i.e. the power (t+2) of the parameter or modulus c= A/ (2k), which division is secured by the successive differ d t+2.

entiations of yo, with which each k is associated, e.g. le t with (T-c) Moreover, the first term of the odd series of k's when divided by the proper power of the parameter, viz. c 3 is small in comparison with the first term of the even series, viz. k, properly referred - divided by c 2 (= 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. 11 The curve of error, even when generalized as here proposed, coincides only with the central of the portion - the body, as distinguished from the extremities - of the actual locus; a greater or less proportion.

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 I d3zo d3zo d3zo d3zo zo - 3 ,, (3,0 k dx3 2 ,1 dx2dy + 31,2 k dxdy2 + k 40) ' where zo is (the normal error-function) I (x2 - 2rxy +y2) i 2) exp. (i - r 2) ' 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; 31 ok is the mean value of the cubes x 3, 211 k is the mean value of the products x 2 y, 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 approximation 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 8 Above, § 133, 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.

9 Journ. Stat. Soc. (1899), p. 55 o, referring to Pearson, Trans. Roy. Soc. (1898), A.

to 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.

11 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.

[[[Laws Of Frequency]] differential to zero (the second differential being always negative). The resulting equation is of the form y-rx-T-ax2-2sxy-yy2=o, where T, a, y 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/ ?) exp -(x-a)2/c12+N(I/s/7fc2) exp - (x - b)2/C22, where a and 1 3 are the proportionate sizes of the two groups (a+# = I); a and b are the respective centres of gravity; and c1, c2 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 prima 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 k 1, k 2, k 3', &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 = i ,f lc. Even in this case k 2 is only -o125; 1: 4 is 0.25 (the odd k's are zero).

Section II. - Laws of Frequency. 165. A formula much more comprehensive than the corrected normal law is proposed by Professor Pearson under the designation of the " generalized probability-curve." - The ground and scope of the new law cannot be better stated than in the words of the author: " The slope of the normal curve is given by a relation the form Curve." dy x y dx c1' The slope of the curve correlated to the skew binomial, as the normal curve to the symmetrical binomial, is given by a relation of the form dy _y dx c i +c2x 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), lxii. 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).

dy_ x y dx c 1 i c2x+c3x2 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." 2 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. 4 " 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 starts from zero, increases slowly or rapidly to a maximum and then falls again to zero - probably at a quite different rate - as the character for which the frequency is measured is steadily increased. This is the almost universal unimodal distribution of the frequency of homogeneous series.. (ii.) 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 y&x measure the frequency falling between x and 'x+Sx:- ' _ 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 dy _ x+a y dx - bo-F-b i x+b 2 x 2 + .. .

Experience shows that the form (x) [" keeping bo, b 1, 2 , only "] suffices for certainly the great bulk of frequency distributions." 5 166. The " generalized probability-curve " presents two main forms S' y= yo(I +I/ai)vai) v tan-lx/a. and y =yo (I +x2/a2)14 e When al, a 2, v 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 1 =a 21 a symmetrical curve, with range limited in both directions. If a 2 = co, the curve reduces to y = Y0(2 +x/al va ll a - vx); representing an asymmetrical binomial with v=2 / ./ 2 44 3 , and 2 1 = 2m 2 2 /m 3 - am 3 /m 2, µ2 and µ3, being respectively the mean second and mean third power of deviation measured from the centre of gravity. In the particular case, when µ3 is small, this form reduces to what is above called the " quasi-normal " curve; and when µ3 is zero, a l 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 FIG. 14. FIG. 15.

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.8 Professor Pearson has successfully applied these formulae to a number of beautiful specimens culled in the most diverse fields of statistics. 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. 381.3 Ibid. p. 360.

4 " Mathematical Contributions to the Theory of Evolution " (Drapers' Company Research Memoirs, Biometric Series II.), xiv. 4.

5 p. 7, loc. cit. 6 Ibid. p. 367.

7 Pearson, loc. cit., p. 364, and Proc. Roy. Soc. 8 A lucid exposition of Professor Pearson's various methods is given by W. Palin Elderton in Frequency-curves and Correlation (1906).

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

method to frequency-loci of two dimensions;' constructing for the curve of regression (as a substitute for the normal right line), in the case of " skew correlation," a parabola, 2 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.3 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.

169. There is possibly an extensive class of phenomena of which frequency depends largely on fortuitous causes, yet not so completely as to present the genuine law of error.' This mixed class of phenomena. might be amenable to a kind of law of frequency that would be different from, yet have some affinity to, the law of error. bify. 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," his type iii. of the form y = yo (1 -{- x/a) ya e - Yx.

The statistics also lend themselves to the Gompertz-Makeham formula for the number living at the age l = S x g x / c. z 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. 6 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 Makeham follows 8 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 Theorie analytique des probabilites. 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, ' " Contributions," No. xiv. (above cited).

2 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.

4 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.

5 " 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.

Generating Functions." Not all parts of the book are as rewarding. as the Introduction (published separately as Essai philosophique des probabilites) and the fourth and subsequent chapters of the second 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 probabilites (1889) abounds in choice examples, while it excels in almost every other branch of the subject. Special mention is also deserved by H. Poincare's Calcul des probabilites (lecons professes, 5893-5894). On local dr geometrical probability Professor 1Vlorgan Crofton is one of the 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 and Mittelworten (1884). Czuber is also to be mentioned as the author of Theorie der Beobachtungsfehler, in which he has reproduced, often 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 leads the beginner easily, yet far. References to other writers are given in Section I. of Part II. above. A list of writings on the cognate 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 Probabilities. (F. Y. E.9)