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LAWS OF

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Originally appearing in Volume V22, Page 395 of the 1911 Encyclopedia Britannica.
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LAWS OF ERRORI See also:Section I.—The See also:Law of See also: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 See also:equation 1 (a-See also:ari'' z -sire more conveniently written (1/11;c) exp-(x-a)2/c2, where x is the magnitude of an observation or " statistic," z is the proportional frequency of observations measuring x, a is the See also:arithmetic mean of the See also:group (supposed indefinitely' multiplied) of similar See also:statistics: c is a See also:constant sometimes called the " modulus"' proper to the group; and the equation signifies that if any large number N of such a group is taken at See also:random, the number of observations between x and x+tx is (approximately) equal to the right-See also:hand See also:side of the equation multiplied by Nox. A graphical See also:representation of the corresponding See also:curve—sometimes called the See also:probability-curve "—is here given (fig. Io), showing the See also:general shape of the curve, and how its dimensions vary with the magnitude of the modulus c. The See also: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 See also:form suited for calculations of probability, the variable, r, being the length of the See also:abscissa referred to (divided by) the modulus.' It may be noted that the points of See also:inflexion in the figure are each at a distance from the origin of 1/See also:J2 modulus, a distance equal to the square See also:foot of the mean square of error—often called the " See also: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 r which corresponds to this point is 04769. . . . Y 0 99. An a priori See also:proof of this law was given by See also:Herschel' as follows: " The probability of an error depends solely on its magni- A See also:prior! tude and not on its direction;" See also:positive and negative proof. errors are equally probable. ` Suppose a See also:ball dropped from a given height with the intention that it should fall on a given See also:mark," errors in all directions are equally probable, and errors in perpendicular directions are See also: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," is for one See also:dimension of the form 4,(x2), for two dimen- ' On this conception see below, See also:par.

122. 2 E.g. in the See also:

article on " Probability " in the 9th ed. of the Ency. Brit. ; also by See also:Airy and other authorities. Bravais, in his article Sur la probabilite See also:des erreurs. . . . " Memoires presentes par See also: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 See also:common tables, and in connexion with the operator, as below, par. 160; k( = lc2) when we investigate the formation of the probability-curve out of independent elements (below, par. 104) ; h(= I/c2) when we are concerned with weights or precisions (below, par. 134).

If one form of the coefficient must be uniformly adhered to, probably, a( =chi 2), for which See also:

Professor See also:Pearson expresses a preference, appears the best. It is called by him the " standard deviation." ' See also:Fuller tables are to be found in many accessible See also:treatises. See also:Burgess's tables in the Trans. of the Edin. See also:Roy. See also:Soc. for 1900 are carried to a high degree of accuracy. Thorndike, in his See also:Mental and Social Measurements, gives, among other useful tables, one referred to the standard deviation as the See also:argument. New tables of the probability integral are given by W. F. See also:Sheppard, Biometrics, ii. 174 seq. ' See also:Edinburgh See also:Review (1850), xcii. 19.

' The italics are in the See also:

original. The passage continues: " And391 sions 4'(x2 + y2) ; and ¢ (x2 + y2) = ¢ (x2) X 4,(y2) ; a functional equation of which the See also:solution is the See also:function above written. A See also:reason which satisfied Herschel is entitled to See also:attention, especially if it is endorsed by See also:Thomson and See also:Tait.' But it must be confessed that the claim to universality is not, without some See also:strain of See also:interpretation,' to be reconciled with common experience. Table of the Values of the Integral I = 10 a i2dx. r I T I r I r I 0'00 0 .00000 •2 •22270 I.3 93401 2.4 99931 •01 •01128 •3 •32863 I '4 '95229 2.5 •99959 •02 •02256 4 •42839 I.5 •96611 2.6 •99976 .03 •03384 .5 .52050 I.6 '97635 2'7 99986 •04 •04511 •6 •60386 I.9 .98379 2.8 99992 •05 •05637 •7 •67780 I.8 •98909 2.9 99996 •o6 •06762 •8 •74210 I.9 99279 3.0 '99998 •07 •07886 .9 '79691 2'0 99532 00 1.00000 •o8 •09008 1.0 •84270 2.1 •99702 •09 •10128 1 I •88020 2.2 •99814 •I •11246 1 •2 •91031 2.3 99886 See also:loo. There is, however, one class of phenomena to which Herschel's reasoning applies without See also:reservation. In a " molecular See also:chaos," such as the received kinetic theory of gases postulates, if a See also:molecule be placed at See also:rest at a given point and the distance which it travels from that point in a given See also: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 See also:Maxwell, in his first approach to the theory of molecular See also:motion, to establish the law of error in that region. tot. The See also:Laplace-See also:Quetelet See also: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 See also:apt to be assumed by a variable magnitude dependent on a See also:great number of independent variables, each of which assumes different values in random See also: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 See also:world of atoms, in organic and in social See also: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 See also:

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 See also:simple a formulas for the See also:action of the causes. The hypothesis seems first to have been stated in all its generality both of mathematical theory and statistical exemplification by See also:Glaisher.9 102. The validity of the explanation may best be tested by first (A) deducing the law of error from the See also:condition of numerous independent causes; and (B) showing that the law is (A) Deducwhich fulfilled in a variety of See also:concrete cases, in (A) from which the condition is probably See also:present. The See also:con- Hypodition may be supposed to be perfectly fulfilled in See also:games thet/ca! of See also:chance, or, more generally, sortitions, characterized by Condithe circumstance that we have a knowledge prior to !ions. specific experience of the proportion of what Laplace calls favourable cases10 to all cases—a See also:category which includes, for instance, the See also:distribution of digits obtained by random extracts from mathematical tables, as well as the distribution of the See also:numbers of points on See also:dominoes. 103. The See also:genesis of the law of error is most clearly illustrated by the simplest sort of " See also:game," that in which the sortition is between two alternatives, heads or tails, See also:hearts or not-hearts, or, generally, success or failure, the probability of a success being p and that of a failure q, where p + = 1. The number of aames of such successes in the course of- n trials may be con- cries sidered as an aggregate made up of n independently varying elements, each of which assumes the values o or 1 with respective frequency q and p. The frequency of each value of the it is on this See also:ignorance, and not on any peculiarity in cases, that the See also:idea of probability in the abstract is formed." Cf. above, par. 6. ' Natural See also:Philosophy, pt. i. See also:art.

391. For other a priori proofs see Czuber, Theorie der Beobachtungsfehler, th. i. Cf. See also:

note to par. 127. a He considered the effect as the sum of causes each of which obeys the simplest law of frequency, the symmetrical See also:binomial. ° See also:Memoirs of Astronomical Society (1878), p. 105. Cf. See also:Morgan Crofton, " On the Law of Errors of Observation," Trans. Roy. Soc. (1870), vol. clx. pt. i. p.

178. 10 Above, par. 2. +x aggregate is given by a corresponding See also:

term in the expansion of (q+p), and by a well-known theorem ' this term is approximately equal to -v2/2npq r2npge ; where v is the number of integers by which the term is distant from np (or an integer See also:close to np) ; provided that v is of (or <) the See also:order ,/n. Graphically, let the sortition made for each See also:element be represented by the taking or not taking with respective frequency p and q a step of length i. If a See also: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 above-written law of frequency, vi being the distance of the stopping-point from npi. Put vi=x and 2npgi2=c2; then the probability may be written (I/J-T c) exp-x2/c2. 104. It is a See also:short step, but a difficult one, from this See also: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 See also: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, z=fr(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 Nzhx, if z = (I/Jxk) exp-(x-a)2/2k; where a is the sum of the arithmetic means of all the elements, any one of which a, = [ f xfr(x)dx] , the square brackets denoting that the integrations extend between the extreme limits of the element's range, if the frequency-See also:locus for each element is continuous, it being understood that [ f fr(x)dx] =1; and k is the sum of the mean squares of error for each element, _ [ f 2 fr(ar+E)dtj , if the frequency-locus for each element is con- tinuous, where a, is the arithmetic mean of one of the elements, and t the deviation of any value assumed by that' element from a„ denoting summation over all the elements. When the frequency-locus 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 See also: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=(1/dr5.83n) exp-(x-3.5)2/5.8n. The See also:

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 P'on. 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 See also: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.8 The necessary and sufficient postulate appears to be that the mean See also:powers of deviation for the elements, the second (above written) and the similarly formed third, See also:fourth, &c., powers (up to some assigned See also:power), should be finite.' 1o6. (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 See also:Stirling's and See also:Bernoulli's theorems, See also:Todhunter, See also:History. . . of Probability. 2 The statement includes the case of a linear function, since an element multiplied by a constant is still an element.

8 E.g. if the frequency-locus of each element were I/r(I+x2), extending to infinity in both directions. But See also:

extension to infinity would not be fatal, if the form of the element's locus were normal. For a fuller exposition and a See also:justification of many of the statements which follow, see the writer's See also:paper on " The Law of Error " in the Camb. Phil. Trans. (1905). 5 Loc. cit. pt. i. § 1. 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 See also:Poisson.' Some idea of this celebrated theory may be obtained from the following See also: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 n=0(), where the centre of gravity is the origin, and 0(+t)-=o(-t); the construction signifying that the probability of the element having a value i; (between say t-iAt and + IAt) is 4(t)0t. Square brackets denoting summation between extreme limits, put x(a) for [S4,(t)e - Ian 0t] where is an integer multiple of At (or Ox) = pOx, say.

Form the mth power of x(a). The coefficient of el/ - IarOx in (x(a))"' is the probability that the sum of the values of the m elements should be equal to rAx; a probability which is equal to Oxy,., where y is the See also:

ordinate of the locus representing the frequency of the See also:compound quantity (formed by the sum of the elements). Owing to the symmetry of the function ¢ the value of y,, will not be altered if we substitute far .x lan.x fore e -- , nor if we substitute i(e+.`'—See also:lar°x+ e _ ~/ _ 1arOx) that is See also:cos arLx. Thus (x(a))'" becomes a sum of terms of the form Axy, cos ardx, where y _,.=y+,.. Now multiply (x(a))'" thus expressed by cos tOxa, where, t being an integer, tix=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 &xy, 2(cos a(r+t)Ox+cos a(r-t)Ox). As every value of r-t (except zero) is matched by a value equal in See also:absolute magnitude, -r+t, and likewise every value of r+t is matched by value -r-t, the See also:series takes the form <1xyrB cos gaZx+Lxy,, where q has all possible integer values from I to the largest value of [r[8 increased by It[; and the term free from circular functions is the See also:equivalent of ~xyr cos a(r+t)Ax, when r= -t, together with &xyr cos a(r-t)Ax, when r=+l. Now substitute for atlx a new See also:symbol 13; and integrate with respect to $, the thus transformed (x(a))m cos tOxa between the limits ti =o and ,3=r. The integrals of all the terms which are of the form Axy,cos q13 will vanish, and there will be See also:left surviving only 1r Xy,. We thus obtain, as equal to it xy,, f { x($/See also:fix) }'"cos 430. Now See also:change the independent variable to a; then as dfl=daAx, Oxy,=Ax—f rl°xda(x(a))r" cos tOxa. r o Replacing tAx by x, and dividing both sides by ix, we have r/Ox y.= f o Sa(X(a))'" cos ax. Now expanding the cos ax which enters into the expression for x(a), we obtain x(a) =[S4,(a)]- 2 [s4,(a)See also:a2]x2+ II [s4(a)See also:a4]x' 4• Performing the summations indicated, we See also:express x(a) in terms of the mean powers of deviation for an element.

Whence x(a)' is expressible in terms of the mean powers of the compound locus. First and See also:

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 I z 1'rl''xdxe —la* cos ax+4l k2 f r/~xdxa4e 2a k cos ax- . where k2 is the coefficient defined below .9 Here r/L,x may be replaced by Go , since the finite difference Ax is small with respect to unity when the number of the elements is large; ° and thus the integrals involved become equateable to known definite integrals. If it were allowable to neglect all the terms of the series but the first the z expression would reduce toN/ (2rk)e-u /k, the normal law of error. 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." 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. '' Laplace, Theorie analytique ' des probabilites, bk. ii. ch. iv.; Poisson, Recherches sur la probabilite des jugements. See also: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. 8 The symbol () is used to denote absolute magnitude, See also:

abstraction being made of sign. " Below, pars. 159, 160. I° Loc. cit. app. I. 11 Loc. Cit. p. 53 and context. 108. (3) De See also:Forest' has given a proof unencumbered by imaginaries of what is the fundamental proposition in Laplace 's theory that, if a polynomial of the form Ao+Aiz+A2z'+ ... +A,nz"' be raised to the nth power and See also:expanded in the form Bo+B,z+B2z'+ ...

+Bm,.Z"" then the magnitudes of the B's in the neighbourhood of their maxi-mum (say B,) will be disposed in accordance with a " probability-curve," or normal law of error. 109. (4) Professor Morgan Crofton's original proof of the law of error is based on a datum obtained by observing the effect which the introduction of a new element produces on the frequency-locus for the aggregate of elements. It seems to be assumed, very properly, that the sought function involves as constants some at least of the mean powers of the aggregate, in particular the mean second power, say k. We may without loss of generality refer each of the elements (and accordingly the aggregate) to its respective centre of gravity. Then if y, =f(x), is the ordinate of the frequency-locus for the aggregate before taking in a new element, and y = ay the ordinate after that operation, by a well-known principle,' y+ay = [S~m()f(x — )AI:], where 71, =~,n(), 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) ay=2[SE''O'"( AtId . From the fundamental proposition that the mean square for the aggregate equals the sum of mean squares for the elements it follows that ISE'4),n(U)At] the mean second power of deviation for the mth element is equal to ak, the addition to k the mean second power of deviation for the aggregate. There is thus obtained a partial See also:

differential equation of the second order dy—,d'y (I) dk A subsidiary equation is (in effect) obtained by Professor Crofton from the See also:property that if the unit according to which the See also:axis of x is graduated is altered in any assigned ratio, there must bea corresponding alteration both of the ordinate expressing the frequency: of the aggregate and of the mean square of deviation for the See also:aggregation. By supposing the alteration indefinitely small he obtains a second partial differential equation, viz. (in the notation here adopted) y+xdx+ 2kdk =o. (2) From these two equations, regard being had to certian other conditions of the problem,' it is deducible that y=Ce-x'/2k, where Cis a constant of which the value is determined by the condition that co ' The See also:Analyst (See also:Iowa), vols. v., vi., vii. passim; and especially vi.

142 seq., vii. 172 seq. ' Morgan Crofton, loc. cit. p. 781, See also:

col. a. The principle has been used by the present writer in the Phil. Mag. (1883), xvi. 301. ' For a See also: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 See also:

reproduction of Crofton's proof as given in the 9th ed. of the Ency. Brit. (art. " Probability," 3 roof Loc. cit. pt. I. § 4; and app. 6. 6 Loc. Cit. p. 122 seq. vertex rises up.

The change in " spread " produced by the See also:

accession of new elements is illustrated by the transition from the high to the See also:low curve, in fig. to, in the case of a sum; in the case of an See also:average (arithmetic mean) by the See also: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 See also:ion fluctuating element; if the function may be expanded Extens in ascending powers of the variables, according to Non-linear See also:Taylor's theorem, and all the powers after the first Functions. may be neglected. The See also:matter is not so simple as it is often represented, when the variable elements may assume large, perhaps See also: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 functions xtension of two variables, x and y, assumes different pairs of to two or values according to a law of frequency rzr=frix, y), themore functions being in general different for different elements.Dimenslons. The frequency with which Q assumes values of the variables between x and +Ax and between y and y+Ay is zAxAy, if 1 _ m(x—a)'—2l(x — a)(q — b) + k(y — b)' z—2xJhm-l' exp 2(km—l') of x and y concurring. If i is the distance from o to I and FIG. it. from I to 2 on the abscissa, and is the corresponding distance on the ordinate, the mean of the values of x for the element—A¢, as we may say,—is s, and the corresponding mean square of See also:horizontal deviations is 412. Likewise Ab=i'; Am=; and Al=e(+iX+i'—iX —i') =;ii'. Accordingly, if n such elements are put together (if n steps of the See also:kind which the See also: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 See also:plane of xy, namely, x=ri and y=ri, will be reached, is given by the equation z=2a 3 exp—3n[(r—n)'i'—(r—n)(r'—n)ii'+(r'—u)i'2]". 115. A verification is afforded by a set of statistics obtained with dice by See also:Weldon, and here reproduced by his permission.

A success is in this experiment defined, not by obtaining a See also:

head when a See also:coin is tossed, but by obtaining a See also:face with more than three points on it when a die is tossed; the probabilities of the two events are the same, or rather would be if coins and dice were perfectly symmetrical' Professor Weldon virtually took six steps of the sort above described when, six painted dice having been thrown, he added the number of successes in that painted batch to the number of successes in another batch of six to form his x, and to the number of successes in a third batch of six to form his y. The result is represented in the annexed table, where each degree on the axis of x and y respectively corresponds to the i and i' of the preceding paragraphs, and i = i'. The observed frequencies being represented by numerals, a general See also:correspondence between the facts and the See also:formula is apparent. 6 Loc. cit. pt. ii. § 7. ' The second by Burbury, in Phil. Mag. (1894), See also:xxxvii. 145; the third by its author in the Analyst for 1881; and the See also:remainder by the present writer in Phil. Mag. (1896), xii. 247; and See also:Comb: Phil.

Trans. (1905), loc. Cit. " 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. f oo ydx=1. I Io. (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 ml elements, fluctuating according to the sought law of error. Let B be the sum of another set of elements m2 in number (ml and m2 both large). Then Q a quantity formed by adding together each pair of concurrent values presented by A and B must also conform to the law of error, since Q is the sum of mi+m2 elements. The general form which satisfies this condition of reproductivity is limited by other conditions to the normal law of error.* i i i. The See also:

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 varietiea the frequency of the compound magnitude spreads out Varini s from its centre. It is otherwise if instead of the simple FunctionL sum we consider the linear function formed by adding the m elements each multiplied by 1/m. The ' spread " of the average thus constituted will continually diminish as the number of the elements is increased; the sides closing in as the where, as in the simpler case, a=See also:Ear, a,. being the arithmetic mean of the values of x assumed in the See also:long run by one of the elements, b is the corresponding sum for values of y, and k = [// x — as)'fr(x, y)dxdy] m [//J''y — br)'f*(x, y)dxdy] = [ JJ (x — ar) (y — br)fr(x, y)dxdy] the summation extending over all the elements, and the integration between the extreme limits of each; supposing that the law of frequency for each element is contin- uous, otherwise summation is to be substituted for integration. For ex-ample, 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 pre- '7 sented 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 in-side probabilites of particular values o 4 a '3 7 8 0 The maximum frequency is, as it ought to be, at the point x=6i, y=6i'.

The See also:

density is particularly great along a See also: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 See also:angle 0 with x and y respectively, the curve which is traced on the plane of zX by its intersection with the See also:surface is of the form z=J exp—X2[k sine 0—21 cos 0 See also:sin 0+m cos' 0]/2(km-l2), a probability-curve which will be more or less spread out according as the See also:factor k sine 0—21 cos 0 sin 0+m cos' 0 is less or greater. Now this expression has a minimum or maximum when (k—m) sin 0—21 cos 20=0; a minimum when (k—m) cos 20+2 lsin 20 is positive, and a maximum when that criterion is negative; that is, in the present case, where k=m, a minimum when 0=;ar and a maximum when 0 ='-,r. 0 1 2 1 3 4 5 6 7 8 9 10 11 12 12 11 I I 5 I I 10 2 6 28 27 19 2 9 I 2 II 43 76 57 54 15 4 8 6 18 49 116 138 118 59 25 5 7 12 47 109 208 213 118 71 23 I 6 9 29 77 199 244 198 121 32 3 5 3 12 51 119 181 200 129 69 18 3 16 2 4 55 100 117 91 46 19 3 3 2 14 28 53 43 34 17 I 2 7 12 13 18 4 I I 1 2 4 I 2 I 0 i 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 See also:imitation of the probability-curve has been obtained by superposing three elements, each obeying a law of frequency quite different from the normal one,' namely, that simple law according to which one value of a variable occurs as frequently as another between the limits within which the variation is confined (y=1/2a, between limits x= +a, x= —a). If the component elements obey unsymmetrical laws of frequency, the compound will indeed be to some extent unsymmetrical, unlike the " normal " probability-curve. But, as the number of the elements is increased, the portion of the compound curve in the neighbourhood of its centre of gravity tends to be rounded off into the normal shape. The portion of the compound curve which is sensibly identical with a curve of the " normal " See also: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 See also: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 See also:

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 i Experiments in See also:part materia performed by A. D. Darbishire afford additional illustrations. See " Some Tables for illustrating Statistical Correlation," Mem. and Proc. See also:Man. Lit., and Phil. Soc., vol. li. pt. iii. 2 Journ. Scat. Soc.

(See also:

March 1900), p. 73, referring to See also:Burton, Phil. Meg. (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 See also: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 See also: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 See also:drawn with respect to some further conditions which the reasoning requires.

A See also:

limitation as to the range of the elements is not necessary when they are already normal, or even have a certain See also: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 See also: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 See also:ordinary experience. 117. Having deduced the genesis of the law of error from ideal conditions such as are attributed to perfectly See also:fair ,B) Verifica games of chance, we have next to inquire how far ttoa ofthethese conditions are realized and the law fulfilled inNorma,Law common experience. 118. Among important concrete cases errors of observation occupy a leading See also:place. The theory is brought to See also:bear on this case by the hypothesis that an error is the algebraic sum of Errors numerous elements, each varying according to a law of frequency special to itself. This hypothesis involves proper. two assumptions: (1) that an error is dependent on numerous independent causes; (2) that the function expressing that dependence can be treated as a linear function, by expanding in terms of ascending powers (of the elements) according to Taylor's theorem and neglecting higher powers, or otherwise. The first See also: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 See also:state of the observer's See also:eye and his physiological condition in general, the state of the See also:atmosphere, of the different parts of the See also: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 See also:

weight from his See also:appearance,• and even higher acts of See also:judgment.5 Aiming at an See also:object is an See also: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 See also:bull's-eye are apt to be distributed according to the normal law, whether in two dimensions on a See also:target or according to their horizontal deviations, as exhibited below (par. 156). A residual class comprises See also:miscellaneous statistics, See also: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 8 and statistics. 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 3 Memoirs of Astronomical Society (1878), p. 105. ' Journ. Stat. Soc. (1890), p.

462 seq. 5 E.g. the marking of the same See also:

work by different examiners. Ibid. 8 Lettres sur la theorie des probabilitis and Physique sociale. 7 E.g. the measurements of See also:Italian recruits, adduced in the Atlante statistico, published under the direction of the Ministero de Agricul- tura (See also:Rome, 1882); and Weldon's measurements of crabs, Proc. Roy. Soc. liv. 321; discussed by Pearson in the Trans. Roy. Soc. (1894), vol. clxxxv. A.

8 Biometrika, iii. 395. Cf. ibid. p. 141. exemplified by A. L. Bowley's grouping of the See also:

wages paid to different classes.' 119. The See also:division of concrete errors which has been proposed is not to be confounded with another twofold See also:classification, namely, A Variant observations which stand for a real See also:objective thing, and Alassifies- such statistics as are not thus representative of something See also:lion. outside themselves, See also: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 See also: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 See also:wind or " See also:drift." 120.

On the other hand, statistics, not purporting to represent a See also:

teal 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 See also:total See also: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.' There is unmistakably suggested a sorlition of the simplest type, with a real ascertainable relation between tha 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 See also:female (or to total) births. Some other instances are given by Lexis 4 and Westergaard.' 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 See also: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 See also:index-numbers. The percentage which represents the change in the value of See also:money between two epochs is seldom regarded as the See also:mere average change in the See also: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 See also:volume of See also:trade, or that of the See also:consumption of the working classes, of which the growth is measured by appropriate index-numbers,' the former due to See also:Bourne and See also:Sir See also:Robert Gillen," the latter to See also:George See also:Wood.10 122. 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 See also: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 See also: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.l3 ' Wages in the See also:United See also:Kingdom in the Nineteenth See also:Century; and art. " Wages " in the Ency.

Brit., loth ed., vol. xxxiii. 2 Phil. Mag. (1900), p. 168. Cf. Journ. Slat. Soc., See also:

Jubilee No., p. 192. Masser erscheinungen. 6 Grundzuge der Statistik.

Cf. Bowley, Elements of Statistics, P. 302. 6 Das Gesetz der kleinen Zahlen. See for other See also:

definitions See also:Report of the See also:British Association (1889), pp. 136 and 161, and compare See also:Walsh's exhaustive Measurement of General See also:Exchange-Value. ' Cf. Bowley, Elements of Statistics, ch. ix. Journ. Stat. Soc. (1874 and later).

Parly. Papers [C. 2247] and [C. 30791- " " Working-Class Progress since 186o," Journ. Stat. Soc. (1899), p. 639. 11 On this conception compare See also:

Venn, See also:Logic of Chance, chs. iii. and iv., and Sheppard, Proc. Lond. Math. Soc., p.

363 seq. '2 Laplace's 6th principle, Theorie analytique, intro. x. " 3 See above, pars. 13 and 14.Without that delicate instrument the See also:

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 App/See also:ica tends to be set up, has a certain speculative See also:interest." ,-The assumption of the law as a hypothesis is legiti- dtheonsaormal See also:mate. • When the presumption is confirmed by specific "fns, experience this knowledge is apt to be turned to Law. See also:account. It is usefully applied to the practice of gunnery,16 to determine the proportion of shots which under assigned conditions may be expected to See also:hit a See also:zone of given See also:size. The See also:expenditure of See also: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 See also:civil service commissioners have founded on this fact some See also:practical directions to examiners.

Apart from such See also:

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 See also:epoch, or more exactly for See also:England, 72, as " Nature's aim, the length of life for which she builds a man, the See also:dispersion on each side of this point being . . , nearly normal."" 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," inferred from the fulfilment of the normal law, may be some See also: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 See also: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?

End of Article: LAWS OF

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