PI|PPSll||l|P»4liiHr'^ 
 
CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 BOUGHT WITH THE INCOME 
 OF THE SAGE ENDOWMENT 
 FUND GIVEN IN 1891 BY 
 
 HENRY WILLIAMS SAGE 
 

 DATE DUE 
 
 
 
 Af^.f^^ ^JP T 
 
 
 
 mm »t 
 
 "' hJoU ftSr JL 
 
 "■"■"^ 
 
 
 MAri-^ 
 
 ttloQ i' r. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r: 
 
 j GAYLORD 
 
 
 
 PRINTED IN U.S. A. ,^ 
 
 * 
 
 Cornell University Library 
 HB201.W22 P9 
 
 The problem of estimation 
 
 olin 
 
 3 1924 032 610 879 
 
B Cornell University 
 M Library 
 
 The original of tliis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924032610879 
 
THE PROBLEM OF ESTIMATION 
 
THE PROBLEM OF 
 ESTIMATION 
 
 A SEVENTEENTH - CENTURY CONTRO- 
 VERSY AND ITS BEARING ON MODERN 
 STATISTICAL QUESTIONS, ESPECIALLY 
 INDEX-NUMBERS 
 
 CORREA MOYLAN WALSH 
 
 author of 
 "thb measuremeht^of geheral exchange-value." 
 
 LONDON 
 
 P. S. KING & SON, LTD. 
 
 ORCHARD HOUSE, WESTMINSTER 
 
 1921 
 
per 
 
 J. 
 
PREFACE 
 
 This little work has been written in the interest of harmony. 
 In mathematics disputes must soon come to an end, when the 
 one side is proved and the other disproved. And where 
 mathematics enter into economics, it would seem that little 
 room could be left for long-continued disputation. It is 
 therefore somewhat surprising that one economist after 
 another takes up the subject of index-numbers, potters over 
 it for a while, differs from the rest if he can, and then drops 
 it. And so nearly sixty years have gone by since Jevons 
 first brought mathetnatics to bear upon this question, and 
 still economists are at loggerheads over it. Yet index- 
 numbers involve the use of means and averages, and these 
 being a purely mathematical element, demonstration ought 
 soon to be reached, and then agreement should speedily 
 follow. 
 
CONTENTS 
 
 
 I. The Controversy .... 
 
 PAGE 
 
 I 
 
 II. Averages and Errors .... 
 
 . i8 
 
 The Criteria of Averages . 
 
 . i8 
 
 Estimates and Observations 
 
 • 36 
 
 The Measurement of Errors 
 
 • 51 
 
 III. Deviations and Variations 
 
 . 68 
 
 Axiometry ..... 
 
 . 69 
 
 Weighting 
 
 . 81 
 
 Index-numbers .... 
 
 • 91 
 
 ProbabiHties and the Median 
 
 . 107 
 
THE 
 
 PROBLEM OF ESTIMATION 
 
 {A Seventeenth-Century Controversy and its bearing on 
 Modern Statistical Questions, especially Index-Numbers.) 
 
 I.— THE CONTROVERSY. 
 
 Among the problems which it was customary for scientists 
 to set to one another two and three centuries ago, an interest- 
 ing one was started at Florence in 1627. A participant was 
 Gahleo, and in his collected works are usually included not 
 only his own letters on the subject, but those of two others, 
 a certain Nozzolini and the mathematician Castelli.^ 
 
 The question propounded was this : If a horse worth 
 100 crowns is estimated by one person at 1,000 and by 
 another at ten, which of these two estimates is the less 
 erroneous, or are they equally erroneous ? The problem 
 may be varied thus : If an article worth 100 crowns is esti- 
 mated by one person at 1,000, what lower estimate would be 
 equally erroneoxis ; or if the first estimate were ten, what 
 higher estimate would be equally erroneous r It admits also 
 of a further variation : one person estimates an article at 
 1,000 crowns, and another estimates it at ten; what must be 
 its value in order that these errors be equal ? Here it is 
 evident that the answer is some sort of a mean between 
 1,000 and ten. In the second form of the problem the mean 
 is given as 100, and the question concerns the extremes. 
 In the original form the question is what relation the proper 
 mean between 1,000 and ten has to the true value 100. 
 Thus the problem of estimation is a question of means, and 
 its solution involves the finding of the kind of mean suitable 
 for equalising errors above and below the true quantity. 
 
 ^ References here are to the Ofere di Galileo Galilei published 
 at Milan in 1832. The dispute occupies twenty finely printed, 
 double-columned, large octavo pages : Vol. I. pp. 57-77. 
 
2 THE PROBLEM OF ESTIMATION 
 
 Evidently, we may say at once, the harmonic mean is 
 ruled out of court ; for if one person estimates the article 
 at half or at less than half its true value, the other person 
 could not equal his error, according to this mean, except 
 by estimating it at infinity or at some imaginary negative 
 value, which has no meaning in the matter before us. For 
 the harmonic mean m between two quantities a and h is thus 
 algebraically expressed, m = 2a6/ {a + b) ; and now, if the 
 mean and one of the extremes be given, the other extreme 
 is thus expressed, a = ot&/(2 h — m), wherefore, when b is the 
 lower extreme, if it is half the mean, we have a = nib jo = 30, 
 and if it is less than half the mean, then 26 — w is a negative 
 quantity, and the quotient is a negative quantity. But a 
 negative quantity, no matter what be its absolute size, cannot 
 be greater than a positive quantity ; and yet it is offered as 
 the greater extreme ; which is absurd. 
 
 The seventeenth-century disputants were well acquainted 
 with the three classic means, the arithmetic, the geometric, 
 and the harmonic, not to mention certain others. Yet not 
 a word do they say about the harmonic mean : their dispute 
 narrowed itself between the arithmetic and the geometric. 
 Galileo and CasteUi advocated the former, Nozzohni alone 
 stood out for the latter. Perhaps it was the unwieldiness of 
 the harmonic mean, or the less frequency of its use in general, 
 that moved them to ignore it. But perhaps, at least with 
 some of them, a consciousness of its unsuitableness, as just 
 pointed out, was a contributory cause of its exclusion. 
 
 Galileo, in a later stage of the dispute, teUs us that when 
 the question was first presented to him, he at once jumped 
 to the answer, that the estimate of 1,000 crowns contained 
 a much greater error than that of ten, since the gain in that 
 case was 900 and the loss in the other only ninety. That is, 
 he was inclined, as most people are, to use the arithmetic 
 mean. But when he was informed that men of intelligence 
 had raised a controversy over this question, he devoted more 
 attention to it, and soon changed his mind, perceiving that 
 there was more to be said in favour of the geometric mean. 
 In his first letter he made a very simple argument. The 
 example offered might seem extravagant enough ; but he 
 considered one still more extravagant. He supposed that 
 one estimate of an article worth 100 crowns was of 200 crowns, 
 and the other was of only one crown. If he employed the 
 arithmetic mean, he would have to regard the former estimate 
 as less erroneous than the latter. Yet the former estimate 
 was only twice as great as the true value, involving- an error 
 
THE CONTROVERSY 3 
 
 not infrequently made, while the latter estimate was loo 
 times less than the true value, committing an error altogether 
 unreasonable. Here, we may remark, Galileo abandoned 
 difference for proportion. In the new supposition, the 
 difference (and, we may say even, the distance) of the first 
 estimate from the true value was a trifle greater than that 
 of the second, but the ratio of the first estimate to the true 
 value was much less than the ratio of the true value to the 
 second. The problem, he concluded, is one to be solved by 
 proportion ; and therefore the geometric mean is the right 
 one for the case. And the original question, which set the 
 one estimate as ten times too great and the other as ten times 
 too httle, he answered by declaring these estimates equally 
 erroneous. 
 
 Nozzolini, unlike Gahleo, stuck to his first inclination to 
 use the arithmetic mean. The estimate of i,ooo crowns he 
 unreservedly pronounced ten times more erroneous than the 
 estimate of ten, since the former departs from the true value 
 by 900 and the latter by only ninety crowns ; and to make 
 their errors equal, he maintained the horse must be worth 
 505 crowns. But knowing of the geometric mean, which 
 was suggested by the statement of the problem itself, and 
 suspecting that some might advocate it, he proceeded to give 
 an argument for the arithmetic mean. He put it on the 
 ground that this is a case for commutative justice and not 
 for distributive justice, and it has been established since the 
 time of Aristotle that the geometric mean is the proper one 
 to use in distributive justice, but in commutative justice 
 (defined as a mean between gain and loss' 2) the proper mean 
 is the arithmetic. But his reason for assigning the affair to 
 commutative justice is by no means clear. He maintained 
 that, as a purchase is a ratified estimate, so an estimate is 
 nothing but an unratified purchase ; and he observed that 
 in purchases and sales all errors are by merchants reckoned 
 arithmetically. Yet he mentions many transactions in 
 which merchants make use of the rule of three (which is a 
 geometric process), and especially he notices that gains and 
 losses are divided by partners in proportion to the capital 
 they invested, wherein the geometric method is used. 
 These, however, he says, are cases of distribution, and there- 
 fore they rightly employ the geometric mean. StiU it 
 remains that, although a question of estimation may not be 
 a subject for distributive justice, it is not shown that it is a 
 
 2 Nic. Ethics, V. iv.. rather between pelf and penalty. 
 
4 THE PROBLEM OF ESTIMATION 
 
 matter of commutation, or balancing of gains and losses ; 
 nor is it proved that the geometric mean is confined to 
 questions of distribution. 
 
 When Nozzolini wrote this first letter, he was unaware of 
 Galileo's opinion. After Galileo's letter was sent him, he 
 returned to the charge. Galileo's still more extravagant 
 case did not bother him in the least . The man who estimates 
 a hundred-crown article at one crown commits, he says, a 
 smaller error than the man who estimates it at 200, as 99 is 
 less than 100. The argument about proportion he rejects 
 on the ground that it changes the measure. In all compari- 
 sons of magnitude, he asserts, we must employ only one 
 measure. We cannot say that two distances are equal 
 because the one is ten yards and the other ten rods. And 
 now when Galileo adduces the fact that in the original 
 example the one estimate is ten times too great and the other 
 ten times too little and therefore holds them to be equal, 
 Nozzolini retorts that he has changed the measure, since the 
 first estimate is distant from the truth by ten large measures 
 of 100 crowns and the second estimate is distant from the 
 truth by ten small measures of ten crowns ; wherefore he 
 likens it to saying that at Florence the campanile is the same 
 distance from the cathedral as is the church of San Giovanni 
 because the latter's distance is ten steps of a giant and the 
 former's is ten steps of a baby. The argument, of course, 
 goes too far ; for by it a considerable amount of our ordinary 
 use of proportion, or the equationing of ratios, would be 
 quashed. Nozzolini has continued to speak of distances, or 
 differences, when the question has been put in the form of 
 ratios. In proportions the measures are eliminated, and 
 therefore, as Galileo will show us in his reply, various measures 
 can be used when we compare the ratios of the errors of 
 estimation committed concerning them. 
 
 In conclusion, to bring some hilarity into the too serious 
 subject, Nozzolini appealed to a forum which he declared to 
 be competent and which every day adjudicates such ques- 
 tions, and never hesitates in its decision. He had noticed, 
 he wrote, that biitchers often laid wagers among themselves 
 and with countrymen about the weight of pigs or calves, 
 and they decided the question always in favour of the one 
 who came nearest to the true weight found when thev 
 weighed the animal. We have a similar proceeding in our 
 country fairs, where often a charge is made for guesses over 
 the weight of some huge hog or pumpkin, and the one who 
 guesses nearest to the truth gets the prize. Here, apparently 
 
THE CONTROVERSY 5 
 
 appeal is made to common sense in favour of the arithmetic 
 mean. 
 
 Castelli now entered the hsts, having seen the preceding 
 letters. He backed up Galileo's position with two very 
 pithy variations on the new example. First, if an article 
 worth 100 crowns is estimated by one person at 200, he 
 asked Nozzolini what should be the estimate below 100 with 
 an equal error. The arithmetic answer must be that the 
 other estimate should be nothing, since the difference between 
 zero and 100 is the same as between 100 and 200. But is it 
 not absurd to adjudge the former error as great as the latter, 
 especially as, if some one estimated the article at 300, and 
 it was asked what the lower estimate must be to match this 
 error, only a very ridiculous answer is indicated ? This, we 
 may remark, is analogous to the argument whereby we at the 
 commencement rejected the harmonic mean. It shows that 
 the arithmetic mean involves the same absurdity from the 
 other side. If one estimate is half or less than half the true 
 value, the harmonic mean can assign no valid higher estimate 
 with equal error. If one estimate is double or more than 
 double the true value, the arithmetic mean can assign no 
 valid lower estimate with equal error. The arithmetic mean 
 is in this respect as faulty as the harmonic. Secondly, taking 
 up Nozzolini's idea of the kinship between estimating and 
 buying-and-selling, and supposing a hundred-crown article 
 to be estimated at 199 crowns and bought for that sum by 
 one party, and another equally valuable article to be esti- 
 mated at and bought for one crown by another party (with 
 equal error, according to Nozzolini), CasteUi pointed to great 
 inequality, in that, as he alleged, in the first transaction the 
 gain and loss are 99 per cent, and in the second they are 
 9,900 per cent. Here he made a mistake through too great 
 conciseness. The seller's gain in the first transaction was 
 99 per cent., but the buyer's loss was less than 50 per cent. ; 
 and in the second the seller's loss was again 99 per cent., and 
 the buyer's gain was 9,900 per cent. Still, though the seller's 
 gain and loss in each case have the same percentage, the loss 
 and gain of the two buyers have entirely different percentages. 
 Compared with the amounts they ventured, their gain and 
 loss are utterly unequal. 
 
 Nozzolini answered Castelli in another letter. We may 
 consider first his reply to the second objection, and carry it 
 to its end. Here, he says, Castelli has unwarrantably altered 
 the terms of the proposition. For at first in reckoning the 
 gain of him who sold the hundred-crown article for 199 crowns 
 
6 THE PROBLEM OF ESTIMATION 
 
 at 99 per cent., he went from the just price to the unjust 
 price ; and then m reckoning the gain of the purchaser of the 
 similar article for one crown at 9,900 per cent., he went 
 contrariwise from the unjust price to the just price. Nozzo- 
 hni contended that we should go in both cases from the just 
 price to the unjust price ; and therefore he relied on the 
 fact that in each of the transactions the seller lost and gained 
 the same percentage, wherein equality is manifested accord- 
 ing to the arithmetic mean. Rather, we might claim, it was 
 Nozzohni who altered the propositions by going in opposite 
 directions ; for he went up from the mean to the higher 
 extreme in the one case, and in the other he went down from 
 the mean to the lower extreme ; whereas Castelli went up 
 from the lower extreme to the mean and continued going up 
 from the mean to the higher extreme, although he happened 
 to perform the latter ascent first.^ 
 
 We have no rejoinder from Castelli, biit Galileo takes his 
 place. While not admitting that gains and losses in actual 
 transactions are the proper measure of error in the estimates 
 upon which they are founded, he replied that the advantage 
 of gaining 99 per cent, on one's capital is by no means equal 
 to the disadvantage of losing 99 per cent, of one's capital. 
 Absolute gains and losses are not proper measures of business 
 capacity, in which the ability to make accurate estimates 
 plays a considerable part ; or else we should have to say 
 that a merchant who increased his capital from 1,000 crowns 
 to 2,000 was a better man of business than one who in the 
 same time increased a capital of 100 crowns to 1,000 ; which 
 is not so, as the former has increased his capital only 100 per 
 cent., while the latter has increased his 900 per cent. To 
 prove that the incapacity of him who reduces his capital 99 
 per cent., from 100 to one, is much greater than the capacity 
 of him who raises his capittil 100 per cent., from 100 to 200, 
 Galileo used this bit of demonstrative reasoning. Evidently 
 the former's incapacity is much greater than the incapacity 
 of one who reduces his capital from two to one ; but this one's 
 incapacity is exactly equal to the capacity of another who 
 raises his capital from one to two ; and this other person's 
 
 ^ Nozzolini added another counter-argument. He said that 
 Castelli's problem was stated in the arithmetic progression i, 100, 
 199, and therefore it was impossible to get a geometric progression 
 out of it (p. 62B). What was meant by this, it is difficult to see. 
 As well might it be said that the original problem was stated in 
 the geometric progression 10, 100, 1,000, and therefore the 
 arithmetic progression could not be got out of it. 
 
THE CONTROVERSY 7 
 
 capacity is equal to the capacity of the person who raises 
 his capital from 100 to 200 ; therefore the first one's incapa- 
 city, being much greater than a capacity equal to this last 
 one's capacity, is much greater than this last one's capacity 
 (p. 69A below). The argument hinges on the equality of the 
 opposite qualities of two persons, of whom the one decreases 
 his capital from two to one and the other increases his capital 
 from one to two. This is evident. In practice, however, we 
 reckon these movements from their starting points, and so go 
 in opposite directions, saying the one has lost 50 per cent, 
 and the other has gained 100 per cent. ; wherein there is no 
 appearance of equality. But the loser here is a person who 
 was 100 per cent, better off than he now is, and the gainer is 
 a person who now is 100 per cent, better off than he was ; or, 
 to turn both movements around, the loser is now 50 per cent, 
 worse off than he was, and the gainer was 50 per cent, worse 
 off than he now is. The appearance of equality is restored 
 by reckoning always in.the same direction, in each case from 
 a higher to a lower position, or from a lower to a higher, as is 
 done in the geometric mode of reckoning. And so in esti- 
 mates, the geometric mean, indicating the true value, stands 
 in a proportion x : my.m : y, in which x and y represent 
 estimates geometrically equally above and below the true 
 
 position ; whence — = — or — = — . The mean m must 
 ^ ' m y X m 
 
 always be either in the numerator or the denominator, and 
 cannot be in one position throughout in the same proportion. 
 Thus an estimate above the truth may be represented by a 
 ratio of the estimate to the mean, and then the estimate 
 below the truth must be represented by a ratio of the mean 
 to the estimate ; or the former may be represented by a ratio 
 of the mean to the estimate, and then the latter must be 
 represented by a ratio of the estimate to the mean. There- 
 fore, in geometrically comparing errors on opposite sides of 
 the truth, we must always go in the same direction, whichever 
 be the direction first chosen, as Castelli and Galileo did, and 
 not in opposite directions, as Nozzolini would have had 
 them do. 
 
 To Castelli's first objection, which, when an estimate is 
 100 per cent, too high, required Nozzolini to place the 
 equally wrong estimate on the other side 100 per cent, 
 below the true value at zero, Nozzolini replied that he would 
 do nothing of the sort. He would still do what he had done 
 in boyhood when he had occasion to meet excessive estimates 
 with the corresponding estimates below the correct ones. 
 
8 THE PROBLEM OF ESTIMATION 
 
 For when in the market he had been asked two farthings 
 for a pear which he knew was seUing for one farthing, he 
 had been in the habit of countering, not by offering nothing 
 for one pear, which he knew would be absurd, but by 
 demanding two pears for one farthing. And so now he 
 maintains that when the owner of the hundred-crown 
 article estimates it at 200, virtually demanding two pay- 
 ments for one article, the corresponding lower estimate of 
 the purchaser should be to demand two such articles for 
 100 crowns, thereby virtually offering one payment for two 
 articles. Here, not in only the one, but in both the cases, 
 he adds, is hidden the zero complained of. For, as the over- 
 estimating seller wants two payments and offers the article 
 for the one and nothing for the other, only so the under- 
 estimating piirchaser wants two articles and offers payment 
 for the one and nothing for the other (p. 62A). 
 
 Here Nozzohni seems to give away his case, since to demand 
 two articles for 100 crowns is the same as to demand each 
 of them for fifty, so that the double estimate is met by a 
 half estimate, as required by the geometric mean. But at 
 the end of the letter, almost as an afterthought, and again 
 incidentally in his reply to Galileo's repetition of the objection, 
 he discloses his meaning. He has converted the question 
 from one of estimation to one of gain and loss in a business 
 transaction, and then verj? lightly jumped back to the 
 question of estimating. Thus, reverting to his idea that 
 an estimate is nothing but an unratified purchase, and letting 
 go of estimation, and considering only what happens in 
 bujdng and selling, he says that when a merchant sells an 
 article for 20 per cent, more than it is worth, the next time 
 he sells a similar article to the same purchaser, if he wishes 
 to make up for his first injustice, and to equalise his price 
 above by a price below the true value, he should now ask 
 20 per cent, less than it is worth — that is, if he had once 
 sold a hundred-crown article for 120 crowns, he should 
 now sell it for eighty ; which is as required by the arith- 
 metic mean. And he points out that it would be absurd 
 for the unjust seller of the hundred-crown horse at 1,000 
 crowns to think that he could equalise the matter and make 
 up for his injustice by selling to the same purchaser another 
 hundred-crown horse for only ten crowns, in accordance 
 with the geometric mean. No, he says, in this case the 
 unjust seller, to ease his conscience, must sell ten such 
 horses for ten crowns apiece (p. 63). This is perfectly 
 evident of purchases and sales ; and now Nozzolini clainK 
 
THE CONTROVERSY 9 
 
 that the same must be true of estimates. He demands an 
 estimate of ten horses at ten crowns apiece in order to have 
 the same total amount of error in these estimates below the 
 true value as was contained in the single estimate ten 
 times above the true value, which cannot be equalled in a 
 single estimate below the truth, even though, he says, the 
 horse were estimated as worth less than a grain of sand 
 (pp. 64A, 73B). 
 
 Thtis Nozzolini admits that the error in an estimate 
 twice, or more than twice, the true value cannot be equalled 
 by the error of a single estimate below the true value ; 
 but he thinks he gets rid of the difficulty by distributing 
 the error over as many estimates as the first estimate was 
 times greater than the true value. We may remark that 
 in this way the harmonic mean might be rehabilitated, since 
 an estimate less than half the true value could be met on 
 the upper side by distributing the error over two or more 
 positive, finite, and not very large estimates. In general, 
 against both these means, we may argue that, although 
 there is some resemblance of an estimate to a piurchase, 
 there is also some difference, and here the difference tells. 
 For, if we allow a single hundred-crown horse to be esti- 
 mated along with nine other imaginary horses supposed to 
 be its duplicates, when the first estimator, thinking of an 
 actiial horse, estimates it ten times too high, and the second 
 estimator, in order to equal his error on the lower side, 
 estimates the actual horse and the nine imaginary ones at 
 ten crowns each, thereby in fact committing a total error 
 of 900 crowns, equal to the other's, there is no reason why 
 the first estimator should not now do the same, and so when 
 he estimates the nine additional imaginary horses at 1,000 
 crowns each, the error of his estimate still is, in this way of 
 computing error, ten times greater than that of the other's. 
 At this rate the second party will never be able to overtake 
 the first in error ; for however many imaginary horses he 
 invokes to estimate below the true value, the same can the 
 first party employ to estimate above the true value. 
 
 Galileo, for his part, after first making a reply which did 
 not altogether reach the mark, as he seems to have failed 
 quite to grasp Nozzolini's argument, made another reply, 
 which left Nozzohni with nothing to say, for he first ignored 
 it, rebutting only the less successful repty, and afterwards 
 refused to consider it. GaHleo claims that the problem of 
 estimation need not be confined to values, but may be made 
 of other quantities in which the question cannot be trans- 
 
10 THE PROBLEM OF ESTIMATION 
 
 ferred to one of gain and loss in an exchange. Thus, alter- 
 ing CasteUi's example, he supposes a tower lOO feet high 
 to be estimated 200 feet high, and asks what estimate 
 below the true height wiU be equal in error to that. Cer- 
 tainly it cannot be an estimate of zero feet ; for that would 
 raze the tower to the ground. And if Nozzolini says the 
 other estimate must be of two towers each 100 feet high 
 so made as to get the total error of 100 feet, he must say 
 either that one of the towers is 100 feet high and the other 
 of no height at all (and so no tower), or that they are each 
 of- them 50 feet high (p. 68b). In the latter case, as the 
 imaginary tower does not concern us, we may accept the 
 estimate of the tower in question, which is as required by 
 the geometric mean. Or, if Nozzolini insists on retaining 
 the additional tower, we may say that he has altered the 
 problem, which was about a single tower. And we may 
 repeat, if the one estimate can deal with two towers, why 
 cannot the other ? 
 
 What has in the above two instances been quoted of 
 Galileo, was made in a postscript to his only other letter. 
 In the letter itself, written before he had information of 
 these, he replied only to Nozzolini's earlier letters. After 
 commenting on NozzoHni's reference to commutative and dis- 
 tributive justice, which had no weight with him, and repeat- 
 ing his own argument from extravagant instances by another 
 extravagant supposition (of a bagful of coins being esti- 
 mated double and very little), and raising CasteUi's first 
 objection in a slightly different form in ignorance of CasteUi's 
 having already made it, and denying that any gain and loss 
 need be considered, as they do not enter when other quanti- 
 ties not values are estimated, he broaches a broader treat- 
 ment of the subject. He points out that NozzoUni's funda- 
 mental error in measuring errors of estimation was in 
 considering only the absolute magnitudes of the errors and 
 in taking no account of the relations between the estimates 
 and the things estimated. This procedure leads to aU 
 sorts of absurdities, among which is the possibility that 
 most ignorant appraisers may have the preference given 
 them over appraisers of the greatest experience and perspi- 
 cacity. For if any one shoiild estimate the value of a nut 
 (ten of which are sold for a farthing) at a crown, he would 
 by everybody be considered a most extravagant appraiser • 
 but if any one should make an error of a crown in estimating 
 the value of a jewel worth four or five thousand, he would bv 
 everybody be considered a most accurate appraiser. Galileo 
 
THE CONTROVERSY ii 
 
 believed that Nozzolini would not judge otherwise ; and 
 yet his reasoning would require him to consider the former 
 the better appraiser. And to show that we cannot really 
 judge errors of estimation without relation to the size of 
 the things estimated, he supposes the case of one who in 
 estimating the height of a mountain makes an error of lOO 
 yards, and of another who in estimating the weight of a 
 bullock makes an error of ten pounds. Here he points out 
 that it is impossible to compare these errors, for the data 
 supplied are insufficient. But if we add that the true height 
 of the mountain is i,ooo yards, and the true weight of the 
 bullock is 100 pounds, then it is possible to compare the 
 errors ; but we cannot say the error of the former is the 
 greater on account of loo being greater than ten, as there is 
 no comparison between loo yards and ten pounds; or if 
 we judge simply by the absolute numbers, we might change 
 the latter statement and say the appraiser of the bullock 
 erred by 120 ounces, and then, as 120 is greater than 100, 
 his error would, from the smaller, become the greater ; 
 which is absurd. The correct answer is that the two errors 
 are equal (provided, we must add, they were on the same 
 side), because in each case the error is of 10 per cent.* 
 
 * Galileo did not make the proviso in the parenthesis, but it 
 is understood in his argument. Of course Nozzolini would agree 
 with him in the above. Nozzolini's real difference with Galileo 
 lay in holding that an equal percentage of error above and below 
 the true quantity makes an equal error. Sometimes, however, 
 he placed the equality in the difference, or distance, from any 
 true quantity. If a single tower 100 feet high is estimated at 
 120 feet and at 80 feet, he says the errors are equal for both these 
 reasons, that the estimates each depart 20 feet from the true 
 height, and that they each depart 20 per cent, from the true 
 height. But these reasons are by no means the same, for, though 
 here coinciding, they may diverge. This may be shown by 
 repeating Galileo's back-and-forth argument, adapted to the 
 present case. If a tower 100 feet high is estimated at 80 feet, 
 and if another tower 80 feet high is estimated at 100 feet, the 
 error in each of these estimations is of 20 feet, and Nozzolini 
 must say they are equal. In fact, in the last paragraph of his 
 last letter on the subject he expressly says that in the arithmetic 
 method of measuring, nothing is considered but the distances 
 (or differences), and that it is as much to go from the more to the 
 less as from the less to the more, there being the same distance 
 from 8 to 4 as from 4 to 8, just as it is the same distance from my 
 house to yours as from yours to mine. According to this, the 
 distance from 100 to 80 being the same as from 80 to 100, the 
 
12 THE PROBLEM OF ESTIMATION 
 
 In measuring errors of estimation against Nozzolim s 
 contrary opinion, Galileo insists that the unit of measure- 
 ment is indifferent, as shown by the above comparison 
 between estimates of a height and of a weight. Taking 
 another instance, Galileo offers as self-evident the proposi- 
 tion that if one person estimates the height of a tower at 
 150 feet, which afterwards is measured and found to be 
 100 feet high, and if another estimates the weight of a calf 
 at 150 pounds, which afterwards is put on the scales and 
 found to weigh 100 pounds, their errors are equal, although 
 the measures are different in kind. He concluded that 
 the measurement of errors of estimation is generically 
 different from the measurement of concrete things, because 
 error is an abstraction. All measures, he maintains, must 
 be of the same species with the things measured, as we 
 measure times by a time, weights by a weight, values by a 
 value. And so errors must be measured by an error, which 
 is a ratio between the estimate and the true quantity, and 
 not a concrete quantity itself. We cannot measure errors 
 by so many pounds, feet, or crowns : we must measure 
 them by the proportions of the pounds, feet, or crowns in 
 the erroneous estimates to the pounds, feet, or crowns in 
 the thing estimated. He therefore defines the measure of 
 error in estimation as, abstractly, the general relation or 
 habit which the false estimate has to the true quantity of 
 the thing estimated (pp. 66, 67). Hence, he repeats, aU 
 reference to commutative and distributive justice is not to 
 the point ; and he opines that, had the question originally 
 been put about errors in estimating the height of a giant, 
 supposed to be 10 feet, and one estimate was supposed to 
 be 100 feet and another i foot, Nozzolini would never have 
 argued as he did, and would have seen that those errors are 
 equally exorbitant. But if Nozzolini persists in his opinion, 
 Galileo warns him, he will have to maintain many ridiculous 
 things — as that he makes a better estimate of a heap of 
 1,000 coins who says he thinks there may be two or three, 
 
 errors must be equal. But again, according to the usual way of 
 reckoning percentages, while the former error is of 20 per cent., 
 the latter is of 25 per cent. Therefore the same Nozzolini must 
 say the errors are unequal — and this time he would be wrong. 
 Into such inconsistency did his application of an arithmetic 
 process to a case of geometric proportion lead him. (Reasoning 
 similar to Galileo's in this matter was employed, without kno\^ 
 ledge of Galileo's precedence, by the writer in The Measurement 
 of General Exchange-Value, pp. 351-2, cf. 236-9, 250-3.) 
 
THE CONTROVERSY 13 
 
 than he who judges them 2,000 ; or worse yet, he must 
 hold that he who estimates Monte Morello to be 10,000 feet 
 high (over twice its height) is a worse judge of height even 
 than he who says it is a hole in the ground ; for Galileo 
 never thought of Nozzolini's scheme of bringing other 
 mountains into the question. In fact, he says that for 
 Nozzolini to equalise, in the original problem, the error of 
 the estimate of the hundred-crown horse at 1,000, he must 
 suppose the horse to have a harness put on worth 810 
 crowns and then the horse and the harness together (worth 
 910 crowns) to be estimated at ten crowns. This suggestion 
 is little more absurd than the method Nozzohni did adopt 
 to avoid the difficulty. 
 
 Lastly, Galileo refers to Nozzolini's appeal to the forum 
 of the butchers, and points out that the butchers used the 
 arithmetic mean through ignorance, as they did not perceive 
 that equal differences above and below the true value do 
 not spell equal errors ; a reason for which was that, being 
 skilled in such matters, they generally made but small 
 mistakes, and in small mistakes the difference between the 
 arithmetic and the geometric means is trifling {e.g., between 
 no and 90, while the arithmetic mean is 100, the geometric 
 mean is over 99) ; whereas, were they confronted with a 
 case where somebody estimates a calf of 100 pounds at 
 200 pounds and another estimates it at an ounce, they never 
 would have given the prize to the latter. Herewith Galileo 
 left the subject. 
 
 But Nozzolini had the last word, writing two more letters, 
 one of them the longest in the lot. Herein, after insisting 
 on Aristotle's authority, he traversed all of Gahleo's criti- 
 cisms, mostly repeating only what he had already said. 
 But to the last criticism about the butchers he replied that 
 no matter how small the difference between the arithmetic 
 and the geometric means in small errors, the right mean 
 ought to be adopted (p. 74B) . It is difficult to see what he 
 was driving at by this ; for it could hardly be maintained 
 that the butchers must have adopted the right mean, and 
 as they adopted the arithmetic mean, therefore this is 
 the right one. The real reason why the butchers, and other 
 people in similar situations, adopt the arithmetic mean, 
 and will continue to adopt it, is because it is the simplest 
 and most convenient.* Simple methods of measuring are 
 
 ' So, to vary the example, children will in all probability always 
 maintain that a lesson-mark of 80 out of a possible 100 is twice as 
 bad as one of 90. Yet their error can easily be shown by con- 
 
14 THE PROBLEM OF ESTIMATION 
 
 constantly used in practice, even by persons who know 
 that some other more complex method is more precise, 
 provided the former is precise enough for practical pur- 
 poses — and Galileo showed no sign of wishing to interfere 
 with the practice of the butchers.^ Where precision is 
 absolutely required is in theory. So it was Nozzolini, and 
 not the butchers, from whom we have a right to demand 
 precision. 
 
 Another rejoinder was a general one to Galileo's method 
 of arguing from extravagant cases. Extravagant cases, 
 very conveniently for his own contention, he would throw 
 out' altogether, notwithstanding that he had admitted con- 
 sideration of them at the beginning, before he perceived 
 the havoc they would make with his side of the question. 
 For justification he appealed to the procedure in dialectical 
 disputes ; for in them, he said, if one party denies that 
 snow is white or that fire is hot, he is regarded as a fool 
 with whom no argument can be profitably made. And so, 
 he says, if any one should estimate a hundred-pound pig 
 as weighing an ounce, or should consider a mountain a 
 hole in the ground, his estimate should be rejected as that 
 of a fool who cannot be argued with. He holds that from 
 extravagant cases we cannot judge of the nature of a thing, 
 citing in example that because a drop of water is round 
 
 sidering the case of eleven scholars, the first of whom receives the 
 mark of go, the next a mark twice as bad, the third a mark three 
 times as bad, and so on. The tenth, according to the children, 
 will receive the mark of zero. What will be the mark of the 
 eleventh ? Again, it is evident that 50 is half as good a mark as 
 100. Then the arithmetician must say that zero is half as good 
 a mark as 50. But obviously the mark half as good as 50 is 25. 
 This is in accordance with the geometric measurement. 
 
 ' In their wagers the only case in which the arithmetic mean is 
 likely to give a different result from the geometric is where two 
 estimates are equally above and below the measured weight ; 
 for then, while the geometric mean would award the prize to the 
 one who made the higher estimate, the arithmetic mean would 
 divide it between them. But here the superiority of the higher 
 estimate would in most cases be so inappreciably small, that it 
 would generally fall short of the probable error in the measure- 
 ment of the true weight ; wherefore even theory would sanction 
 the practice of using the arithmetic mean in such cases. And if 
 the condition is first stated, as is usually done, as being that the 
 one who comes nearest to the measured weight shall be considered 
 the winner, there is no room for dispute about any other mean 
 but the arithmetic. 
 
THE CONTROVERSY 15 
 
 (a very small instance), we cannot infer that a barrelful 
 of water has the nature of roundness (p. 72A). This is a 
 rather unfortunate example, as we should do exactly what 
 he denies : we should infer that a barrelful, or a whole 
 oceanful, of water, if as free in space as is a drop of rain in 
 the air, would assume the spherical shape, just as, and for 
 the same reason that, the drop does. He admits that he 
 had himself entertained the supposition of some such 
 extravagant estimates, as of the hundred-crown horse being 
 estimated at only ten crowns ; but now he says he had all 
 along supposed the estimator had some reason for his 
 estimate, as that he suspected some disease in the animal. 
 A more humiliating admission is another which he now 
 makes, to the effect that the person who should estimate a 
 heap of 1,000 coins on the table before him at two or three 
 makes a greater error than the one who estimates them at 
 2,000, although the former's estimate is nearer the truth. 
 But he demands the privilege of throwing out all such 
 foolish estimates, and of considering only the more judicious 
 closer ones. He overlooks that Galileo did not merely 
 make such suppositional estimates himself, but had shown 
 that >Jo2zohni's principle required him, Nozzohni, to main- 
 tain that such foolish estimates, because nearer the truth, 
 are better than others that are not foolish, and had pre- 
 dicted that Nozzolini could not hold fast to his principle. 
 When one disputant employs against another the rediictio 
 ad absurdum of his views, it would be strange if the other 
 could avoid its effect by simply demanding that his opponent 
 must not introduce absurdities. But Nozzolini, rejoicing 
 in this method of rebuttal, went on and likewise threw out 
 all Galileo's use of other quantities, in which there is no 
 gain or loss, on the ground that he had to consider only 
 the problem as originally proposed, which concerned itself 
 only with values, forgetting that he himself had been the 
 first to introduce other quantities with his reference to the 
 butchers.' 
 
 Beside these rejoinders, Nozzolini makes three or four 
 new arguments, which are left for us to tackle. One is an 
 appeal to a more technical forum than that of the butchers — 
 to the fact that when two or more offtcial appraisers of 
 
 ' But he would allow no reciprocity on Galileo's part. His 
 incomprehensible argument referred to in Note 3 was ignored by 
 Galileo. He now repeats it, says it is the best, and he will stand 
 on it alone, and claims the victory because it was not refuted 
 (P- 75)* 
 
i6 THE PROBLEM OF ESTIMATION 
 
 some property in litigation differ, the arithmetic mean 
 between their appraisals is everywhere, and always has 
 been, the one legally employed (p. 71 b). To this no different 
 reply need be made than to that other appeal to other 
 people's opinion and practice. Then he cites the case of 
 archery and of similar contests (rifle practice to-day) to 
 hit a mark, and finds confirmation of his view in the fact 
 that the person who comes nearest to the buU's-eye (whether 
 above or below it) is considered the best shot. Here he 
 cites a case where practice is theoretically correct ; but it 
 is one essentially different from the subject of estimation. 
 For estimates can err only above and below the mark, while 
 shots may go astray in any direction ; and, which is the 
 principal thing, the limits of error in shooting at a mark 
 are the same in aU directions, being nowhere ; but in estima- 
 tion, while the errors above may go up without limit, the 
 errors below quickly reach a limit when the estimate falls 
 to zero. Now, it is precisely the geometric mean which 
 sends its extremes at each remove in such wise that the 
 lower one never reaches zero before the upper one reaches 
 infinity ; whereas the arithmetic mean sends its lower 
 extreme down to zero as soon as the upper extreme has 
 reached double the quantity aimed at. Thus, while in 
 errors of archery and the like, where the limits on both, as 
 on all, sides are equally distant, or distanceless, the arith- 
 metic mean is the proper one to employ ; in errors of estima- 
 tion, which have a definite lower limit and none but an 
 infinite upper limit, the proper mean, in theory at least, 
 whatever be the practice, is the geometric mean. Again, 
 he argued that if a thing weighs 60 pounds, and both the 
 wrong estimates are on the same side, say 55 and 50, we 
 give the decision in favour of the nearer, and so judge them 
 by their distance ; and just so, when the estimates are on 
 opposite sides, say 55 and 70, we should do the same, judge 
 them by their respective distances, — or rather, he asks, 
 why should we not ? Here he is betraying signs of dotage ; 
 for he overlooks that Galileo had over and over again shown 
 why we should not ; and even in the first supposition, where 
 the difference of proportion accompanies the difference of 
 distance (wherefore NozzoHni cannot rightly say we use 
 the latter only), there is a begging of the question in his 
 assertion; for, according to the opposite contention, we 
 can no more correctly say that an error of 20 per cent, 
 above the truth is twice as great as an error of 10 per cent.' 
 above the truth, than we can rightly say that the latter 
 
THE CONTROVERSY 17 
 
 error is equal to an error of 10 per cent, below the truth. 
 Lastly, in a supplementary letter, he introduces a principle 
 never heard of before and never again since, which he can 
 hardly word clearly, but which is something to the effect 
 that in all questions of greatemess we go from the greater 
 to the smaller, and in all questions of smallerness we go 
 from the smaller to the greater ; and so we must do in a 
 question of estimates, that is, we must go, as he before 
 contended, from the true value in opposite directions to the 
 greater and to the lesser extremes ; whereby he seeks to 
 
 convict Galileo of having to maintain that is = — , 
 
 ° 100 100 
 
 whereas, of course, Galileo had maintained that is 
 
 100 
 
 = — . It is a principle invented as an afterthought for 
 
 the purpose of supporting the conclusion desired ; and it 
 deserves no further consideration. 
 
 Nozzolini had commenced with great moderation, but 
 he ended by betting his eyes that the geometric proportion 
 has nothing to do with measuring errors of estimation. He 
 displays the signs we expect in a man who argues on the 
 wrong side — of floundering into greater and greater absur- 
 dities as he persists in his course. 
 
II.— AVERAGES AND ERRORS. 
 
 So lengthy a dispute over so useless a problem may seem 
 puerile. But a problem may be useless in itself, and yet 
 be profitable as an exercise in reasoning. The controversy 
 we have been reviewing was not a mere logomachy : it may 
 be viewed as a methodological essay. It can serve both for 
 a model and for a warning, the former provided by Galileo, 
 the latter by Nozzolini. And there are indications to-day 
 that the one as well as the other is needed. Even some of 
 its direct results have had to be rediscovered in our own 
 time, as will be seen more than once ; and it is a pity they 
 were ever forgotten. Hence that controversy by no means 
 deserves the contempt with which Todhunter alluded to it, 
 as having been a waste of time, and still being without 
 "any scientific interest or value." ^ To draw the lessons 
 that may be learnt from it, let us first inquire into the nature 
 of averages, and then into the nature of estimates and their 
 errors, after which we may extend the investigation to more 
 practical problems. 
 
 The Criteria of Averages. 
 
 In drawing a mean or average the most definite purpose 
 we have is to simplify our quantitative conception of things 
 by obtaining one figure that can take the place of two or 
 more and give the same result as they, numerically speaking. 
 This may be illustrated in four examples, (i) Of two 
 provinces with equal territory, if the population of the one 
 
 ' History of the Theory of Prohahility, p. 6. He referred to 
 Libri's Histoire des Mathematiques en Italie, Vol. IV. pp. 288-g, 
 as saying that Galileo was occupied with this question (" non 
 encore resolue ") a long time. Galileo, though he spent some 
 time over methods of estimating distances and elevations with 
 the eye, did not linger long over this problem of measuring 
 estimates themselves. It is, indeed, a pity that he did not reply 
 to Nozzolini's last letter. He might then have pointed out the 
 distinction between shooting at a mark and guessing at a value 
 and have saved succeeding generations from error through 
 ignoring it. 
 
AVERAGES AND ERRORS 19 
 
 is 10,000 and of the other 20,000, this is the same for the 
 state as if the population of each were 15,000 ; for the sum 
 of the population is 30,000, which is distributed over two 
 provinces.* Thus the mean population of any country is 
 measured by the arithmetic mean or average. It is not even 
 necessary to know the individual items : if we know the 
 whole population of a country and the number of provinces 
 (or of square miles) in it, we get the average by dividing the 
 former by the latter. (2) If the population of a country has 
 increased at irregular rates, but with the result that ft has 
 increased 100 per cent., i.e., has doubled, in 100 years, 
 the same result would have been obtained if it had increased 
 regularly at the rate ^^^2 = 1-007, '^^ 0'7 P^r cent, every year ; 
 for whatever may have been the 100 individual rates of 
 increase (or even of decrease at times), their product is two, 
 and this product is likewise obtained by i'007 multiplied 
 by itself 100 times. Here, then, the average increase is 
 the geometric average of the individual increases ; and 
 again it is not necessary to know the individual items, all that 
 is needed being to know their number and their product. 
 
 (3) If a merchant sells one article for 190 doUars and another 
 for ten, the net result is the same as if he sold each for 100. 
 Here again the mean to use is the arithmetic. If the articles 
 were each worth (or their proper selling prices were) 100, his 
 gain and loss would be equal and would nullify each other ; 
 that is, the result to him would be the same as if he neither 
 gained nor lost on either article, as indicated by that mean . 
 
 (4) If a merchant alternately gains and loses 90 per cent, in 
 his successive business transactions, in every case venturing 
 his whole business capital, what mean gain or loss will have 
 the same result ? Here the model is twofold, according as he 
 begins by gaining or by losing. It may be thus tabulated : 
 
 100 190 19 36-1 3-61 6-859 0-686 
 100 10 19 1-9 3-61 0-361 0-686 
 
 It is seen that of every second venture the result is the same, 
 being an ever growing loss, and the intervening divergent 
 figures rapidly converge. As there are two lines of develop- 
 
 » The same numerically, be it remembered ; for in other 
 respects it may be much better for a state that its population be 
 distributed in one way than in another. And so when we com- 
 pare unequal territories by their average populations (on equal 
 territories), we are doing no more and no less than what we do 
 when we compare equal territories simply by their populations. 
 The same remark applies to the other examples. 
 
20 THE PROBLEM OF ESTIMATION 
 
 ment, it is impossible for the mean to have more than a 
 recurrently equal effect, while in the intervals it must lie 
 between the divergent figures. Evidently the arithmetic 
 mean is inappropriate for this case ; for the arithmetic mean 
 between 190 and 10 is 100, which indicates that the capital 
 would remain 100 after every venture, contrary to what does 
 take place. The geometric mean between 190 and 10 is 43-6, 
 indicating a loss of 56-4 per cent. ; and now if every trans- 
 action showed such a loss, the result v/ould be as follows : 
 
 100 I 43-6 19 8-284 3-6i 1-57 0-686 
 Thus at every second remove the figures are the same as 
 before, and in the intervening stages they are at the geometric 
 mean between the other two ; but ultimately this series 
 approaches the same limits as do the others, and before long 
 it will be practically indistinguishable from them even in the 
 intervals where it does not exactly coincide with them. It 
 may be objected that the supposition should be that the 
 merchant divides his capital and alternatively gains and 
 loses on the halves, oppositely in each case. Then the 
 figures would be : 
 
 50 95 9-5 18-05 1-805 3-43 0-343 
 50 5 9-5 0-95 1-805 0-18 0-343 
 
 100 
 
 100 19 19 3-61 3-61 
 
 Again at every second period the result is the same as in the 
 preceding suppositions, but in the intervals the results are 
 the arithmetic means between the first divergences ; which 
 gives a jerky progress, with two stages on a level, followed by 
 steeper falls. If it be further objected that at every venture 
 the merchant should re-divide his capital into two halves, 
 whereupon his capital would remain intact at 100, in accord- 
 ance with the arithmetic mean ; the reply is that this returns 
 the case to the second example, of the merchant making 
 equal gains and losses on articles of equal value, which of 
 course gives the same result however many times it be 
 repeated. But we have desired to make a distinct example. 
 And so, while this case is kept as first stated, the proper mean 
 to use on it is the geometric.i" 
 
 1" Also the harmonic mean may be the proper and necessary 
 one in certain cases. Thus in the ancient problem of the fountain 
 (found in the Greek Anthology, XIV. 135— the statement here is 
 slightly altered) the time required to fill the basin through two 
 pipes together which singly half-fill it in different times is 
 obviously a mean between these times ; and calculation shows 
 
AVERAGES AND ERRORS 21 
 
 In all these cases the right choice of a mean is demonstra- 
 tive, and not subject to dispute ; for the kind of mean chosen 
 is determined by its yielding a known result. But in many 
 cases there is no such definite criterion at hand. Then the 
 choice may be determined by the resemblance of the subject 
 to some demonstrative case ; which resemblance may be 
 found in the conditions set to, or the observed behaviour of, 
 the extremes, or the terms in general. The mean to be 
 chosen in the case of estimation can be determined only in 
 this way. Thus it is to be noted that Nozzolini always 
 identified this problem with our third example, and Galileo 
 assimilated it to something like our fourth ; for Nozzolini 
 converted it into a question of gain and loss, and Galileo 
 treated it as a question of business capacity. But Galileo 
 went further, and pointed out an underlying distinction 
 between these two kinds of subjects for averaging. In the 
 one there are no limits on either side of the true position, and 
 in the other there is a limit at zero on the one side and on the 
 other side the only limit is infinity. In the former, any 
 figure above or below the true position may have a corre- 
 sponding figure on the opposite side equidistant from the true 
 position ; and we may add, if there is a limit on the one side, 
 there must be an equidistant limit on the other. In the 
 latter, the limits zero and infinity are equally related to the 
 true position only as extremes in a proportion, 0: m = m:x ; 
 wherefore similar equality must be sought in all the inter- 
 mediate terms that correspond to each other, x : m = m : y. 
 This we may generalise, and say the former condition is 
 suitable for the arithmetic mean or average, the latter for the 
 geometric." 
 
 that it is the harmonic mean. Note that the lower extreme (the 
 shorter time in which one of the pipes half-fills the basin) can 
 never be qviite as small as half the mean (the time in which it in 
 conjunction with the other pipe will fill it) ; and so the harmonic 
 mean is applicable to the case, while the unsuitableness of the 
 arithmetic and geometric means is easily seen by using extra- 
 vagant suppositions, as of one pipe taking a day to do what the 
 other will do in an hour. Here the only purpose the harmonic 
 mean can serve is to facihtate the calculation the next time we 
 are confronted with such a problem. 
 
 ^^ We might likewise work out some corresponding condition 
 for the harmonic mean. This is omitted as not needed. Some 
 hints of it may be found in The Measurement of General Exchange- 
 Value, Chapter VTII. For the criteria see there especially 
 pp. 245-8. 
 
22 THE PROBLEM OF ESTIMATION 
 
 Thus in our fourth example it would have been possible to 
 suppose the merchant at every other venture to gain iio per 
 cent., or any higher percentage, but we cannot suppose him 
 in the intervening ventures to lose any more than lOO per 
 cent., remembering that we are making these suppositions 
 as indications of his business capacity ; for this cannot sink 
 below nothing, and if he were supposed to lose lOO per cent, 
 of his capital, he would have no capital to start with for 
 another venture, and the series would come to an abrupt end 
 after the first or second venture. But in the third example 
 there is no such Umit. We might change the figures and 
 suppose that the merchant sells the hundred-dollar article 
 for 300, making a gain of 200, and then we can ask how he 
 should dispose of another hundred-dollar article to make an 
 equal loss, or to make up to the purchaser for his first unjust 
 sale. Here, to be sure, there is no price that will satisfy the 
 question ; but it is easy for us to answer it by saying he 
 shovild give away the second article and 100 dollars to boot.^^ 
 There is nothing ridiculous here. But if any one has made an 
 error of 200 dollars in estimating at 300 an article worth 100, it 
 would be ridiculous to say he could equal this on the other 
 side by estimating the same or an equally valuable article at 
 100 dollars less than nothing. There are no more limits to 
 losses than to gains ; for a man can get himself into the position 
 of having less than nothing by running into debt. But there 
 is a limit to errors of estimation below the true value, as no 
 man can estimate a valuable article at less than nothing. 
 
 We may note, further, that Nozzolini always obtained his 
 result by slightly altering the problem, converting it from 
 one of estimation into one of gain and loss. In many cases 
 shght changes in the way of stating a problem, or in the 
 attitude assumed toward it, or in the use expected to be made 
 of it, will involve a change in the solution. Another instance 
 may be cited. Consider two cities each of 10,000 inhabitants, 
 and suppose 9,000 migrate from the one to the other. The 
 one has lost 9,000, the other has gained 9,000. The one has 
 been reduced from 10,000 to 1,000, the other has been 
 increased from 10,000 to 19,000. Which of these changes is 
 the greater ? and what average are we to use in measuring 
 them ? Now, if these two cities are both in the same country, 
 and we have in mind the effect of the changes, not upon the 
 cities themselves, but upon the country at large, it is evident 
 
 12 If in this transaction the price be conceived as — 100, the 
 arithmetic mean of the sales, I (300 H 100), is still = 100. 
 
AVERAGES AND ERRORS 23 
 
 that in mere numbers the country at large has neither gained 
 nor lost ; and so, no effect being produced, the changes have 
 nullified each other, and therefore are equal. So conceived, 
 the case comes under the first example above given and 
 resembles also the third, and the arithmetic mean is the right 
 one to employ. But suppose again that the cities are in 
 different countries, or in different ages, and, without migra- 
 tion between them, the one has dwindled from 10,000 to 
 1,000 and the other has grown from 10,000 to 19,000. We 
 now have to consider, not the combined effect upon any 
 whole of which the cities are parts, but the effects upon the 
 cities themselves, individually and independently. And now 
 we cannot say the changes are equal. Equal changes in 
 opposite directions would be that of a city dwindling from 
 10,000 to 1,000 and that of a city growing from 1,000 to 
 10,000 ; but this last is equal to that of a city growing from 
 16,000 to 100,000 ; and as the second city in the supposition 
 has grown from 10,000 only to 19,000, it has not grown as 
 much as the first has dwindled. Therefore the mean by 
 which to judge these changes is the geometric. 
 
 A little over a century after the controversy we have been 
 reviewing, Daniel Bernoulli in his Specimen Theories Novee 
 de Mensura Sortis ^^ and Buffon in his Essai d'Arithmeiique 
 morale ^* made a similar use of the geometric mean. 
 Bernoulli, to whom the distinction between " moral " and 
 " mathematical " expectation is due, and Buffon argued to 
 this effect : if two men of equal fortunes stake half their 
 fortunes on a throw of dice, the one loses more than the other 
 gains. Now, from the standpoint of the wealth of the 
 country of which they are citizens, there is evidently no gain 
 or loss, numerically considered ; and so, when we draw the 
 average wealth of the parties after the play, we must use the 
 arithmetic mean. But from the point of view of the indivi- 
 duals the case is different ; for the one who has lost half his 
 fortune, compares his present state with his previous state, 
 and his previous state was twice as good ; and so does the 
 other, but his previous state was two-thirds as good as his 
 present state, and therefore he looks upon (or feels) his gain 
 as a gain of only one-third, since what he has gained is the 
 third of his present state.^^ In other words, the esteem- 
 
 13 Arts. 5 and 13. This work was written in 1730-1, and pub- 
 lished in the Commeniarii AcademicB Petropolitana, 1738, Vol. V. 
 
 '* Sections 11-13. This work was written in 1760, and first 
 published in 1777. 
 
 1* Todhunter pointed out that Buffon at least did not do full 
 
24 THE PROBLEM OF ESTIMATION 
 
 value of the money which a man owns or commands has an 
 inverse relation to its amomit : as his fortune increases, the 
 esteem- value of his money falls, and as his fortune decreases, 
 the esteem- value of his money rises, and it rises faster than it 
 falls at every arithmetically equal remove from the same 
 initial position ; so that losses are estimated in money of 
 greater esteem-value than are gains. If the losses and gains 
 (or the stakes in a play) are small compared with the person's 
 fortune, the difference is imperceptible ; but it becomes 
 apparent when they are comparatively large." Hereby is 
 
 justice to his case. The loser has lost a fortune equal to his 
 present fortune, while the winner has gained only one-third his 
 present fortune. Todhunter says that, representing the original 
 fortune by a and the stake by b, Buffon estimated the gain by 
 
 the expression — — =- and the loss by -, whereas the loss should 
 a +0 ■'a 
 
 be estimated by r '■ History of the Theory of Probability, 
 
 p. 345. Bernoulli and Buffon have been followed, among others, 
 by Laplace, who introduced the distinction between a person's 
 " moral " and " physical " fortune, Thdorie analytique des Pro- 
 babilites, 1812 (in his (Euvres Computes, 1847, Vol. VII. 
 pp. 474-88, cf. p. xxii. of the Introduction) ; Lacroix, Traiti 
 elementaire du Calcul des Probabilites, 2nd ed., 1822, p. 127 ; 
 Quetelet, Lettres sur la Theorie des Probabilitds, 1846, let. viu. ; 
 Fechner, who brought the principle under Weber's law, Psycho- 
 physik, 2nd ed., Vol. I. pp. 236-8, Vol. II. pp. 549-50 ; Jevons, 
 Theory of Political Economy, 1871, p. 154. On the other hand 
 Venn in his Logic of Chance, 2nd ed., 1876, pp. 135-7, 408-12, does 
 not accept this reasoning. He points out that the pleasure of 
 the game has been overlooked, as also the temperament of some 
 people, who are bored with the monotony of mediocrity and 
 delight in excitement. This explains why gambling always has 
 been, and probably always will be, popular, in spite of the loss 
 of " moral fortune " which attends it ; but it does not disprove 
 that loss. That loss is a payment for the amusement obtained— 
 and it may be an overpayment through ignorance on the part of 
 the players. Cajori, in his History of Mathematics, 2nd ed., 
 p. 223, remarks that Bernoulli's theory has become classic, " but 
 no one ever makes use of it." So much the worse for those who 
 might but do not. 
 
 " In all the above by a man's fortune must be meant, not 
 merely his actual capital, but in addition to that the capitalised 
 value of his earning power. A man possessing property worth 
 20,000 dollars, who earns 10,000 a year, may lose his whole 
 capital without feeling much loss. As Bernoulli pointed out 
 
AVERAGES AND ERRORS 25 
 
 explained the well-known phenomenon that people grieve 
 more over losses than they rejoice over arithmetically equal 
 gains ; and cynics are mistaken when they sneer at this. 
 In the case of our two gamblers, individually considered, as 
 the one has lost more than the other has gained, the net efiect 
 is a loss ; and this is shown by drawing the geometric mean 
 of their final positions. Bernoulli calcinated the loss of what 
 Laplace, who accepted his reasoning here, called their " moral 
 fortune " at 13 per cent., because V^SO X 5o = 87 nearlyj 
 and this lies 13 below 100.^' Bernoulh, Buffon and Laplace 
 reasoned as Gahleo had done. 
 
 Different ways of viewing a problem, however, need not 
 lead to the use of different averages. An instance in point 
 is that of drawing the average height of the men of a country 
 or of a race. Our purpose may be merely to compare the 
 stature of the men of one country or race with the stature of 
 the men of other countries or races, or of men to-day with 
 men of earlier times. Such a use of the average has most 
 resemblance to that in the first model above given ; for we 
 may suppose that all the various heights of all the men of 
 one country are to those of another as they would be if all 
 the men of each country were uniformly tall and yet together 
 had the same combined height that they actually have. We 
 may conceive of all the men in each country standing on top 
 of one another, forming two upright columns, and then, 
 provided their numbers are equal, the comparison of their 
 heights is the comparison desired, and the same result is 
 given by comparing the arithmetically averaged heights of 
 
 even a beggar has some fortune, which is the capitalised value of 
 the gifts he annually receives. Without this correction it is 
 wrong to say that 10,000 is as great a sum to a man owning 
 20,000 as is half a million to a man owning a million. The former 
 is likely to be much better able to replace his loss than the latter. 
 Hence Franklin was not quite right when he said, in 1768, in 
 connection with universal suffrage, that " the all " (evidently 
 meaning all the property) of a poor man is " as dear to him " as 
 the all of a rich man is to him. Works, Spark's ed., Vol. IL 
 p. 372 (and again so reported by J. Baynes in Sir Samuel RomiUy's 
 Memoirs, 3rd ed., Vol. L, p. 447). Franklin's dictum was con- 
 troverted on this ground by Home Tooke in his Letter to Lord 
 Ashburton, quoted in the Edinburgh Review, Vol. XXXI. p. 183. 
 
 ^' Going back to the point of view of the country at large, we 
 now see that it too has suffered a loss, as it has got a worse dis- 
 tribution of its wealth. But this cannot be shown in the merely 
 numerical calculation of its wealth. 
 
26 THE PROBLEM OF ESTIMATION 
 
 the individuals in the columns ; and if the numbers are 
 unequal, the proper correction is made by using these 
 averages. In the case of races, the total heights are beyond 
 our powers of ascertaining, and then we must use a number 
 of each race sufficiently large to serve as samples, and must 
 average them arithmetically. Most statistical averages are 
 drawn for this purpose of comparison, especially in time 
 series, to see whether there has been improvement or 
 deterioration, with a view to finding causes, whereby we may 
 check the latter and forward the former. And whenever 
 such combination of the items is possible, the arithmetic 
 average is properly indicated. But an entirely different 
 purpose for drawing anthropometric averages was invented 
 by Quetelet. Quetelet wished to obtain the type of the men 
 of a country or of a race. This type is conceived as something 
 really existing, and if we could know it otherwise (by revela- 
 tion, say,) we might use it to see which kind of average fits it, 
 and thereby determine our choice of the average proper for 
 this case. But as we can learn it only by first drawing an 
 average, the kind of average to use is not determinable in 
 this way. We are thrown back on the other criterion 
 provided us by Galileo and Nozzolini in their controversy. 
 If the heights of men behave like estimates, the geometric 
 average should be used. If they behave like gains and 
 losses, the average to use is the arithmetic. The heights of 
 men have one resemblance to estimates : they never fall to 
 zero ; but they are very different from estimates in another 
 respect, in that they have never been found to exceed a 
 certain figure, which is hardly more above the general run of 
 human stature than the lowest recorded limit is below it. 
 They come under the class, therefore, of things with limits 
 equidistant on the opposite sides ; and the proper average 
 to use for finding the typical or normal height of man would 
 still seem to be the arithmetic. 
 
 The last example discloses a defect in part of the reasoning 
 above employed, which needs to be corrected. We have 
 found the resemblance of these later examples to our definite 
 models to consist in the behaviour, so to speak, of the terms. 
 If they behave like the arithmetic terms around their average, 
 or if they behave like the geometric terms around their 
 average, the arithmetic or the geometric average is to be 
 chosen accordingly. But this implies that we know the 
 average already, which sometimes is contrary to the suppo- 
 sition ; for in other cases, notably those of change or varia- 
 tion, knowledge of the total effect brings them under one of 
 
AVERAGES AND ERRORS 27 
 
 the models. Now, in the last example we suppUed for the 
 average what we there called " the general run of human 
 stature." This is something knowable when we have a large 
 assortment of data. In mathematics it has been called, in 
 our language, the " mode," which is a meaningless word. 
 The German Fechner named it the ' dichtester Wert," 01 the 
 " thickest value " ; and it is a pity we have not some equally 
 significant word — the " mean of greatest thickness " has 
 been suggested, which, however, is too clumsy. But in the 
 absence of a better word we must put up with this one. The 
 mode, then, is found by arranging the quantities under 
 treatment according to their sizes, and if, as is generally the 
 case when great numbers are collected, they cluster most 
 thickly around some one position, this is the mode. Now, it 
 may be found that the numbers in the different groups of 
 quantities thus arranged taper off equally, or about equally, 
 on each side of the mode. This is the behaviour of arith- 
 metic terms, and indicates that the arithmetic average of 
 them should be employed. But it may also happen that they 
 taper off more thinly and spread out more above than below 
 the mode ; and this is the way geometric terms around their 
 average behave. In theory, for arithmetic terms, if they be 
 grouped in numbers that fall within certain equal intervals — 
 for instance, if the heights of men be measured only by inches, 
 so that all heights falling say between 4 feet ii| inches and 
 5 feet I inch are grouped together ; then, on the supposition 
 that the group just mentioned is found to be the mode for 
 some people or race, the intervals equidistant above and 
 below it will contain equal numbers of items. But this will 
 not be the case with geometric terms. To form groups con- 
 taining as many items above as the corresponding groups 
 contain below the mode, the upper intervals must be gradually 
 enlarged as they recede from the mode. On the lower side, 
 theri, the intervals must be contracted, in order that they may 
 still be equal geometrically. For instance, on the supposition 
 that the stature of men followed the geometric arrangement, 
 the mode remaining the same as before, if the lower interval 
 around 2 feet 6 inches has shrunk to 2 feet 5I inches — 2 feet 
 6i inches, then the corresponding upper interval would have 
 
 its position and bounds set thus : its lower bound = ^A- 
 
 = 9 feet II inches, and its upper bound = -^ = 10 feet 
 
 I inch approximately ; so that this interval would be twice 
 as far removed and four times broader. 
 
28 THE PROBLEM OF ESTIMATION 
 
 What we have been doing, is to study the dispersion of 
 the data, and we find that there may be an arithmetic and 
 a geometric dispersion." The stature of men is found to 
 conform rather to the arithmetic dispersion ; and this is 
 the reason why the arithmetic average is the proper one to 
 use in averaging the heights of men (also for Quetelet's 
 purpose). Were it possible to measure our mental capacities 
 quantitatively with any exactness, Galton's investigations 
 of human intelligence indicate that it more hkely follows 
 the geometric dispersion ; for his researches disclose that 
 men of the greatest genius are to men of mediocre ability as 
 these are to idiots or witless men.^® If this be so, the proper 
 average for averaging degrees of intelligence would be the 
 geometric. To prove that actuality conforms with theory 
 in this subject, an infinite number of data would have to be 
 collected. In practice not even a very large number, mathe- 
 matically speaking, is ever collected. The data actually 
 collected, therefore, can be expected to conform only 
 approximately with the theoretical requirements. If the 
 mode in our collections agrees more closely with the arith- 
 metic average, this average should be employed ; and the 
 geometric, if it agrees more closely with the geometric. 
 It may, however, happen that the mode can be only roughly 
 outlined, and then if (as is likely) the two averages fall close 
 together, the accuracy of the discrimination may be greatly 
 impaired. It may also happen (which is not unlikely with 
 a small assortment of data) that the dispersion is so irregular 
 that no mode is discernible, or there may be no mode to 
 discern, according to the nature of the subjects. Then this 
 criterion has failed us, and we must do the best we can without 
 any, employing analogy with known cases where we can, 
 or else in practice adopting the most convenient average, 
 which is the arithmetic. Instances are known in which 
 two or more modes appear, which may be taken to indicate 
 that the data belong to different species, e.g., to different 
 
 1" By examining the behaviour of harmonic terms we might 
 find also a harmonic dispersion. But with this we are not 
 concerned. 
 
 ^' Cf. Hereditary Genius, end of Chapter III. Galton expresses 
 the matter in an arithmetic progression ; but his whole work 
 shows that the progression is rather the geometric. When he 
 wrote that work in 1869, he had not knowledge of geometric 
 dispersion. Cf. William James : " As the genius is to the 
 vulgarian, so the vulgar human mind is to the intelUgence of a 
 brute," Psychology, II. p. 348. 
 
AVERAGES AND ERRORS 29 
 
 races of men, when their stature or other measurements are 
 studied.^" 
 
 Doubtful cases are of frequent occurrence in statistics. 
 Statisticians frequently have to deal with ratios. Now, it 
 might be thought that because, when we deal with differences, 
 we must use the arithmetic average, therefore when we 
 deal with ratios, we should always have to use the geometric 
 average. But this is not necessarily so. If ratios express 
 variations — ^increases or decreases, — then the geometric 
 average must be used. Even in the case of human stature, 
 if our object is to measure its growth, as in the case of a 
 boy who has grown irregularly in five years from 4 feet to 
 6 feet, the proper average of the individual rates of increase 
 is the geometric. In other cases perhaps there is a pre- 
 sumption in favour of the geometric average ; but necessary 
 evidence is lacking. 
 
 To study ratios, we must first of all distinguish them 
 from mere fractions, since they are expressible in the same 
 form as fractions. Fractions proper express the compara- 
 tive number of parts in a common whole. Ratios express 
 the comparative relations between distinct quantities. 
 With fractions there is no difficulty. If we have occasion 
 to use the arithmetic average between 6, 8, and 9 inches, 
 we still must use the arithmetic average between these 
 quantities when expressed as fractions of a foot, viz. \, f, 
 and I ; and the result wUl be the same. But in social science 
 there is need, among other things, of expressing the ratio 
 of the annual deaths in a country's population to that 
 population, thereby obtaining what is called the country's 
 death-rate, which is usually taken to be the numerator when 
 the denominator has been reduced to 1,000. Here the deaths 
 are not parts of the population ; wherefore the death-rate, 
 though expressed in the form of a fraction, or as so many 
 thousandths, is not a mere fraction, but is a ratio proper. 
 Again, in anthropometry there is need of expressing the 
 ratio of the breadth of the head to its length, which ratio, 
 in a similar manner generally expressed as so many hun- 
 dredths, is called the cephalic index. Such cephahc indices 
 as 70, 80, 90, and the like are not parts of 100, but they are 
 relations to 100. Now, evidently the death-rate must lie 
 
 "" So Quetelet, of. cit., p. 143. Adolphe Bertillon has been 
 able thereby to distinguish two races in the Department of Doubs, 
 near the eastern border of France, La Theorie des Moyennes en 
 Statistique, in the Journal de la Sociiti de Statistiqne de Paris, 
 1876, pp. 289-93. 
 
30 THE PROBLEM OF ESTIMATION 
 
 between o and i, or as usuall}^ expressed between o and 
 1,000; for the deaths cannot sink below zero and cannot 
 rise above the population. As for the cephalic indices, it is 
 conceivable that they might rise above 100, as we know 
 no reason why the breadth (from right to left) of somebody's 
 head should not be greater than its length (from front to 
 back) ; but no such cases are found, nor are cases found of 
 the cephalic index falling much below 60. Thus there are 
 limits above and below, within which the death-rate and 
 the cephalic indices range, the latter having only empirical 
 limits, and the former having additional extreme limits. 
 These cases, therefore, belong to the class where the proper 
 average may be either the arithmetic or the geometric, 
 according to the position of the mode in a large collection 
 of such figures. Statisticians have not yet taken the trouble 
 to find which of the two is the proper one, and perhaps 
 many of their subjects will not yield clear results. They 
 have by common consent adopted the arithmetic average, 
 as being the easiest, like the butchers to whose example 
 Nozzolini appealed. Moreover, at least in the case of 
 cephalic indices, they have been bothered by having two 
 different methods at their disposal. 
 
 To calculate the cephalic indices of thousands of skulls, 
 and then to draw the arithmetic average of them, is a labo- 
 rious task. To shorten the labour, many anthropologists 
 have adopted the method of adding together all the breadths 
 and all the lengths, and of then drawing the ratio between 
 these as the one common or average ratio of the lot. This, 
 however, gives a slightly different result from the preceding, 
 sometimes above, sometimes below, though it may some- 
 times agree. Bertillon has given instances. Using only 
 two skulls for simplicity, he pointed out that if the one is 
 180 mm. broad and 200 long, and the other 112 broad and 
 160 long, the index of the former is 90 and of the latter 70, 
 between which the arithmetic mean is 80 ; but if we 
 divide the sum of the breadths by the sum of the lengths, 
 292 by 360, and multiply by 100, we obtain for the mean 
 index a higher figure, 81 -i. Again, if the one has the measure- 
 ments 144 and 160, and the other 140 and 200, the indices 
 are Hkewise 90 and 70, with the mean at 80 ; but if we 
 divide their combined breadths 284 by their combined 
 lengths 360, and multiply by 100, we obtain for the mean 
 index a lower figure, 79. Lastly, if the measurements be 
 144 and 180, and 136 and 193, jaelding the same indices 
 and the same mean, the latter method also yields the mean 
 
AVERAGES AND ERRORS 31 
 
 index 80. The proper method, Bertillon asserts, is to average 
 the indices separately, in the first way ; for then, he says, 
 their dispersion can be studied. ^^ It would seem more 
 correct to say that we should obtain as many indices sepa- 
 rately calculated as we can, and by arranging them in the 
 order of their magnitudes study their dispersion for the 
 purpose, first of aU, of discovering which is the proper 
 average to draw. It might be found that the proper average 
 is not the arithmetic, which Bertillon has simply assumed, 
 but the geometric. In the instances cited the geometric 
 mean between 90 and 70 is 79-3, which perhaps is the best of 
 all.22 But in practice the geometric average, even with the 
 aid of logarithms, is a very troublesome one. Therefore 
 even if it were the theoretically correct one, it might be 
 advisable in practice to employ one of the other two methods, 
 since these in practice give results very slightly different 
 not only from each other but from the geometric average. 
 Still, as the true one, the geometric average could serve 
 as a norm for deciding between the other two ; for " mathe- 
 maticians and anthropologists," says one of the most recent 
 writers on statistics, " differ as to which method is more 
 correct. "2^ Wliat, then, should be done ? 
 
 / // in 
 
 Letting jp> -uTf yr,, ■ ■ • represent the cephalic indices 
 
 (in their original form) of n number of skulls, we express the 
 three methods in these three formulae : 
 
 I fa' , a" , a'" , ' . \ 
 
 nW'^F'^W + •••to« termsj , («) 
 
 a' + a" + a'" + . . . to n terms 
 
 b' + b"+ &'" + . . . to « terms' (°) 
 
 nja' a" a'" r~ 
 
 V P ■ F ■ P^ ■ ■ terms, (c) 
 
 The first is the simple arithmetic average of the several 
 ratios, and the third is the simple geometric average of them. 
 The second is of a rather recondite nature. It can be shown 
 to be two things and very approximately a third thing. 
 
 "■ La Thdorie des Moyennes en Statistique, pp. 304-5. 
 
 "2 Indeed, the cephalic indices 90, 80, 70, etc., have a resem- 
 blance to the lesson-marks of school-children, which, we have 
 seen in Note 5, ought to be averaged geometrically. 
 
 ^^ Zizek, Statistical Averages, W. M. Persons's translation. New 
 York, 1913, p. 19. 
 
32 THE PROBLEM OF ESTIMATION 
 
 (i) It is the weighted arithmetic average of the ratios with 
 
 weights according to the figures in the denominators ; for 
 
 it is equal to 
 
 a' a" a'" 
 
 |-, . 6' + ^, . 6" + Ittt . V" + . . . 
 
 h' + h" + h'" + . . . ' ^'^^ 
 
 which is the formula for such a weighted average. (2) It is 
 the weighted harmonic average of the ratios with weights 
 according to the figures in the numerators ; for it is equal to 
 
 a' + a" + a!" + . . . 
 
 a' , a" a'" ' ^^1 
 
 a' IV '^ a" lb" '^' a'" lb"' + • ' • 
 
 which is the formula for such a weighted harmonic average. 
 
 And (3) it can be shown, both by theory and by trial, to be 
 
 in ordinary cases very approximately equal to this formula, 
 
 2 "^^j (a'\ V^ /a"\ V"'-" /a"'\ V^''^ 
 
 V WJ • W') -[w) • • • w 
 
 which is the weighted geometric average of the ratios with 
 weights according to the geometric means of both the 
 numerators and the denominators in the individual ratios.^* 
 It is of course also equal to 
 
 ^ {a' + a" + a'" + . . . to n terms) 
 
 I {b' + b" + b'" + . . . to M terms) 
 
 which is the ratio of the arithmetic average of the numerators 
 to the arithmetic average of the denominators : something 
 which need not further concern us, any more than the fact 
 that the third is equal to 
 
 ^/a' . a' 
 
 Vft' . b" . b'" ... to M terms 
 
 " It should be noted that if we change the figures in some of 
 the ratios, or in all of them differently, but without altering the 
 
 2 1 XCI d 
 
 ratios themselves (substituting for instance — for — , or — r-for-ri 
 
 and the like, but variously), this mere change of the figures does 
 not affect the result in the first and third formulae, but it does 
 affect the result in the second, since it alters the weights of the 
 ratios differently. Again, of course, if all the figures are changed 
 in the same proportion, no effect on the result is produced even 
 in the second ; lor weights are relative. 
 
AVERAGES AND ERRORS 33 
 
 which is the ratio of the geometric average of the numerators 
 to the geometric average of the denominators. The first 
 three facts have generally not been recognised by mathe- 
 matical statisticians, if even by mathematicians. Fechner, 
 for instance, has named the second formula the " siommary 
 mean," as if it were something distinct from the other 
 means, although he perceived that in it the ratios with 
 greater figures have greater weight.^^ It is extremely 
 important that the true natiire of this expression should be 
 recognised. 
 
 Now, in the case of the cephalic indices, when the second 
 method is employed, leaving aside the fact that its advocates 
 are employing, quite unconsciously, the harmonic average 
 of the ratios weighted according to the breadths of the heads, 
 and also (approximately) the geometric average of them 
 with weights combining the breadths and the lengths, but 
 confining our attention to the arithmetic average, we see 
 that they are really using this average of the ratios weighted 
 according to the lengths of the heads ! Evidently there is 
 no rhyme or reason for using such weighting as this. But 
 if the arithmetic average itself is a wrong one for the case, 
 perhaps it with a wrong weighting may give a better result 
 than it would give with the right weighting, since two 
 
 ^^ KollekUvmasslehre, 1897, pp. 164, 353, 357. On pp. 359-61 
 Fechner compared the second formula with the third ; but it 
 did not occur to him to compare it with /. Likewise Walras, 
 making no inquiry into weighting, distinguished a formula for 
 index-numbers of prices like the second, as a " multiple-standard 
 combination," from methods of averaging price-vaiiations 
 arithmetically or geometrically, as if it were something distinct, 
 D'une Methode de Rdgularisation de la Valeur de la Monnaie, 
 pp. 15-16 of the offprint from the Bulletin, Vol. XXI., of the 
 Society Vaudoise des Sciences Naturelles, 1885. And many other 
 economists have done the same. Even the mathematician 
 Cauchy, in his Cours d'Analyse, 1821, Vol. I., ist theorem, 
 although he demonstrated that the above second formula is an 
 average of the given ratios, yet did not point out what kind of an 
 average it is. This was done in The Measurement of General 
 Exchange-Value, pp. 504, 511, 519 (resting the proof of the 
 approximation of formula /to formulae h, d, and e on the fact that 
 the geometric average is always less than the arithmetic and 
 greater than the harmonic, between the same terms with the 
 same weights). But already Professor Irving Fisher had noticed, 
 in connection with price-variations, the identity of 6 with d only, 
 The R6le of Capital in Economic Theory, Economic Journal, 
 December, 1897, pp. 517, 520. 
 
34 THE PROBLEM OF ESTIMATION 
 
 wrongs may neutralise each other, while one wrong alone 
 remains wrong. We shall have occasion for this remark 
 later, whether it be applicable here or not. What statis- 
 ticians should do, therefore, is to find whether the simple 
 geometric average is the right one for these cases, and if it 
 is, then let mathematicians find which of the two practical 
 methods gives results in the long run closer to the results 
 given by the geometric average ; and if it should turn out 
 to be the second method, the second method should be 
 adopted, in spite of its curious weighting, for practical pur- 
 poses. Or if it turns out that the simple arithmetic average 
 is the right one, even then the second method may continue 
 to be used, as the most convenient, and not appreciably 
 erroneous for ordinary practice. At all events statisticians 
 will then know exactly what they are doing. 
 
 Thus the theoretically true method, even though not 
 adopted in practice, may serve as a guide to practice. It 
 is only by first getting the theoretically true method that 
 we can know how near to it our practical methods come. 
 The theoretically true method, therefore, is important, even 
 though we do not employ it in practice. ^^ 
 
 "' In De Morgan's Essay on Probabilities, 1838, pp. 156-7, 
 occurs a curious passage with regard to the death-rate and 
 similar ratios (wMch he speaks of simply as " fractions "). 
 Different calculations of the death-rate may give different results, 
 which need to be averaged. He supposes three such results to be 
 
 — , — , and -^, or in decimals 0-6, 0-591, and 0-567. The simple 
 
 arithmetic average of these is 0-586, and their average by the 
 second formula above, which he did not recognise as an arithmetic 
 average with weights according to the denominators (the deaths), 
 is 0-583, being smaller merely because it happens that the smaller 
 ratios have received the greater weights. Then he asserts that 
 neither of these is right, and the right (or most probable) result is 
 obtained by multiplying each numerator and denominator by 
 the denominator and dividing the sum of the new numerators by 
 the sum of the new denominators, which in this case gives 0-581, 
 being still smaller for the same reason enhanced. Now this, 
 though he did not recognise it, is the arithmetic average with 
 weights according to the squares of the denominators (the deaths). 
 No reason is apparent for such weighting, any more than for the 
 weighting in the second average. Note that if the calculations 
 of the deaths and of the population were made by six different 
 operators, the ratios can be combined in six different ways 
 
 {e.g.. one other would be ||, g, and g), and each of the three 
 
AVERAGES AND ERRORS 35 
 
 The upshot of tliis inquiry into averages is the discovery 
 of two criteria for their use, the second of which has some 
 subdivisions. The first criterion is a known result, with 
 which the mean or average must agree. The second criterion 
 comes into play when the result is the object sought, and it 
 consists in a resemblance between the ways the terms behave 
 and the ways terms behave around arithmetic and geometric 
 averages when these are known. Here there are three 
 different ways of behaviour : either (i) the terms extend 
 both above and below without any conceivable or assignable 
 Hmits, and then the arithmetic average must be used ; or 
 {2) there is a definite lower limit at or above zero and no 
 upper conceivable or assignable limit, and then the geo- 
 metric is the right one to use ; or (3) the terms either cannot, 
 or at least so far as we find in experience do not, fall below 
 some figure, and do not, or cannot, rise above some other 
 figure, which two figures serve as limits either necessary in 
 the nature of the things or found in practice to exist ; and 
 then, when enough data are provided for examining their 
 orderly dispersion, if their mode is found nearer to the arith- 
 metic average, the arithmetic average is the proper one to 
 use, and if their mode is found nearer to the geometric 
 average, the geometric average is the proper one to use ; or 
 else this criterion fails, and we are left to be guided by 
 analogy or by convenience. 
 
 It is important to note that the examination of the dis- 
 
 averages just used would most likely give a different result in each 
 case. But if we use the simple geometric average, it is indifferent 
 which combination be used, the same result being given in all. 
 This result for the above figures is 0-585. Another peculiarity of 
 De Morgan's treatment of this matter is that he offers the method 
 he so dogmatically recommends, as " the method of least squares " 
 (and so again in his article on Least Squares in the Penny Encyclo- 
 pcedia and in his Theory of Probabilities, § 124, in the EncyclopeeMa 
 Metropolitana, Vol. II., p. 452), although it is the first average 
 (the simple arithmetic) which always gives the least sum of the 
 squares of the residual errors in such cases, while the method he 
 recommends gives, in this case, the greatest sum of all ! What his 
 method does, is to give the least sum of the squares, not of the 
 errors simply, but of the errors multij lied by the denominators. 
 This, however, may be argued for on the analogy of the reduction 
 of errors in measuring, say, a mile ; for here we do not draw the 
 arithmetic average of the errors expressed as fractions of a mile, 
 but of the errors expressed absolutely (or as those fractions 
 multiplied by their denominators) . But it is questionable whether 
 this analogy holds. 
 
36 THE PROBLEM OF ESTIMATION 
 
 persion for this purpose is needed only in these last ambiguous 
 cases. If the data behave in either of the first two ways, 
 the average is determined by the second criterion with as 
 much certainty as it is by the first criterion, and there is no 
 need of reference to the mode in their dispersion. Take for 
 instance the second model above given : if all the lOO rates 
 of increase that have contributed to the doubling of the 
 population in a century were known, we could arrange them 
 in the order of their magnitudes and find the number of 
 rates falling in each magnitude, and it might turn out that 
 they exhibited the geometric dispersion ; but if they did 
 not, we shoiold still use the geometric average for finding the 
 average annual rate of increase. And it is quite possible 
 that with so small a total number of terms as loo, their 
 dispersion would not contain a clearly defined mode, or one 
 nearer to the geometric than to the arithmetic average ; 
 but we have no concern whatever it may happen to be. The 
 same certainty as to the appropriateness of the geometric 
 average appertains to the fourth model, and likewise to the 
 case of estimates — the estimates we have so far been dealing 
 with, — although this belongs under the second criterion. 
 It is possible that if a thousand persons should independently 
 estimate the length of a given line, we might find their 
 estimates to fall in geometric dispersion. But it is not 
 necessary to wait for this experiment before deciding that 
 the geometric average is the proper one for averaging esti- 
 mates of this sort. The appeal to the mode in the dispersion 
 of the data is only a last resort in dubious cases — and of 
 course only in case of sufficient importance to make it worth 
 while. 
 
 Estimates and Observations. 
 
 Coming back to estimates and their errors, we must note 
 first of all that the word " estimate " is used in two different 
 meanings. In the one meaning it refers to valuations men- 
 tally made, or drawn at sight or by the aid of other senses— 
 in short, to guesses, at least to such as are made with care. 
 In the other meaning an estimate is a measurement, or the 
 result of many measurements, or in astronomy they are 
 observations, since astronomical observations are frequently 
 accompanied by measurements. And the errors committed 
 in these two kinds of estimates may be distinguished as 
 errors of estimation and errors of observation, the former 
 being errors of guesses, the latter errors of measurements. 
 
AVERAGES AND ERRORS 37 
 
 Errors of estimation are the ones we have been dealing with, 
 and Galileo has shown that they must themselves be measured 
 or estimated geometrically. But errors of observation are 
 entirely different : they must be measured arithmetically. 
 For, although the former can and do go much further above 
 than below the true position, the latter need not deviate 
 more above than below. Surveyors in measuring and re- 
 measuring a given distance, say about a mile, let them 
 measure it no matter how carefully, never get results that 
 are not slightly discrepant, within limits of at least a few 
 inches ; but there is no reason to suppose their errors are 
 more above than below the true distance ; " .wherefore 
 when they average their measurements in order to eliminate 
 their errors as much as possible, they very properly use 
 the arithmetic average. These considerations show that 
 while estimates proper are to be averaged geometrically, 
 measurements are to be averaged arithmetically. If you 
 measure a line with instruments many times, you should 
 use the arithmetic average of the measurements ; if you 
 estimate a line by the eye many times, you should use the 
 geometric average of the estimates. 
 
 Whether Galileo had occasion to draw an average between 
 different observations of the same quantity, and which 
 average he used, it is not easy to gather from his works ; 
 but there is one passage which indicates that he would not 
 have hesitated to use the arithmetic average. In his 
 Dialogues, pubhshed in 1632, near the beginning of the 
 third, he maintains that in adjusting various discordant 
 calculations of the distance from the earth of a new star, 
 
 " In measuring a line with a ruler our errors, so far as caused 
 by the expansion or contraction of the ruler under changes of 
 temperature, may be greater in excess than in defect, because 
 expansion and contraction are subject to the geometric scheme, 
 since there is an absolute zero temperature and no superior limit. 
 But our position in the heat scale is so high compared with the 
 variations to which our rulers are exposed, that the difference in 
 their expansion and contraction above and below the standard 
 chosen is infinitesimally small and quite negligible. For a similar 
 reason, if whenever we look at a thermometer we first guess the 
 temperature, our guesses would probably be no greater (appre- 
 ciably) above than below the true figure. Here we should be 
 practically ainung, so to speak, at a single mark, the other end 
 of the scale being out of our experience ; wherefore the arithmetic 
 average should be used of such guesses. Thus guesses also are 
 of a two-fold nature. 
 
38 THE PROBLEM OF ESTIMATION 
 
 based on twelve observations in different latitudes, the inost 
 probable result is obtained by retaining aU the observations 
 and by employing the smallest possible emendations that 
 will bring them into agreement ; and he based the last on 
 the consideration that the errors of the observers were as 
 likely to be above as below the truth, and as great on the 
 one side as on the other, and more likely small than great.^s 
 All these assumptions underlie the principle of using the 
 arithmetic average in correcting observations. But whatever 
 his practice or want of it, his successors soon had to employ 
 an average in reducing their observations, and they rightly 
 adopted the arithmetic average. And they used the arith- 
 metic average so much, and their example was followed 
 by civil engineers and others to such an extent, that it came 
 to be forgotten that there were or might be cases that call 
 for the geometric average.^^ The suitableness of the geo- 
 metric average for estimates proper seems to have been 
 re-discovered in 1879 W Galton. Galton started from 
 Weber's law, which had twenty years before been thoroughly 
 elaborated by Fechner. Because nerves react, from a certain 
 initial position (after crossing the threshold of consciousness) 
 in arithmetic progression as the stimuli increase in geometric 
 progression, Galton argued that our estimates of weights 
 by handling bars, or of shades of colour by looking at them, 
 should be averaged geometrically. He remarked that " the 
 ordinary law of frequency of error based on the arithmetic 
 mean," which " asserts that deviations in excess must be 
 balanced by deviations of equal magnitude in deficiency," 
 requires that " if the former be greater than twice the mean 
 itself, the latter must be less than zero, that is, must be 
 negative " ; but " this is an impossibility in many cases, 
 to which the law is nevertheless applied by statisticians 
 with no small success, so long as they are content to confine 
 its application within a narrow range of deviation." He 
 would, therefore, extend the use of the geometric average 
 to sociological subjects as well,*" thus reversing the course 
 
 2' Opere, Vol. II. pp. 302 A and b, 311, and 312 A. 
 
 ^» The want of knowledge of the distinction between errors of 
 observation and errors of estimation is especially conspicuous in 
 the opening passage of Chapter VII. of De Morgan's Essay on 
 Probabilities. 
 
 '° The Geometric Mean in Vital and Social Statistics, in Pro- 
 ceedings of the Royal Society, 1879, XXIX. pp. 366-7.' Galton 
 won precedence by a narrow margin ; for, apparently in inde- 
 pendence of him, the next year in Italy, in a paper on // Calcolo 
 
AVERAGES AND ERRORS 39 
 
 of history, since Weber's law of vital phenomena had received 
 its first 'hint from Bernoulli in the field of sociology and 
 economics. Galton was followed by Professor Edgeworth, 
 who a few years later wrote : " The estimates which different 
 persons (or the same person at different times) might make 
 of a certain weight would be likely to err more in excess 
 than in defect of the true weight, and in such wise as to 
 render the geometric mean of such a series of estimates 
 the proper method of reduction." '^ But the re-discovery 
 has not yet made its way into general recognition and 
 acceptance.^^ 
 
 dei Valori Medii e k sue AppUcazioni staiistiche, in the Archivio 
 di Statistica, Anno V., 1880, also as a pamphlet, Rome, 1883, 
 pp. 31-5, Messedaglia reviewed the Galileo-Nozzolini controversy, 
 and accepted Nozzolini's solution for the purpose of measuring 
 gain and loss in buying and selling and Galileo's for the purpose 
 of measuring the precision of estimates, since estimation can err 
 with equal facility more in excess than in defect ; and immediately 
 thereupon he noticed the distinction between this proper geometric 
 way of measuring estimates and the usual arithmetic way of 
 measuring observations, and explained it by the subjectivity of 
 the former and the objectivity of the latter, and also by invoking 
 Fechner's exposition of Weber's law. But there Messedaglia 
 left the subject. 
 
 '' Memorandum to the first Report of the Committee on Index- 
 Numbers of the British Association for the Advancement of 
 Science, published in the Report of the Fifty-seventh (1887) 
 Meeting of the Association, 1888, p. 283. Again, in his Memo- 
 randum to the second Report of the Committee, in the Asso- 
 ciation's Report of the Fifty-ninth (i88g) Meeting, 1890, p. 201, 
 alluding to a hypothetical error in certain data of 25 per cent., he 
 adds in a parenthesis " (an error which, if possible in excess, is 
 almost inconceivable in defect)." He could not, however, find 
 proof of geometric dispersion in examination-marks, although he 
 expected it there, and suggested the use of the geometric average 
 for averaging the marks of different subjects, The Statistics of 
 Examinations, in the Journal of the Royal Statistical Society, 
 September, 1888, pp. 607, 625. 
 
 »" An example illustrating how the distinction continues to be 
 overlooked is furnished by Merriman, who in his text-book on 
 The Method of Least Squares, § 138 (still in the latest revised 
 edition, 1911), sets a problem about 1,600 persons guessing the 
 contents of a vessel, and after giving the (presumably arithmetic) 
 average of their guesses expects his students to solve a question 
 in connection therewith in the same way as all the other problems 
 in the book, all the others being about observations or measure- 
 ments. 
 
40 THE PROBLEM OF ESTIMATION 
 
 We however, having seen abundant reason provided 
 by Galileo and Nozzolini, for this distinction, are m a posi- 
 tion to accept it unreservedly. And now some further 
 study of the difference between the behaviour of observa- 
 tions, dispersing arithmetically, and of estimates proper, 
 dispersing geometrically, needs to be made. 
 
 Gait on gave this subject to the mathematician Dr. 
 McAlister to work out mathematically, which was done, 
 and the result was published in sequence to Galton's own 
 paper, under the title of The Law of the Geometric Mean. 
 Whether Dr. McAlister performed the task completely 
 and satisfactorily, does not appear to have been examined 
 by other mathematicians. It cannot be examined here. 
 All that can be done is to note the difference between 
 the curve of observations and the curve of estimates. 
 
 For by representing obser- 
 vations and estimates as 
 abscissae and their fre- 
 quency as ordinates, a 
 curve may be drawn ; and 
 such a curve, in the case 
 of observations, is well 
 known because of its bell- 
 shape or resemblance to a 
 gendarme's hat. It is more 
 or less elongated accord- 
 ing to the accuracy of the 
 observations. A common form of it is shown in Fig. i. 
 It is a symmetrical curve around the axis drawn vertically 
 upon the point t, which represents the true magnitude, 
 so drawn that the horizontal lines relate to each other 
 thus : 
 
 at = tb ; 
 
 for these lines represent, on each side of the axis, the amounts 
 of the errors, which in the case of errors of observations are 
 assumed to be equal on both sides of the true magnitude. 
 This magnitude tiss. distance from some point on the base 
 line to the left, which need not be indicated in the figure, 
 because it has no influence on the proportions of the curve. 
 But in drawing the curve of estimates it is necessary to intro- 
 duce this point (with its vertical line), because all the esti- 
 mates have reference to it. The curve of estimates wiU 
 therefore be like that drawn in Fig. 2, on the supposition 
 that the errors in defect of the true magnitude are somewhat 
 
 n' 
 
 ^ 
 
 -^ 
 
 ■.b' 
 
 a 
 
 / 
 
 
 f 
 
 \ 
 
 6' 
 
 nl 
 
 A 
 
 
 
 t" 
 
 
 \ 
 
 d"' 
 
 y 
 
 
 
 
 t" 
 
 
 
 \ 
 
 t 
 
 Fig. I. 
 
AVERAGES AND ERRORS 
 
 41 
 
 like those in the preceding case. The difference now shows 
 itself most strikingly in the curve on the right side of the 
 axis, since on this side are represented the errors in excess, 
 which extend as much beyond the true magnitude as the 
 errors in defect fall short of it, proportionally to 0, having the 
 relation 
 
 oa: ot = ot: ob. 
 
 It will be noticed that, in Fig. 2, if the magnitude which 
 is estimated be enlarged by pushing the zero point further 
 to the left, then, on the supposition that the curve on the 
 left remains the same by the estimates becoming more 
 accurate, the curve on the right will be drawn in toward 
 
 
 
 a 
 
 
 ->„_^ 
 
 ^' 
 
 'y 
 
 
 f 
 
 \ 
 
 ^ 
 
 ay 
 
 
 
 t" 
 
 
 \^6" 
 
 ^ 
 
 
 
 
 t" 
 
 
 
 
 Fig. 2. 
 
 the axis, and if that magnitude be increased infinitely by 
 removing the zero point altogether, the curve on the right 
 will become in all its points equidistant from the axis as the 
 curve on the left, reducing the whole to the same symmetrical 
 shape as in Fig. i ; for now we have virtually changed the 
 estimates into observations around a single point, having 
 done away with any reference to the line ot, and having only 
 the position of t to consider.^' In observations, it is true, 
 we always measure some magnitude ot, but on examining 
 our instruments in the neighbourhood of t, as also in com- 
 paring our observations afterward, we have no use for the 
 line ot ; but in forming and comparing estimates we must 
 always keep the reference to the Une ot. An estimate, in 
 fact, is always a ratio, since it has no meaning except in 
 relation to the true magnitude (determined by measure- 
 ment) ; but a measurement has independent validity, since 
 
 *' Gf. above, f note 27. 
 
42 
 
 THE PROBLEM OF ESTIMATION 
 
 it can be corrected only by another measurement.^ Com- 
 mon parlance bears out this distinction ; for we mostly 
 dissociate errors of measurement from the magnitude 
 measured, but in errors of estimation we always relate the 
 errors to the magnitude estimated.** 
 
 We have seen above that in theory the dispersion around 
 the mode, which in the figures is obviously the axis, gives 
 equal numbers of items at corresponding intervals on each 
 side, the difference between the arithmetic and geometric 
 dispersions consisting in the fact that in the former the 
 intervals are equal in breadth, but in the latter the intervals 
 above (to the right) are broader than the intervals below 
 (to the left) and progressively so as they depart from the 
 mode.*® In the case of observations we must suppose 
 
 '* Although a measurement does not give the exactly 
 true magnitude, it is so much more precise than an estimate, 
 that its indication is always conceived as the true magni- 
 tude in comparison with the estimate. Similarly, however, 
 measurements made with a rough instrument can have their 
 average controlled by measurements made with a more precise 
 instrument. " We can appeal from kitchen scales to an 
 atomic balance," as Professor Edgeworth has well said, in 
 the Journal of the Royal Statistical Society, September, 1888, 
 p. 601. 
 
 ^^ We may add the curve of harmonic dispersion : 
 
 
 
 
 0, 
 
 
 ^ 
 
 M 
 
 Qj 
 
 / 
 
 
 f 
 
 
 K 
 
 ./ 
 
 
 
 t" 
 
 
 
 ^ 
 
 
 
 f" 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 Fig. 3. 
 
 This must be drawn so that 
 
 oa : oh = at — oa : ob — ot ,- 
 
 ., . . , oa X ot 
 
 whence, if oa is given, ob = ^^^ _ ^ - ^ . The figure gives the pro- 
 portions correctly, but cannot vouch for either side separately. 
 
 3» In the calculus of the probabilities of errors of observation 
 use is made for various purposes of the areas contained between 
 the base and vertical Hnes and the curve, because the observa- 
 tions are equally distributed on both sides. As this is not the 
 
AVERAGES AND ERRORS 43 
 
 (i) that errors are equally likely to be above as below the 
 true magnitude, so that in the long run there will be an 
 equal number of them on opposite sides of the true position, 
 and (2) that equal errors are equally likely on the opposite 
 sides. We must believe these two things because we have 
 no reason for the contrary, and because the supposition is 
 supported by experience of the fact that in large numbers 
 of observations so-called residual (or reputed) errors do so 
 behave around the average of the observations.*' In the 
 case of estimation the first supposition still holds : we have 
 no reason to believe that estimates will be more numerous 
 on one side of the true position than on the other. But in 
 estimates the second supposition must be interpreted as 
 referring to geometric equality, or else be stated in the form 
 that greater errors in excess are equally as numerous as the 
 (arithmetically) lesser errors in defect to which they (geo- 
 metrically) correspond. As for the first supposition, which 
 is unequivocally common to them both, this involves that 
 the mode must coincide with what is called the " median," 
 or midmost item in the series, and both the mode and the 
 median must coincide with either the arithmetic average, 
 in arithmetic dispersion (of observations), or with the geo- 
 metric average, in geometric dispersion (of estimates). 
 This, remember, is the theoretical requirement. But the 
 theory itself requires that this condition shall be fulfilled 
 only in an infinite series. Short of infinity, the theory 
 declares that it will be very closely obeyed in a very large 
 series or collection of data, and that the discrepancy of fact 
 from the theoretical condition will increase as the number of 
 the data collected decreases. In practice, therefore, we 
 cannot expect the arithmetic average in its proper subjects 
 or the geometric average in its proper subjects in every 
 particular case exactly, and in some cases even approxi- 
 mately, to conform to the mode and still less to the median ; 
 or, which is the same statement reversed, we cannot expect 
 the mode or the median to coincide exactly with the proper 
 average of the subjects, or even with each other. Now, when 
 they do not coincide, it is these averages, the arithmetic 
 
 case in geometric dispersion, in which the estimates are pro- 
 gressively thinner from the left to the right, no such use can be 
 made of the areas in calculating the probabilities of eirors of 
 estimation. 
 
 " Residual errors are not true errors ; but their behaviour in 
 conformity with the theory of true errors adds inferential strength 
 to the theory. 
 
44 THE PROBLEM OF ESTIMATION 
 
 and the geometric, each in its proper subjects, which are 
 the summarising quantities, the single ones that may stand 
 for and take the place of the manifold quantities, because 
 yielding the same numerical results as they yield, or the 
 indicative quantities, as yielding most nearly what we wish 
 our calculations to yield (this last may be affirmed on the 
 analogy of the first) ; and it is not the mode or the median 
 that can serve these purposes of the averages. The mode, 
 and the median too, can serve, in dubious cases (and only 
 in such is their service needed), to help us find which of 
 the two averages is the correct one to use ; and then their 
 work is done. They cannot be properly used to oust and 
 to replace those other averages. Yet in practice, since the 
 mode and the median diverge as a rule but slightly from 
 the proper averages, they can, and have been, used for 
 convenience in their place (just as the arithmetic average 
 has been used in place of the geometric) ; but with them, 
 on the whole, the saving of labour is so small, and they, 
 especially the median, are so capricious, that it is poor 
 policy on the part of statisticians to recommend their 
 
 use 
 
 38 
 
 '' That is, as substitutes of the other averages ; for they may 
 
 serve other purposes. It may be questioned whether even 
 
 mathematicians have done well in choosing the median of the 
 
 errors to measure the " probable error " in observations, instead 
 
 of the arithmetic average of the errors. The fact that in a series 
 
 of observations there are as many errors beyond as within a 
 
 certain error does not import that in a repetition of observations 
 
 the errors will fall equally around that one rather than around 
 
 the arithmetic average of the previous errors ; and the fact that 
 
 mathematicians have chosen the arithmetic average of all the 
 
 observations to get the one most probably true, would seem to 
 
 require them to use the same average to get the error most likely 
 
 to be exceeded and to be fallen short of. Otherwise they might 
 
 as well have chosen the median of the observations as giving the 
 
 most probable true value ; for they could have argued thus : 
 
 The observations hitherto made are equally numerous on each 
 
 side of a certain magnitude, therefore it is probable they will 
 
 continue to be equally numerous around that magnitude; but 
 
 it is probable that observations fall in equal numbers on opposite 
 
 sides of the true magnitude ; therefore that (median) magnitude 
 
 is most probably the true magnitude. But they do not argue 
 
 this way in this case. Then why should they argue so in the 
 
 other ? But of course, as they have almost unanimously chosen 
 
 to use the median for the " probable error," too little advantage 
 
 could be obtained to make the change advisable. 
 
AVERAGES AND ERRORS 45 
 
 Still another point with respect to observations and esti- 
 mates, especially the former, needs to be cleared up before 
 we pass on. When dealing with the criteria of averages, 
 we could not speak of the criterion of the average proper 
 for observations without anticipating the distinction here 
 drawn between them and estimates ; and now having drawn 
 this distinction, there is need of reverting to the criteria 
 of the averages proper in them. The criterion of the geo- 
 metric average proper for estimates — ^that it must be used 
 of them because they have a lower limit and no upper limit 
 and consequently disperse more above than below their 
 mode — has never been disputed, since it has rarely been 
 considered at all. But in the case of observations, the 
 criterion enjoining the arithmetic average because their 
 limits are equidistant from their mode and consequently 
 they disperse equally above and below, has apparently 
 been thought too simple ; for something else has sometimes 
 been put in its place. 
 
 Errors of observation have been found to disperse them- 
 selves around their mode or median like the chances of 
 black and white balls coming out together in various com- 
 binations, from half-and-half up to all-of-one-kind, from 
 an urn containing an infinite number of black and white 
 balls in equal proportion — a condition which can be pro- 
 duced by putting the ball back in the urn after every draw, 
 as then there is no limit to the drawing. Mathematicians 
 have investigated the law of such chances, and have found 
 them to agree with the numbers in one of the lines of Pascal's 
 arithmetical triangle and with the co-efficients in the develop- 
 ment of a binomial raised to a power equal to the number of 
 balls in the combination ; and Gauss discovered the formula 
 of the curve following the dispersion of these combinations 
 around the central most probable half-and-half combina- 
 tion. And the law of such combinations, transposed, has 
 been called the " law of error." It is a pity it has been called 
 so, because it really is a law of chances, and it is a law of 
 error only in a different way. It is a law of chances, sub- 
 jectively, belonging to chances, as resulting from the con- 
 sideration of chances ; while it is a law of error only objec- 
 tively, as governing errors, because these, when accidental 
 and unbiassed, can be conceived as resembling even chances. 
 It is, too, a law of only one kind of errors, whereas we have 
 seen that there are two kinds, and the other is not subject 
 to this law. The law of the one kind of errors was originally 
 worked out for astronomical observations, which in practice 
 
46 THE PROBLEM OF ESTIMATION 
 
 conform to the law of chances ; for as measurements they 
 in the long run fall equally and equidistantly on both 
 sides of their mode, and their small residual errors are more 
 frequent than their large residual errors. The latter fact 
 is not esseatial to the propriety of applying to them the 
 arithmetic average ; but the former fact is. The fact that 
 the Umits of errors of observations are equidistant on 
 opposite sides of the true position, is the whole and sole 
 justification for the application of the arithmetic average 
 to observations and their errors. 
 
 Now, this conception of the subject has not always been 
 entertained ; for another justification or so-called proof of 
 the propriety of the arithmetic average for this purpose 
 has been thought to be supplied by what is known as a 
 " doctrine, principle, or method of least squares." Observa- 
 tions of a single magnitude are reduced by the simple arith- 
 metic average. But two or more magnitudes often have 
 a definite relation to each other, as for instance the internal 
 angles of a triangle, which together must equal two right 
 angles, so that when surveyors measure the three angles of 
 such a figure they know what the sum of their measurements 
 ought to be. Or, again, two ways of measuring the same 
 thing ought to agree, as when the elevation of a place above 
 the sea is measured from another place the elevation of 
 which has been measured, and then is measured from a 
 second place the elevation of which has been measured, 
 these two sets of measurements ought to agree. Such 
 measurements (individually made with several observations 
 reduced by their arithmetic average) hardly ever do agree 
 or fit together exactly. Therefore they need to be adjusted 
 by correcting, obviously not one of them alone, but all of 
 them. In such adjustments an average cannot be used. 
 But the arithmetic average has three properties— this 
 may be said notwithstanding that one of them belongs to 
 it only as it agrees with the median, (i) The sum of the 
 differences between the arithmetic average and aU the items 
 above it is equal to the sum of the differences between it 
 and all the items below it (or the algebraic sum of all the 
 differences is zero). (2) The sum of the differences between 
 the median (and consequently the arithmetic average so 
 far as it agrees with the mechan) and all the items, taken 
 absolutely, is a minimum. (3) The sum of the squares of 
 the differences between the arithmetic average and all the 
 items is a minimum. Each of these properties of the arith- 
 metic average (or the median) can be used in adjustments ; 
 
AVERAGES AND ERRORS 47 
 
 for the aim may be to make the differences required by the 
 corrections to behave in any one of these three ways. 
 Reversely, when any of these methods of adjusting the 
 measurements of several magnitudes is applied to the many 
 observations of a single magnitude, it reduces to the arith- 
 metic average (or in one case to the median) ; so that the 
 arithmetic average is safe in any case (except that in the 
 second it depends upon its agreement with the median). 
 Legendre and Gauss discovered the third method. Before 
 them the first had mostly been used, even by Laplace, who, 
 after Legendre's publication of the third method in 1806, 
 adopted and elaborated it. Which is the best, or the most 
 probable ? A good illustrative example has been given by 
 Whewell. He supposed a quantity known to increase 
 uniformly (as the rising of a star) to be observed at equal 
 intervals, and the measurements to be, say, 4 1/8, 12, 
 14 1/8. These must be reduced to an arithmetic progression, 
 and he instances three, viz., 6, 10, 14, and 4, 9, 14, and 
 5, ID, 15. In the first the differences between the measure- 
 ments and the corrected figures (that is, the reputed or 
 residual errors) are such that their algebraic sum is zero ; 
 in the second, their absolute sum is the least of the three ; 
 in the third, the sum of their squares is the least.*® The 
 first correction implies that the observer has made two large 
 errors and one very small one ; the second, that he has 
 made one very large error and two very small ones ; the 
 third, that he has made one large error and two small ones. 
 Obviously, on the assumption (always required) that all 
 the observations were made with equal care,*" the last 
 distribution of errors is the most probable one. Here is a 
 conjunction of the most probable result with the least 
 squares. If mathematicians can prove that this conjunction 
 is not accidental here, but is necessary, occurring in all 
 possible cases, they prove the method of least squares. In 
 effect, the method of least squares introduces a more even 
 low range of corrections, rather than many very small ones 
 and a few great ones ; and so it conforms with the assump- 
 tion that equally careful observations are more likely to 
 contain small than large errors. But mostly the mathe- 
 
 *' Philosophy of the Inductive Sciences, 2nd ed.. Vol. II. pp. 
 408-9. The figures are slightly altered. 
 
 *" If the observations were made with unequal care, they would 
 have to be given unequal weights ; and then the three methods 
 require that the sums of the differences so weighted should behave 
 as stated ; which is only an extension of the simple case. 
 
48 THE PROBLEM OF ESTIMATION 
 
 matical proofs have rested upon a mathematical need of 
 converting negative errors or negative observations into 
 positive quantities, which can be done first by squaring 
 them ; which need goes back to the formula of the curve 
 around the Y-axis, since in it there must be a square the 
 root of which can be both positive and negative.*^ All this 
 is not very convincing. But aU the proofs have presupposed 
 the very kind of dispersion which calls for the arithmetic 
 average. Herschel criticised Gauss for basing his proof 
 of the method of least squares upon the use of the arithmetic 
 average in reducing observations of a single magnitude, 
 which, he says, is a "a thing to be demonstrated, not 
 assumed." He then proceeded to give a proof of his own, 
 and based it upon three assumptions, the third of which was 
 the very assumption upon which the use of the arithmetic 
 average in reducing observations of a single magnitude directly 
 rests." Others have argued that because the method of least 
 squares is applicable to cases to' which the arithmetic average 
 is not applicable, and because wherever the arithmetic 
 average is applied, there the method of least squares is ipso 
 facto also applied, therefore the method of least squares is 
 the more comprehensive, and the proof of it is the desired 
 and long-missing proof of the arithmetic average.*^ They 
 forget that either of the other methods likewise comprehends 
 the use of the arithmetic average (at least the first directly, 
 and the second through the median). The arithmetic 
 average, on the other hand, comprehends aU these three 
 methods (or at least two of them). Each of them, therefore, 
 is only denotatively more comprehensive than it, while it 
 is connotatively more comprehensive than either of them. 
 
 *^ The figures given above are curves of observations and 
 estimates, all positive, and the Y-axis, or zero origin of the abscissae, 
 is on the left, and the secondary axis erected on t is the axis of 
 the true position of the magnitude observed or estimated. But 
 as curves of errors (in which form Fig. i is generally given), the 
 axis t is the Y-axis, or zero point of no error, the errors to the 
 right being positive and those to the left negative. It is to fit 
 this conception of the curve that Gauss's formula for errors of 
 observation was invented. 
 
 « Essays, 1857, pp. 397, 398. 
 
 *' " There is no doubt whatever," says Jevons, " that the 
 method of means [i.e.., the use of the arithmetic average] is only 
 an application of the method of least squares," Principles of 
 Science, 3rd ed., p. 386. Cf. Venn, Logic of Chance, 2nd ed., 
 p. 336 n. 
 
AVERAGES AND ERRORS 49 
 
 They forget, too, that the rendering of the least squares is 
 only a property of the arithmetic average, so that when they 
 give precedence to it, they commit a hysteron proteron, 
 very much as if metaphysicians should speak of a property 
 possessing a substance, or, to use a more homely similitude, 
 as when humorists speak of the tail wagging the dog. The 
 use of the arithmetic average, when it is properly used, 
 according to the criteria examined in the preceding section, 
 is not based upon, proved by, or even confirmed by the 
 method of least squares : ** quite the contrary, upon it is 
 based, by it is confirmed, and, if it be proved, by it (with 
 something else) is proved the method of least squares.** 
 
 ** Merriman, in his Method of Least Squares, § 80, says this 
 method " confirms " the use of the arithmetic average, and 
 refers to § 44 as doing so. But that article merely argued back 
 from the method to the average, going over in reverse order the 
 argument in § 41 which proved " the principle of least squares " 
 by resting it on " a fundamental law " proved in § 26 by a demon- 
 stration which rests on the primary supposition of equal dis- 
 persion, stated in § 24, which is independently the foundation 
 upon which the use of the arithmetic average rests. It is plain 
 that the method of least squares confirms the use of the arithmetic 
 average only as the conclusion of a long chain of arguments 
 confirms an earlier conclusion much more evidently proved 
 already. 
 
 " Better than that in the preceding note is the statement of 
 H. Jacoby, in An Elementary Lecture on the Method of Least 
 Squares, The School of Mines Quarterly, Columbia University, 
 Vol. XpCV. (1904), p. 294, that because Legendre's method of 
 adjusting observations of several magnitudes, when applied to 
 the observations of a single magnitude, reduces to the arithmetic 
 average, " this is a further strong addition to the evidence of 
 plausibility attaching to Legendre's theorem." But even so, of 
 course, the plausibility of Legendre's method is not increased 
 over that of the other two (or at least the first), but needs some- 
 thing else to prove it. It looks as if the method of least squares 
 was invented before it was known that the least sum of the 
 squares of residuals is a property of the arithmetic average ; and 
 when this property of the arithmetic average was discovered 
 through the method of least squares, it gave the appearance that 
 this method was primary and provided the basis supporting the 
 arithmetic average in its use in reducing errors of observation. 
 To substantiate this suggestion would require more research than 
 can be given to it by the writer. The subject of averages has 
 received little attention on the part of mathematicians, and still 
 less on the part of the historians of mathematics. 
 
50 THE PROBLEM OF ESTIMATION 
 
 And when the arithmetic average is directly applied, that 
 is, to single magnitudes of any sort — observations or any- 
 thing else, — the fact that it gives the least sum of the squares 
 of the differences between itself and the items has nothing 
 to do with the propriety of using the arithmetic average 
 when it is proper to use it. The principle of least squares is 
 observed when we average the population of a country 
 per square mile. As well say it underlies, supports, and 
 strengthens the use of the arithmetic average in that case, 
 as to say it does so in the case of averaging observations.** 
 
 It has been necessary to go into this matter so fully in order 
 to clear the ground of misrepresentations. The use of the 
 arithmetic average in reducing observations needs nothing 
 so complex and mysterious as the method of least squares to 
 rest on. And the use of the geometric average for reducing 
 estimates and other things like estimates, also needs nothing 
 
 " Fechner noticed that the median has the property that the 
 absolute sum of the differences (or the first powers of the differ- 
 ences) between it and the items is the least possible ; and the 
 arithmetic average has the property that the sum of the squares 
 of the differences between it and the items is the least ; and some 
 other average (the formula of which he could not find, but a 
 method for obtaining it he worked out) has the property that the 
 cubes of the differences between it and the items is the least ; 
 and there may be still another whose biquadratics give the least 
 sum, etc., etc. Hereupon he inferred that as the arithmetic 
 average is better than the median, therefore the average giving 
 the least cubes is better than the arithmetic, and the average 
 giving the biquadratics still better, and so on. And to get an 
 application for this sequence, he remarked that the median is 
 good enough when the observations are inexact, and the arith- 
 metic average is required for more exact measurements ; where- 
 fore he concluded that in general the averages giving the higher- 
 powered least sums are the proper ones for observations of greater 
 and greater accuracy: Ueber den Ausgangswerth der kleinsten 
 Abweichungssumme, in the Abhandlungen der koeniglich sachsischeti 
 Gesellschaft der Wissenschaften, Vol. XVIII., Math.-Phys. Classe, 
 Vol. XI., Leipsig, 1878, pp. 51-3- There is no sense in all this, 
 except that in adjustments possibly the methods using the least 
 sums of those higher powers might smooth off the corrections 
 still better than the method of least squares, although this is not 
 evident, as they might smooth them off too much. Neither 
 Fechner nor any one else has ever shown what the property of 
 yielding the least sum of differences raised to any power has to 
 do with the probability, in reducing observations of the same 
 magnitude, of approximating to the true magnitude. 
 
AVERAGES AND ERRORS 51 
 
 of that sort to rest on. Their uses are perfectly simple and 
 plain. 
 
 The Measurement of Errors. 
 
 The averaging of observations and estimates is one thing, 
 and the averaging of their errors is another ; for of course 
 errors are different from the observations and estimates 
 which contain them. Yet, as observations and estimates 
 are different from each other, and are to be averaged differ- 
 ently, so the errors contained in them are different, and are 
 not only to be averaged differently, but are to be measured 
 or reckoned differently. An account, then, of the methods 
 of reckoning errors is necessary. 
 
 Errors can be conceived in at least two ways — a geometric 
 and an arithmetic. The geometric is the one exhibited by 
 Galileo, being involved in his usage. According to it the 
 definition of error is this : A {geometric) error in an estimate 
 is the ratio of the difference between the true value and the 
 estimate, the true value being the subtrahend, (i) to the true value 
 when the estimate is less than the true value, and (2) to the 
 estimate when the estimate is greater than the true value, or 
 always to the greater figure, whether it be the estimate or the 
 true value. If we represent the error by R, the estimate by 
 S, and the true value by V, the definition is expressed thus : 
 
 i? = ^-^-^whenF>5,andi? = ^?-^-^whenS> V." 
 
 For example, something having the true value of 14, an 
 
 estimate of 13 contains an error = — = or 
 
 ■5 14 14 
 
 — 7-14 per cent. ; an estimate of 15. an error = — — — ^ 
 
 = -j or + 6f per cent. ; an estimate of 7, an error = ' 
 
 = Z or — 50 per cent. ; an estimate of 28, an error 
 
 = — ^ — i = -] — 4 or + 50 per cent. ; an estimate of i, an 
 
 *' If only the absolute figures are desired, the first should 
 
 V —S 
 read, R = — jr- when V > S. 
 
52 THE PROBLEM OF ESTIMATION 
 
 error = ?-~ ^"^ = — ^ or — 02 -8 per cent. ; and an estimate 
 14 14 ^ '^ 
 
 106 — 14 , 182 , „ 
 of 196 (= 14''), an error = ^ ^^^ = + i0 °^ + 92-8 per 
 
 cent.*® Here the same or opposite signs indicate the same or 
 opposite sides ; and equaUty or inequahty of percentage, in 
 absolute figures, indicates equality or inequality of error, but 
 not proportional inequality (as will be shown later), according 
 to this'geometric method of reckoning error, although it may 
 do so according to other methods of reckoning it. 
 
 The arithmetic method of reckoning error was similarly 
 outlined for us by Nozzolini, who gave it in two different 
 forms, the one of which is wrong except when it can be 
 reconciled with the other. The full arithmetic definition of 
 error is this : An {arithmetic) error is the ratio of the difference 
 between the true value and the estimate, the true value being the 
 subtrahend, to the true value. It is represented thus : 
 
 Nozzolini, we have seen, sometimes treated it as simply a 
 comparison of differences, represented thus : 
 
 R^S -V; 
 
 which means that an arithmetic error is the difference between 
 the true value and the estimate. This is correct when we are 
 dealing with errors around only one true value, as then this 
 one true value is the common denominator in all the fractions, 
 and may be dropped. It is sufficient in practice when the 
 difference between the measurement and the true value is the 
 
 ^' That this method of reckoning geometric error is correct, 
 is shown by the fact that an estimate of 13 has to the true value 
 
 JO 
 14 the ratio -^ = 0-9286, which being subtracted from unity 
 
 indicates a diminution of 7-14 per cent., which is represented as 
 an error by — 7-14 per cent., as above ; and the true value 14 has 
 
 to an estimate of 15 the ratio -^ = 0'9333, indicating (a diminu- 
 tion of the true value or) an augmentation of the estimate of 
 6f per cent., that is, an error of + 6f per cent. In general, with 
 
 the symbols above used, for a lesser estimate we have^ — i 
 
 s —V ... V s ~v 
 
 = — y^, and for a greater estimate i — -^ = — =; — . 
 
AVERAGES AND ERRORS 53 
 
 main thing (as when engineers dig a tunnel from opposite 
 sides of a mountain and wish the two sections to meet in the 
 middle ; for here it is the absolute difference which concerns 
 them, and not its relation to the length of the tunnel). But 
 the second definition always rests on the first, which contains 
 it. With the same example as before of a thing having a 
 true value of 14, the estimates below the true value have the 
 same errors as before, but above the true value the estimates 
 
 have different errors, one of 15 having an error = ^ ~ ^ 
 
 I I r J. r o 28 — 14 
 == H or + 7'i4 per cent., one of 28 an error = 
 
 = + I or + 100 per cent., and one of 196 an error = -§ 
 
 = + 13 or + 1300 per cent. Here, too, the same or opposite 
 signs indicate the same or opposite sides ; and equality or 
 inequality of percentage, in absolute figures, indicates 
 equality or proportional inequality of error, according to this 
 arithmetic method of reckoning error, proper of course only 
 when the arithmetic average is the proper one to use.*' 
 
 Another definition of arithmetic error, however, has been 
 given by Dr. Bowley in his Elements of Statistics.^" He 
 speaks simply of error, having in mind only the arithmetic 
 method of conceiving of error. His definition amounts to 
 this : Error is the ratio of the difference between the estimate 
 and the true value, the estimate being the subtrahend, to the 
 estimate. His formula, with the symbols here used, is 
 
 He adduced the example we have been using. The true 
 value being 14, an estimate of 13 contains an error = -^-fT^ 
 = 4- -i or 7-69 per cent., and an estimate of 15 an error 
 — -"-^ ~ ^5 _ or — 6-66 per cent. He unfortimately 
 
 *» We might also make a definition of harmonic error. Cf, 
 above, notes 11 and 18. 
 
 "SO Third ed., p. 201. The definition was given in an article of 
 his in the Journal of the Royal Statistical Society, December, 1897, 
 p. 856. 
 
54 THE PROBLEM OF ESTIMATION 
 
 neglected to take the precaution of trying his method on 
 extravagant instances. An estimate of 7 would have an 
 
 error = '''^ ~ ^ = + i or + 100 per cent., and an estimate 
 
 of 28 would have an error = — — 5 — = — - or — 50 per 
 
 cent. Obviously these results are wrong. When an esti- 
 mate decreases from the true value to half the true value, 
 it has decreased to 50 per cent, of the true value, and its 
 error has grown to 50 per cent. ; and when an estimate 
 increases from the true value to double the true value, it has 
 increased to 100 per cent, above the true value, and its error 
 has grown to 100 per cent. In the former case, an error of 
 100 per cent, below the true value is evidently the error of 
 an estimate at zero, but, according to Dr. Bowley, this would 
 have to have an infinite percentage ; while an estimate of 
 one, instead of an error of — 92-8 per cent., would have an 
 
 error = — = + 13 or + 1300 per cent. This scheme 
 
 also does not permit of dropping the denominator when 
 deahng with several estimates about the same object. 
 Moreover, the signs are contrary to the usual practice ; for 
 we conventionally treat excess as positive and deficiency as 
 negative. 
 
 Dr. Bowley goes on and gives some examples of the uses 
 that can be made of his formula of error. It may be profit- 
 able for us to follow him at least through the first. He 
 supposes two trade unions to return respectively 555 and 45 
 members as out of work when the true figures are 565 and 50, 
 
 so that the errors are in the first — and in the second - 
 
 according to his definition of error. He desires " the error 
 in the sum " of these estimates ; and he lays down a rule 
 that " the error in an estimated sum [ = a sum of estimates] 
 is equal to the sum of the errors in the parts when each is 
 multiplied by the ratio of the corresponding part to the 
 sum " (p. 203). Accordingly " the error in the sum " here 
 is obtained so, 
 
 — X 1^ + - X ^ = — , or + 2-5 per cent. 
 Ill 600 ' 9 600 40 ' -^ ^ 
 
 This is not satisfactory. The total estimates are 600 and 
 the total true values 615, so that, according to Dr. Bowley's 
 
AVERAGES AND ERRORS 55 
 
 way of reckoning error, the total error is ^ ^ "^ == —> 
 
 and the two methods tally ; for what is desired is, obviously, 
 the error in the two estimates added together compared with 
 the two true values added together, which is the same as the 
 error in the arithmetic mean of the two estimates compared 
 with the arithmetic mean of the two true values. But instead 
 of operating directly in this latter way, Dr. Bowley has sought 
 the true result by using only the errors and the estimates. 
 In reality he has been doing something which he does not 
 
 I 2 1 
 
 state. The resultant — lies between — and — , and so is 
 40 III 9' 
 
 a mean of some sort between them ; and what Dr. Bowley 
 is really after is to find the way of obtaining this mean 
 between the two errors, or what kind of a mean it is. Here 
 we are dealing with oversights, which can hardly err above 
 the truth, but below the truth may be admitted to err in 
 arithmetic progression, so that no question need be raised 
 about the use of the arithmetic mean. But the simple 
 
 211 
 arithmetic mean between — and - is not — . Evidently 
 
 III 9 40 ^ 
 
 uneven weighting must be used. Now, the operation 
 performed by Dr. Bowley is the same as this, 
 
 6^ (1-^x555 + ^X45) =4^; 
 
 and this is nothing but the arithmetic average between the 
 errors with weights according to the estimates. It is as if 
 
 Dr. Bowley argued : there are 555 errors of — because the 
 
 estimate with this error is 555, and there are 45 errors of - 
 
 because the estimate with this error is 45, wherefore the 
 weight of the former is 555 and of the latter 45. Dr. Bowley 
 is one of those who think that uneven weighting is of little 
 importance in drawing averages except when the figures are 
 few and very divergent (pp. 13-18, 205) — and he is more or 
 less right with regard to practice. The exception occurs 
 here, and uneven weighting is obviously a necessity. Only 
 Dr. Bowley has kept concealed (or is it that he did not know ?) 
 that he was using a weighted arithmetic average. But 
 though these two operations agree, it does not follow that 
 
56 THE PROBLEM OF ESTIMATION 
 
 they must be correct. Were only one mistake involved, there 
 would be little likelihood of agreement. There is need of 
 two mistakes to make agreement in error possible. It may 
 be shown that, apart from the wrong use of the signs, which 
 does not affect the case, another mistake has been committed 
 in addition to a wrong measure of error, namely a wrong 
 system of weighting. 
 The total error, according to the proper arithmetic method, 
 
 is T— — -1= , or — 2-44 per cent. ; and the single 
 
 errors are respectively ^^^ = — — and '^^ ~ ^ 
 ^ ■^ 565 113 50 
 
 = . The problem, then, is to find the weighted arithmetic 
 
 21 I 
 
 average between and that is . This is 
 
 '^ 113 10 41 
 
 given in the following operation. 
 
 Here the weights are according to the true values (not 
 according to the estimates), and this would seem to be the 
 right system of weighting, or at least a better one ; for the 
 errors are not on the estimates, but on the true values. 
 Otherwise the weights would vary with the estimates and 
 might be smaller on the larger object, whereas it seems 
 reasonable to hold rather that the weights should be fixed 
 according to the true values, an error on a larger figure being 
 more important than an error on a smaller, no matter what 
 the estimates may be.^'^ If, however, we think the estimates 
 ought to count in the weighting as well as the true values, 
 there is no tallying between these two methods of reckoning 
 the same total error. 
 
 The correct arithmetic process may be generahsed. Let 
 the true values be represented by v^, v^, V3, . . . to n terms, 
 and the estimates by s^, s^, S3, ... to w terms.^^ Then the 
 
 ""^ Dr. Bowley himself remarks : " The greater error in the 
 returns of the smaller union has little effect on the total " ; 
 which seems to imply that the effect on the result should be 
 according to the true values. He has said nothing about the 
 sizes of the unions, but only about the sizes of the returns. He 
 must mean here "the greater error in the smaller returns." 
 
 ^* There may be a larger number of estimates than of true 
 values, if one or more values have two or more estimates. The 
 true values must then be repeated for every additional estimate. 
 
% — Vs 
 
 AVERAGES AND ERRORS 57 
 
 arithmetically combined arithmetic errors of the totals is 
 (^1 + 52 + ^3+ •• ■ tow terms) — (t)i + V2 + V3+. ■ .tow terms) 
 Vi+ V2 + V3+ . . . to n terms 
 
 The individual arithmetic errors are -* -, — , 
 
 and so on. And the arithmetic average of these, 
 
 "3 
 
 weighted according to the true values, is 
 
 C-^0 -^ + (^1 "- + (^^) ^3 + • ■ • to . terms 
 
 '''i + 5^2 + ^3 + • • ■ to M terms 
 It is evident that this formula is equal to the preceding 
 formula. 
 
 When, now, we try Dr. Bowley's scheme on geometric 
 errors, we no longer have such plain sailing. It is impossible 
 to construct geometric formulaa of averages corresponding 
 to the two just given. We must try the two methods on 
 numerical examples. We shall now find agreement only in a 
 special case. 
 
 Let us suppose two towers, 500 and 600 feet high, to be 
 estimated as 600 and 500 respectively. The' individual 
 
 errors, geometrically reckoned, are 7 — - — = + -^ or 
 
 + i6f per cent, and - — ^ = ~ a °^ ~ ^^f P^"^ ^^^t. 
 
 We have seen that (i) the arithmetic mean between these, 
 which is 0, indicating per cent, of error, or no error, is 
 correct. In effect (2) the geometric mean between the 
 
 estimates, \ 7 — X = -, indicates that the mean esti- 
 
 V 600 500 I 
 
 mate has the ratio to the truth of i to i, showing that it is 
 correct and there is no error ; for the equal errors (geome- 
 trically reckoned) on opposite sides (geometrically) nullify 
 each other.*^ The same result is obtained (3) if we seek the 
 geometric error in the geometric mean of the two estimates 
 compared with the geometric mean of the two true values, 
 
 for this is 54772 ~ 54772 _ q Evidently the agreement 
 54772 
 
 ^2 Here, too, it happens that the comparison of the combined 
 heights and the combined estimates, the two sums being equal, 
 indicates absence of error. 
 
58 THE PROBLEM OF ESTIMATION 
 
 of these three ways of obtaining the desired result will always 
 take place when there is such alternation of two values and 
 two estimates. For, using the same symbols, we have the 
 condition s^ = v^ and s^ = Vi, and if v^ be the greater value, 
 Sg must be the greater estimate, and the two errors are 
 
 ~ 1 and ^ ^ , which on substitution of equals become 
 
 ~ i and — -, between which two (i) the arithmetic 
 
 Wl Vi ^ ' 
 
 mean is o, indicating no error ; and (2) the geometric mean 
 
 of the estimates, \/-^^ =\/ -^^ = i, likewise indicates no 
 V VjV^ V WiSi 
 
 error ; and of course (3) Vs^s^ — VvjV^, whatever be its 
 denominator, is = o under the condition, for this makes 
 S1S2 = fiy2- And if v^ be the greater value and Sj the 
 greater estimate, the result is the same after some mere 
 inversions of the terms. It can be shown also that the 
 condition may be enlarged to the extent of requiring only 
 S1S2 to be = v^v^, and then all the three methods wiU agree 
 in indicating the absence of error in the whole result. Under 
 this broader condition the common indication of no error 
 is obvious in the case of the two last methods, and needs to 
 be proved only of the first. We desire, then, to prove that, 
 S1S2 being = VjV^, the arithmetic mean of the (geometric) 
 errors is zero. Again there are two cases, for (unless all the 
 terms are equal, in which case there is no difficulty) either 
 Vi is greater than s^, and then s^ must be greater than V2 
 and the geometric means are as above, or v^ is greater than 
 Sg, and then Sj is greater than Vi, and the geometric errors 
 
 are ^ ~ ^ and ^ ~ '^^ . In the first case the arithmetic 
 
 Si V2 
 
 mean between the errors is = ^^^^ ^i i : \ i ^t-i^ 
 
 2 1^152 
 
 and in the second case it is (^i - '"il^i + (^2 " ^2)^1 , But 
 
 2112 51 
 
 from the condition we get s^ = -*-^ and v^ = ^-^. By 
 
 Si v^ 
 
 substitution of these equivalents, and with further help 
 
 from the condition, the formulae reduce to — ^ ^ ^ 
 
 2V-, 
 
AVERAGES AND ERRORS 59 
 
 and ^^~ ^'^7''"^~ ^^ both of which are = o. Further, 
 
 with three or more values and estimates, subject to this 
 condition (viz. s^s^s^ . . . = ViV^Va . . .), although the 
 second and third methods will evidently continue to agree, 
 the first will not agree with them, and so wiU be incorrect. 
 
 In other cases (the more usual complex ones) the first 
 method fails entirely, and the third agrees only partially with 
 the second, which remains the correct method. Suppose the 
 same two towers, 500 and 600 feet high, to be estimated as 
 550 and 450 respectively. The first geometric error is 
 
 — TTT — ~ "^ ii ^"^ "^ ^^^ P^^ cent., and the second is 
 
 450 — 600 I ^ . ^T_ -xl 
 
 g = — - or — 25 per cent. The arithmetic mean 
 
 between these is — 7-95 ; but this does not correctly indicate 
 the geometrically reckoned whole error.^^ The correct result 
 is obtained in the second way, by drawing the geometric 
 mean between the estimates stated in terms of their true 
 
 values, thus W — X J^ = V'o-825 = 0-9083, or — 9-17 per 
 
 cent. This can again be obtained as follows. Reduce both 
 
 the true heights to unity ; the corresponding estimates then 
 
 a re i-io and 75, a nd the geometric mean between these is 
 
 Vi-io X 075 = Vo-825, as before. The reason why the 
 
 arithmetic mean between the geometric percentages of errors 
 
 does not in such cases give the right result is because, as 
 
 already stated, the geometric method of reckoning error does 
 
 not correctly give the proportion of one error to another. 
 
 This is easily seen ; for the estimate 121 is known to be twice 
 
 as erroneous as the estimate no of a true value 100 (for 100 : 
 
 no = no : 121} ; but the percentages of error, viz. 10 and 21, 
 
 do not correctly indicate the proportional magnitudes of the 
 
 errors. Wha,t is equal here is the excess of no above 100 
 
 ,^, j: -u i no — 100 121 — no 
 
 and the excess of 121 above no, for = . 
 
 no 121 
 
 Errors are to be compared by analysing them in this way.^^^ 
 
 '3 Nor is it correctly indicated by the added heights compared 
 
 1000 
 with the added estimates, since --— = 0-90909, or 9-09 per cent., 
 
 the same (so it happens) as the first error alone. 
 
 "" Cf. The Measurement of General Exchange-Value, pp. 250-3. 
 
6o THE PROBLEM OF ESTIMATION 
 
 Hereby another proof of the above result is obtained. The 
 estimate of the second true value below 600 that equals in 
 error the given estimate of the first true value above 500, is 
 the estimate 545 '45, for 550 : 500 = 600 : 545-45. But the 
 given estimate of the second true value is still lower, so that 
 
 its unbalanced error is 45 ^^^ ^^ = — 0'i75, or an error 
 
 545 "45 
 of — I7'5 per cent. But this error is distributed over two 
 estimates, the other of which is correct, since it has been 
 balanced. This error is the error of an estimate 0-825, when 
 the true value is unity. The other estimate, being correct, 
 is unity. The geometric mean between these is Vi x 0-825 
 = 0-9183, as before. Furthermore, the correct result in the 
 instance before us, and in all instances in which the product 
 of the true values is greater than the product of the estimates, 
 is given by the third method also, which consists in seeking 
 the geometric error in the geometric mean or average be- 
 tween the estimates compared with the geometric mean or 
 
 average between the true values ; for this is 497 4 ~ 54/ 7 
 
 ^ 54772 
 
 = — 0-0917, or — 9-17 per cent., as before. We may generalise 
 this, continuing to use the same symbols. The formula of 
 the second method, giving the correct result in geometric 
 
 percentage of error, is 100 iyj -i-^ ~^) > ^^^ ^^^ ^^^ method, 
 
 when the product of the true values is the greater, has the 
 
 formula 100 /'^^i^a ~ ^I'^^zN ^ which is evidently equal to 
 
 that 5* ; but when the product of the estimates is the greater, 
 
 the formula is 100 /^^iSa - Vt^ii^z') ^ ^^ that in these cases 
 
 this method does not agree with that. The two agree, of 
 course, also when s-^s.^ = v-^^ ; for this is the condition in the 
 preceding paragraph over again. What is here proved of 
 two values and two estimates can evidently be proved of any 
 number of values and the same number of estimates. But 
 beyond these methods and the cases to which they are 
 applicable the correct result cannot be obtained by any 
 
 ^*- For in these cases, with the product of the true values the 
 greater, the geometric method of reckoning error is the same as 
 the arithmetic. 
 
AVERAGES AND ERRORS 6i 
 
 assignable mean or average drawn merely between the 
 (geometric) errors, with any weighting suggested by and 
 confined to the figures given. No geometric mean can be 
 drawn between a positive and a negative quantity. If we 
 replace the negative error by its estimate, the geo metric 
 mean between i-ogog and 075 (= Vi-5^ x 075 = Vo-8i) 
 = 0-9045, indicating an error of — 9-55 per cent., which is 
 wrong, though only to a trifling extent. 
 
 Thus only in the simple but rare cases where there are two 
 estimates of two values such that the product of the estimates 
 is equal to the product of the values, can we correctly get the 
 geometrically combined result by drawing an arithmetic 
 mean between the geometric errors — for a geometric mean 
 between them is impossible. Here is a difference between 
 the behaviour of the geometric average and of the geometric 
 mean. The geometric mean when it is applicable (between 
 two sets of two figures whose products are equal) gives a 
 correct indication of the result. But the geometric average 
 (between two unequal sets of two figures, or between three 
 or more sets, whether equal or unequal, provided they do 
 not form two equal groups) does not give a correct indication 
 of the result. This breakdown of the geometric average 
 (between many quantities), in spite of the correctness of the 
 geometric mean (between two evenly weighted quantities), 
 will trouble us again. 
 
 But there is a problem in connection with errors of esti- 
 mation involving unequal quantities and uneven weighting, 
 in which the weighted geometric average works as smoothly 
 as does the simple geometric mean. This is a problem in 
 which the different quantities or magnitudes are joined to 
 each other endwise on opposite sides of a single true position, 
 and so form one geometric progression. Such a problem 
 is presented when the estimates of a single magnitude are 
 given and their comparative errors are supposed to be known, 
 and the true magnitude is sought. This is a widening of one 
 of the problems included in the seventeenth-century contro- 
 versy. The original question did not leave room for variety 
 of weighting. For, as posited, the problem was to apportion 
 the errors of known estimates around a single known value ; 
 and no question of any but even weighting could come in. 
 And when it was altered into the form of deriving the true 
 value from the estimates and the errors, the errors were 
 assumed to be equal, so that it still could employ only even 
 weighting. We may now, however, assume the errors to be 
 unequal, wherefore we shall have to weight the estimates 
 
62 THE PROBLEM OF ESTIMATION 
 
 unevenly. In this connection there is no failure of the 
 weighted geometric average to give the correct result. 
 
 To begin with the simplest case, let us suppose B.'s estimate 
 of a certain value to be twice as erroneous as A.'s estimate of 
 the same. This means that B.'s error is twice as great as A.'s, 
 or that it takes two of A.'s errors to equal B.'s one. Therefore 
 if we suppose another person to repeat A.'s estimate (and 
 consequently A.'s error), we have three estimates, two of 
 which are alike ; and the errors of these two together (not 
 necessarily their sum, yet the last being conceived as begin- 
 ning where the first left off) are equal to B. 's one error. Thus 
 A's estimate must be weighted as two to B.'s estimate as one, 
 that is, estimates must he weighted inversely according to their 
 errors [or directly according to their accuracy). This rule, of 
 course, holds for this kind of problem, and not necessarily for 
 others. It is different from the system of weighting we 
 found necessary for Dr. Bowley's problem also when applied 
 to estimates. There we found that the weights should be, 
 not according to the estimates or their errors (although those 
 were supposed to be known), but according to the true values ; 
 for that problem involved more than one true value. So 
 again in the problem of reducing observations (and the same 
 in the case of reducing estimates) in order to get the most 
 probable value (the niost probably nearest approach to the 
 true value), the different observations are weighted in some 
 proportion to the care with which they were made, and not 
 according to their errors ; and so, if they are all made with 
 equal care, they are weighted evenly, notwithstanding that 
 some may be much more erroneous than others.^*' 
 
 The weighting being as described, if we suppose the two 
 
 ^*' In most practical cases, if, after the probably truest value 
 has been ascertained in this way, the observations (or estimates) 
 were to be weighted inversely according to their probable errors 
 and the operation of averaging them thus weighted were to be 
 repeated, so little would be gained in accuracy as to make this 
 additional operation not worth the very much enhanced trouble. 
 Theory shows that in an infinite series of observations (or esti- 
 mates) the weighted reductions would not differ at all from the 
 unweighted ; and when there are many observations (or 
 estimates), the difference between them would be very slight. 
 When the observations (or estimates) are few, it may be ques- 
 tioned whether this second operation ought not to be performed. 
 And yet in these cases, as no precise result can be obtained, it 
 would be difficult to prove that the second operation yields a 
 nearer approach to the true value than the first alone. 
 
AVERAGES AND ERRORS 63 
 
 errors of estimation to lie on the same side of the true 
 value, there is little difficulty. It is merely a question of 
 geometric progression, and not of geometric averaging, and 
 so does not interest us. We are interested in cases where 
 the estimates he on opposite sides of the true value, so that 
 the true value, lying between them, is some average between 
 them— between the estimates, not (of course) between the 
 errors. Let us suppose A.'s overestimate of an article to be 
 200 and B.'s underestimate of it to be 50 ; and the supposi- 
 tion still being that the error in B.'s estimate is twice as 
 great as the error in A.'s, the problem is to find the true 
 value of the article. If we were dealing with observations 
 and the arithmetic average were employed, the answer 
 
 would be -- (200 + 200 + 50) = 150, indicating that 150 
 
 is the true value. According to this, A.'s error would be 
 + 50 and B.'s — 100, which at once yields the doubleness 
 of B.'s error over A.'s, in absolute differences. Or if we 
 reckon them in percentages, we should have A.'s error 
 
 200 — 150 ,1 , . , J T> . 
 
 = — - — = -f- - or -f 333 per cent., and B. s error 
 
 = = or — 66| per cent., Hkewise indicating 
 
 the doubleness of B.'s error. But we are dealing with esti- 
 mates proper, and we have learnt that a lower estimate 
 contains an error greater than its absolute figure or per- 
 centage arithmetically reckoned indicates, so that if the true 
 value were 150, B.'s error would be more than double A.'s. 
 
 The geometric average is \/200 X 200 X 50 = 126 nearly, 
 
 whereby A.'s error is shown to be + 74 and B.'s — 76, or 
 nearly equal in absolute figures ; but as the error of the 
 lower estimate counts for more than the other, it may be 
 twice as great. When we reckon the errors in percentages 
 
 by the geometric method, we get A.'s error = 
 
 = -f — or + 37 per cent., and B.'s error = § 2 — 
 
 100 I ->/ i' 126 
 
 = closely, or — 60 per cent., which is not double the 
 
 other, though nearer to being so. But again we have seen 
 that, while equality in this reckoning indicates equality of 
 
64 THE PROBLEM OF ESTIMATION 
 
 error, inequality does not indicate proportional inequality 
 of error. We must now find how to calculate the doubleness 
 and other manifoldness, geometrically reckoned, of one 
 estimate over another on opposite sides of the true value, 
 when their comparative erroneousness is known. 
 
 For this we must begin with the easier task of comparing 
 estimates on the same side of the true value. Let v as before 
 be the true value, s a given estimate, and x (which we desire 
 to find) an estimate twice as erroneous, that is, containing 
 an error twice as great. These three He in geometric pro- 
 
 gression m the order named, v, s, x, in which x = —. Now 
 
 V ' 
 
 two equally erroneous estimates (with the same symbols 
 
 accented) on opposite sides of the true value form the 
 
 7,2 
 geometric progression s', v, x', in which x' = —.. In this 
 
 s 
 
 last any value of s' may be substituted. Hence an estimate 
 
 s 
 
 2 
 
 — on one side of the true value is equalled in error by an 
 
 estimate on the other side -jj- = -„. Therefore, as this 
 
 s'/v s^ 
 
 estimate - is twice as erroneous as the estimate s on the 
 
 V 
 
 same side, and as it is equalled on the other side by the 
 
 estimate -g, an estimate —^ on either side of the true value 
 
 is twice as erroneous as the estimate s on the other side. 
 Applying to our example, we replace v by 126 and s by 200, 
 and we find in effect that an estimate on the other side twice 
 
 126^ 
 as erroneous as an estimate 200 is — ^ = 5^. which agrees 
 
 with the supposition, and shows that the error — 76 con- 
 tained in the estimate 50 is twice as great as the error 
 + 74 contained in the estimate 200. 
 
 To generaHse the problem, we may proceed with the 
 supposition that B.'s estimate is thrice as erroneous as A.'s. 
 Then on the same side the estimates form the geometric 
 progression v, s, x, y. Here we already have the value of 
 X in s, and by substituting that value for x in the progression 
 
 s, X, y, and resolving, we obtain y = -j. If B.'s estimate is 
 
AVERAGES AND ERRORS 65 
 
 four times as erroneous, we may abbreviate the progression 
 to this, V, X, z, and by substituting the value of x in s, we 
 
 £_ 
 
 iti-i 
 
 obtain ^ = ^. We now have sufficient basis for the mathe- 
 matical induction that the powers of s and v increase by 
 unity with every additional erroneousness supposed. In 
 general, then, on the same side of the true value an estimate 
 
 n times as erroneous as a given estimate s is -^. And 
 now, by the same reasoning as before, this estimate 
 on either side of the true value is equalled in error by an 
 estimate — ^ on the other side ; and therefore, to a given 
 estimate s on either side of the true value, the estimate on 
 
 the other side that is n times as erroneous is —r-.^^ Here. 
 
 s 
 
 of course, n may stand for a broken number, or fraction, as 
 well as for an integer. 
 
 Now, when an estimate s^is n times more erroneous than 
 estimate s^, these estimates are the same as if we had n 
 estimates like Si, and one estimate Sg, or w + i all told ; and 
 the geometric average between these, which yields the true 
 
 value of the article, is ys{^s^. In effect, the estimate n 
 
 yit + i 
 times more erroneous than the estimate s^ being — — , the 
 
 "+1 /" j;» + i 
 
 geometric average between these, -J ^1 • — s". is = ^• 
 
 V Sj 
 
 But the extension of this problem to the averaging of 
 three or more estimates again leads to confusion. When 
 the numbers are odd, it is impossible to have equal numbers 
 of estimates on the opposite sides (unless one is correct, 
 when it may be ignored). Even when the numbers are 
 
 *^ E.g., a lower estimate three times more erroneous than the 
 estimate 200, when the true value is 126, is 31^ ; and one four 
 times more erroneous is igf, while one equally erroneous is 79^. 
 All these form the geometric progression 19I : 31^ : 50 : 79J : 
 126 : 200. The figures going from 126 to the left represent 
 estimates respectively one, two, three, and four times as erroneous 
 as the estimate 200. 
 
 P.E. F 
 
66 THE PROBLEM OF ESTIMATION 
 
 even and the estimates faU equally on the opposite sides, 
 it is impossible to get the proportional weights for more 
 than two of them at a time. But the difficulty is the same 
 in the case of observations, even with the use of the arith- 
 metic average.^^ The fact is, however, that any two alone 
 of the estimates (or observations) are sufficient, with the 
 known comparison of their errors, to indicate the true value, 
 and the indication yielded by any other two, provided their 
 comparative errors have been correctly stated, wiU be the 
 same. As there is no need of using more than two opposite 
 estimates (or observations), there is no need of incurring the 
 difficulty. 
 
 Indeed, this whole problem, like the one cited from 
 Dr. Bowley's book, seems futile, since we never can know 
 the erroneousness of estimates (or of observations) without 
 first knowing the true value itself. The method of solving 
 it can be of use only in cases where we have some reason for 
 believing the errors of two estimates (or observations) to 
 have a certain proportion to each other ; for then the result 
 obtained for the true value wo\ild have the same amount 
 of plausibility in its favour. In practice, however, all that 
 we can know is that two or more sets of estimates (or 
 observations) of the same magnitude have different " prob- 
 able errors," that is, errors of certain amounts equally likely 
 to be exceeded as not whenever other estimates (or observa- 
 tions) are made in the same ways. These probable errors 
 cannot be treated like true errors, as they have a law of 
 their own. Now, to repeat, in averaging many estimates, 
 just as in averaging many observations, of the same magni- 
 tude, we must attach to them equal weights, provided they 
 have been made with equal care and with equally good 
 instruments, and independently of one another. But if we 
 possess two calculations of estimates of the same magnitude, 
 the one set made by experts and the other by less experienced 
 persons, we shall probably find less dispersion in the former 
 than in the latter, which would be a sign of the greater 
 accuracy of the former. In the analogous case of observa- 
 tions, when we possess two calculations on the same magni- 
 tude, the one set made with instruments of greater precision 
 than the other, the dispersion in each set is used to calculate 
 
 ^« But if the errors are all equal (arithmetically or geometri- 
 cally) in pairs, every one on one side having another corresponding 
 to it on the other, there is no difficulty about averaging them all 
 together simply, either arithmetically or geometrically. 
 
AVERAGES AND ERRORS 67 
 
 the probable error of its result ; and now in drawing the 
 average between the results of two such sets the usual 
 practice is to weight them inversely according to the squares 
 of their ■probable errors.^'' This operation should rest on the 
 assumption that the two sets err on opposite sides of the 
 true position. Such a distribution cannot be known, but 
 the assumption can be made on the general principle that 
 errors are as likely on the one side as on the other ; and, 
 too, if they do happen to be on the same side, we do not 
 know on which side it is, and so even in this case the chance 
 is as good of decreasing as of increasing the error, while in 
 the case of their happening to be on the opposite sides 
 (equally likely), the chance is improved of decreasing the 
 error. When there are three or more sets with different 
 weights, they must still be averaged, on the general principle 
 of probabilities, in spite of the fact that this cannot be done 
 in the case of true errors ; but there is no difficulty about 
 averaging any number of equally good sets. As for estimates, 
 they ought to be treated in the same manner, with substitu- 
 tion of geometric methods. Only in these cases there is 
 little need of such treatment, since for greater accuracy we 
 had better betake ourselves to measurements. 
 
 " Cf. Merriman, The Method of Least Squares, §§ 63, 88. 
 
III.— DEVIATIONS AND VARIATIONS. 
 
 If the treatment of errors of estimation were the only use 
 we had for the geometric method, its difficulties would be 
 of little moment. But this is not the case. Characteristic 
 of errors is it that they are deviations from a central posi- 
 tion, either known or conceived as the true position. Allied 
 to them are variations, which differ from them only in 
 that variations are changes from one position to another 
 equally good. Deviations are conceived as going in opposite 
 directions from a centre. Variations may meet and cross 
 one another while moving in opposite directions between 
 distant positions. But just as deviations may have their 
 starting points separated, so variations may have their 
 starting points brought together, and then they move like 
 deviations in appearance, but with the important difference 
 just noted, that in deviations the central position is viewed 
 as their proper resting place, while in variations we know 
 no reason why the position from which they start is better 
 than the position in which they end. 
 
 Deviations and variations are common subjects in statis- 
 tics. Here the central position is viewed, for deviations as 
 a type or norm from which the objects depart, and for 
 variations merely as a common starting point artificially 
 made. And the purpose of drawing averages may be, for 
 deviations to find the type or norm, and for variations, not 
 to find the common starting point, which is known, but to 
 find the average amount of the deviations, in order to learn 
 how much the objects have varied aU taken together or as 
 a whole. And all the deviations and variations dealt with in 
 statistics fall into at least the two kinds we have been 
 discoursing upon. There are deviations and variations 
 which depart from a starting position (roughly indicated 
 for the former by the mode or median, and afterward more 
 exactly by the average adopted) about equally on both 
 sides ; and there are others which depart from it more 
 above than below it, as reckoned in the usual arithmetic 
 way. Deviations and variations that disperse equally on 
 the opposite sides are mostly deviations of single quantities ; 
 those which disperse more above than below are mostly 
 
DEVIATIONS AND VARIATIONS 69 
 
 ratios between independent variables, although this is not 
 a necessary or proved distinction. It is obvious that the 
 arithmetic average is the right one to use of deviations and 
 variations that move in the former way, and the geometric 
 of those which move in the latter way. 
 
 A great many of the deviations and variations dealt with 
 in statistics, like the heights of men already noticed, behave 
 in the former way, and are fit subjects for the arithmetic 
 treatment. In ignorance of any other kind of errors, statisti- 
 cians have generally likened all the deviations and variations 
 they deal with to errors of observation, the principles of 
 which have been elaborated to their hand, as we have seen, 
 by astronomers and mathematicians. Astronomers have 
 rightly ignored any other kind of error, since their science 
 is confined to observations, in which the errors move in 
 arithmetic dispersion. Statisticians likewise have little 
 use for errors of estimation, but they are concerned with 
 other objects of which the deviations or variations may 
 move in geometric dispersion. Hence, in confining them- 
 selves to the use of the arithmetic average, they have done 
 what astronomers have not done : they have applied it to 
 cases to which it does not belong. On account of their usual 
 practice hitherto of using only the arithmetic average, it is 
 difficult as yet to enumerate many of the subjects which ought 
 to be treated geometrically. But at least one subject 
 immediately suggests itself. 
 
 AXIOMETRY.^8 
 
 The subject which immediately suggests itself is very 
 prominent in statistics and economics. It is the deviations 
 of prices at a later period from what they were at an earlier 
 period, when they had a level which it is desirable to main- 
 tain — and it is desirable to maintain any price-level that 
 is more or less settled. It is, however, the whole price-level 
 that it is desirable to maintain, not the prices of individual 
 commodities, which should be allowed to vary according 
 to the rule of demand and supply, the former determined by 
 people's needs, the latter by their ability to produce the 
 commodities. Therefore price-changes are variations rather 
 than deviations, and their individual positions in the price- 
 level which it is desirable to maintain, are merely the starting 
 
 68 This term is proposed for the measurement of general 
 exchange-value, especially of money. 
 
70 THE PROBLEM OF ESTIMATION 
 
 points of their possible and even desirable variations. We 
 often speak of averaging prices, as if prices were single 
 objects (like the heights of men) ; which is perhaps one 
 reason for the tendency to average them arithmetically. 
 But prices are not single objects ; they are ratios — the 
 ratios of the values of commodities to the value of the money- 
 unit. Strictly speaking, there is not even a level of prices, 
 as an absolute object, like the level of the ocean in a calm. 
 A level of prices exists only in comparison with another level 
 of prices at another period, which other level of prices may 
 be the same with, or different from, the level at the first 
 period.^* Jevons has said that " there is no average of 
 prices at any one period " ; s" this is too broad a statement. 
 But the truth is, we do not really average prices in axiometry ; 
 and the only meaning of an average of prices would be to 
 determine the amount of preciousness of commodities, or 
 the relation of their value to their weight or bulk — an 
 object of little interest.®^ In axiometry what we really do 
 is to average the variations of prices. The common starting 
 point, if we have constructed it, from which we begin to 
 measure the deviations of prices, is purely arbitrary. At any 
 period of time all commodities have various prices ; but we 
 can find of the commodities various amounts that have the 
 same price. Then at the next period these amounts of the 
 commodities have more or less differing prices. To average 
 these second prices is really to average the deviations of the 
 prices from the common starting price. These deviations of 
 the prices are nothing but the variations of the prices ; so 
 
 ^« Cf. The Measurement of General Exchange-Value, pp. 158-9. 
 The process of comparing price-levels is rather like trying, on an 
 always billowy sea, which has a flat bottom, and in which the 
 quantity of water constantly varies, to obtain its levels (such as 
 they would be if it were still) by averaging the various heights 
 of different sections of the waves above the bottom (the zero base). 
 The waves would vary in extent according to the winds, and their 
 averages above the bottom would vary according to changes in 
 the quantity of water. If the proper averages were correctly 
 drawn, their variations would represent variations in the never- 
 existing level or plane surface of the water, and yet would be just 
 as truthful as if the level did exist. 
 
 *" A serious Fall in the Value of Gold ascertained, 1863, reprinted 
 in Investigations in Currency and Finance, p. 23. 
 
 ^^ Cf. The Measurement of General Exchange-Value, pp. 165 n., 
 200-1. For brevity this work will hereafter be referred to as 
 M. G. E.-V. 
 
DEVIATIONS AND VARIATIONS 71 
 
 that, without performing the labour of reducing all commo- 
 dities to a common price (which would properly have to be 
 repeated every year, or whatever other interval is chosen «2), 
 the same result is obtained by simply averaging the price- 
 variations. 
 
 Again, however, suppose we have performed that labour of 
 reduction, and starting with all prices alike, we average their 
 deviations at the next period. This process must not be 
 hkened to the process of averaging observations. The 
 purpose of averaging observations is to get an average 
 observation that is most probably nearest to the true position 
 (a magnitude of some sort) of the object observed. The true 
 position is conceived as, so to speak, the bull's-eye at which 
 the observations have aimed (as the astronomer aims his 
 telescopic instruments at a star) ; but the observations have 
 not exactly hit it, and have deviated, if unbiassed or acci- 
 dental, almost equally on both sides, and the average of them 
 is believed to bring them back most closely to the true posi- 
 tion aimed at. But in averaging price-deviations we do not 
 desire to get back to the starting point ; for that we know 
 already. We desire to get the average deviation. It has 
 some resemblance to the practice indulged in by observers of 
 averaging their errors — that is, their reputed errors, as 
 measured from the average (not from the real and true, but 
 otherwise unknown, position). Astronomers and surveyors 
 perform this, to them secondary, operation only to get the 
 " probable error," as they call it, of their averaged observa- 
 tion, by which they mean the probable amount of accuracy 
 of their observations. All this they do with observations 
 the errors of which are believed to be unbiassed and only 
 accidental. And yet this similitude does not go on all fours, 
 as in calculating the " probable error " the errors on opposite 
 sides are averaged separately. A better analogy is the 
 following. 
 
 Suppose a great number of shots are aimed directly at a 
 distant buU's-eye while a strong wind is blowing sideways 
 across the path of the bullets. The wind deflects them all to 
 a more or less equal extent toward the side of the buU's-eye 
 to which the wind is blowing. It thereby extends the deflec- 
 tions of the bullets that are already accidentally deflected on 
 that side, and it lessens the deflections of the bullets that are 
 
 «2 Otherwise a various and haphazard weighting is introduced 
 for every comparison made between subsequent periods, contrary 
 to the intent of the operator. Cf. M. G. E.-V., pp. 188-9. 
 
72 THE PROBLEM OF ESTIMATION 
 
 already deflected on the opposite side, and may even reduce 
 their deflections to nothing or carry them over to the other 
 side. When, therefore, we average the lateral positions of 
 the hits, the average will no longer approximate to the bull's- 
 eye, but will lie at a greater or less distance to the lee side of 
 it according to the strength of the wind ; and this distance 
 of the average of the hits, or the average hit, from the centre 
 of the bull's-eye, will measure approximately the bias caused 
 by the wind of the given intensity. The same result would 
 be obtained by averaging directly the lateral errors of the 
 hits, this average giving the same bias of the wind in a much 
 shorter process.** This operation, in either of its two forms, 
 of measuring the bias that has deflected all the hits more or 
 less alike, is the only one in connection with errors of observa- 
 tion that has complete resemblance to the statistical problem 
 in economics of averaging price-variations, in either of its 
 two corresponding forms. The level of prices or the exchange- 
 value of money remaining the same, prices must vary by 
 both rising and falling equally (but remember there ard 
 different ways of reckoning equaUty) : their rises and falls 
 must compensate one another, or else the price-level and the 
 exchange-value of money have not remained unchanged. 
 If they vary more on the one side than on the other — falling 
 more than they rise, or reversely, rightly reckoned — they 
 may be said to show a bias, which is the change of the price- 
 level, or inversely of the exchange- value of money, caused by 
 whatever is the cause of change in the exchange-value of 
 money (roughly a change in the quantity and mobility of 
 money relatively to the quantity and mobility of commodi- 
 ties). It may be added that riflemen measure the bias 
 caused by various intensities of the wind blowing in various 
 directions, in order to be able to allow for it, after measuring 
 the intensity and direction of the wind when they shoot, by 
 aiming the ascertained angle to the windward side. In 
 economics what corresponds to this is an inverse operation, 
 or one like measuring the deflecting power of the wind by 
 measuring the bias of the hits on the target. For here the 
 causes are more complex even than the effects, so that we 
 
 «' Only approximately correct is the result because it is plus 
 or minus what would have been the true error of the average 
 from the centre of the bull's-eye, had there been no wind. But 
 this error may be small on the supposition of a great number of 
 fairly accurate shots. In astronomical observations the bias of 
 the wind is replaced by the bias of refraction, according to the 
 height of the star above the horizon. 
 
DEVIATIONS AND VARIATIONS 73 
 
 cannot argue from them to the effects, but we must measure 
 the effects first and directly. The deviation of the whole 
 pnce-level being thus measured, it may now be allowed for, 
 by altenng contracts accordingly ; or an attempt may be 
 made to counteract it— to prevent its increase, and to bring 
 it back to the starting point — by decreasing or increasing the 
 currency as soon as a rise or a fall of the price-level manifests 
 itself. But this is a use to be made of the measurement of 
 variations in the exchange-value of money, supposed to be 
 already accompHshed, and lies beyond our subject ; as also 
 does the consideration of the causes of the variations that are 
 measured. The averaging of price-variations in order to 
 measure the inverse variation in the exchange-value of 
 money, is one measurement ; and the measurement, what- 
 ever rnay be its nature or procedure, of the causes of such 
 variations, is another distinct measurement. ^^ The former 
 is our present subject. 
 
 But properly the similitude of averaging prices is not to 
 what is done in connection with observations, but to what is 
 or should be done in connection with estimates. Only these 
 have not been so thoroughly studied by others ; wherefore 
 it is incumbent on statisticians to study, not so much the 
 averaging of estimates, which is not their affair, but directly 
 the averaging of price-variations. 
 
 Still, to the extent that the subject of estimation has been 
 developed, it may be of some use as a model. For price- 
 deviations have many of the pecuHarities possessed by the 
 aberrations of estimates. Thus, as it is possible for an esti- 
 mate to be made as high as any quantity whatever, so it is 
 possible for a price to rise to any quantity whatever ; and 
 as it is impossible for an estimate to be made as low as zero, 
 and certainly not lower, so it is impossible for any price to 
 
 °* It has escaped the attention of many economists that if 
 they are to argue from the causes of price-variations to the price- 
 variations, they would have to be able first to measure those 
 causes quantitatively. But, with the possibly single exception 
 of Professor Irving Fisher, they have not yet even reached the 
 stage of entering upon the subject of this kind of measurement, 
 since they have not yet come to agreement about the nature of 
 those causes. Dif&cult as is the measurement of price-variations, 
 much more difficult would it be to measure their causes with 
 anything like equal accuracy. We may, therefore, place much 
 greater reliance upon the direct measurement of price-variations, 
 than we could gain by bringing in any reference to their causes. 
 Cf. M. G. E.-V.. pp. 22-3. 
 
74 THE PROBLEM OF ESTIMATION 
 
 sink to less than zero and even to zero, for then it would 
 cease to be a price and the article would go out of the 
 market.®^ Now, when in the case of two articles at first 
 priced the same (for some quantity of one article is always 
 equivalent to some quantity of another) the price of the one 
 rises and the price of the other falls, we have almost exactly 
 the same problem and its modifications that those seven- 
 teenth century disputants had with their hypothetical case of 
 estimates higher and lower than a true value. For we may 
 wish to know whether the rise of the one is greater than the 
 faU of the other ; or, the rise being given, what the fall must 
 be to equal it, and reversely ; or again, the two deviations 
 being given, what is their combined effect upon the price- 
 level, that is, what common variation of the two prijces 
 together would have the same effect — which common varia- 
 tion must evidently be a mean of some sort between the two 
 given variations. These are not idle questions like those 
 debated so violently by Nozzolini and reluctantly by Galileo ; 
 for if the price-level has changed, we want to know it, so as 
 to adapt ourselves to it, and, too, to counteract it, if possible. 
 Thus that academic dispute of nearly three centuries ago has 
 a direct bearing upon a very important problem, with which 
 not only scientific but also practical people are to-day much 
 concerned. 
 
 If we confine our attention to only two prices, of two articles 
 of equal importance, almost everything that was advanced 
 in that old controversy would seem, mutatis mutandis, to 
 have application. If one price rises to more than double 
 what it was, it cannot be equalled by anj^ fall < the other 
 price, according to the arithmetic mean, just as, according 
 to the harmonic mean, if the one falls to less than half its 
 previous price, it cannot be equalled by any rise of the other ; 
 but whatever the rise or the fall of the one, it can be equalled 
 by some fall or rise of the other, according to the geometric 
 mean. Again, if the one price rises from 80 to 100 and the 
 other falls from 100 to 80, these obviously are an equal rise 
 and fall ; but equahty is assigned to them only by the 
 
 •5 Above, in note 12, it was said that we might suppose a boot 
 to be given with an article and conceive this boot as a negative 
 price. But that supposition does not take place in ordinary 
 commercial transactions. Jevons noticed that we pay to get 
 rid of some things, and such things he considered to have negative 
 value. Theory of Political Economy, 4th ed., p. 127. But such 
 things do not appear in the market, nor are they consumed. 
 Cf. M. G. E.-V., p. 248. 
 
DEVIATIONS AND VARIATIONS 75 
 
 geometric mean. The question is not one of equal differ- 
 ences or distances, but of equal ratios, equality of difference 
 coinciding with equality of ratio only in cases like the last. 
 No one can read the arguments of GaUleo and Castelli without 
 being convinced of their superiority to the arguments of 
 Nozzolini. And the same arguments apply here. They 
 apply here, however, only on the assumption above made — 
 an assumption involved in that other problem itself, but here 
 not easily carried out (we shall return to it later) — ^that the 
 two deviations, and consequently the two articles, are equally 
 important, so as to permit them to be weighted evenly.*^ 
 
 It wUl be objected that in the case of prices we are not 
 concerned with only two articles, but with a whole marketful 
 of them, and now the contingency exists that any rises and 
 falls of some prices can be balanced by the falls and rises of 
 other prices even according to the arithmetic average (and, 
 it may be added, according to the harmonic, too), since now 
 any one extravagant rise (or fall) can be met by falls (or 
 rises) of several other articles. Yet even so there is a possible 
 case, which would admit of no balancing, as when all the 
 prices happen to come under two groups (or under three, the 
 third being of prices that have not changed), in the one of 
 which the prices rise (or fall) extravagantly, and in the other 
 the prices fall (or rise), but cannot fall (or rise) so far as is 
 required by the arithmetic average (or the harmonic). 
 Apart from this, which must be allowed to be an improbable 
 case, the argument against the arithmetic average (and the 
 harmonic) is that it would be strange if, the geometric mean 
 being the right one to use in the case of two prices, the 
 arithmetic mean or rather average (or the harmonic) suddenly 
 became the right one to use in the case of three or four or 
 more prices — or at what number is the jump to be made ? 
 The dispute we have reviewed over errors of estimation might 
 very readily have been extended to more estimates than two. 
 Can we believe that Galileo, after demonstrating (for this is 
 not too strong a word) the correctness of the geometric mean 
 for measuring the errors of two estimates, would have 
 adopted the arithmetic average for measuring the errors of 
 many estimates ? The law of continuity, which Galileo 
 recognised, forbids. 
 
 It would seem as if it could be laid down as axiomatic, that 
 the same mean or average must be applicable to any possible 
 number of items. Two price-deviations are possible items. 
 
 «» Cf. M. G. E.-V., pp. 248-9. 
 
76 THE PROBLEM OF ESTIMATION 
 
 Therefore a mean or average right or wrong for two such 
 items, it would seem, must be right or wrong for any other 
 number of them. But now a curious phenomenon shows 
 itself, of which a hint has already appeared. This is, that 
 in the case of the geometric average, although it would seem 
 to be the theoretically right one for any number of price- 
 variations as well as for two, yet it works with perfect 
 accuracy only of two (equally important) price-variations. 
 The axiomatic test, then, can be stated only as a negative 
 one. A mean or average not applicable to the case of two 
 price- variations cannot be the right one for the case of many 
 price- variations — ^that is, the theoretically right one, although 
 in practice it may approximate to a theoretically right one. 
 The justification for trying a problem of averages on two 
 objects first, as is so often done by statisticians and others 
 (we have seen Bertillon and Bowley doing so), is of course 
 that such are the simplest cases. If the average in question 
 will not work with two objects, it is not the theoretically 
 correct average. But if it wiU work with two objects, we 
 need to go on and apply it to the more complex cases of many 
 objects. Two objects form a perfect negative test : they 
 may disprove an average. They are not a positive test : 
 they cannot prove an average — or of any kind of average it 
 must first be proved in general that they can. This is 
 provable of the arithmetic and harmonic averages,^' but not 
 of the geometric average.*^ The geometric average, if in any 
 subject it passes the first ordeal, needs to be examined 
 further in application to many objects. 
 
 «' In a paper on the method of least squares in the Berliner 
 Astroiiomisches Jahrbuch, 1834, Enclce tried to prove that method 
 by demonstrating that because the arithmetic mean is correct 
 for reducing two observations, therefore the arithmetic average 
 is correct for reducing three or more observations. A generation 
 later Venn in his Logic of Chance, Chapter XIII. § 13, expressed 
 the opinion that, although we must adq)t the arithmetic mean 
 for reducing two observations, we have a wider choice of inter- 
 mediate values when we come to reducing three or more. It was 
 probably to allay such doubts as this in his day, that Encke made 
 his demonstration. It needs hardly to be said that he had little 
 difficulty. 
 
 »8 This distinction was first pointed out in M. G. E.-V., pp. 
 239-42, cf. 519-20. That work may be said to be founded on 
 this distinction, since otherwise it would have proved thegeometric 
 average, and the subject would have been disposed of in short 
 order. Cf. p. 255. 
 
DEVIATIONS AND VARIATIONS tj 
 
 It is well known that when the problem of measuring 
 variations, or deviations, of prices came up for scientific 
 investigation, the arithmetic average was at first adopted 
 without hesitation and without examination, and continued 
 to be so used till Jevons perceived that it was a case for 
 proportions instead of differences, since variations of prices 
 are changes of ratios, and accordingly recommended the 
 geometric average.^* It is clear from Jevons's first state- 
 ments on the subject, that he had a correct idea of the 
 geometric deviation involved. He said that not the arith- 
 metic but the geogaetric average must be used in his inquiry 
 about the change going on in the value of gold, because the 
 alterations of prices with which he was dealing had a wide 
 range, " varying from more than 50 per cent, of decrease 
 to more than 100 per cent, of increase " ; and he instanced 
 as almost exactly such, cocoa and cloves, the former of which 
 had risen from 100 to 200, and the latter had fallen from 
 100 to 50, so that, in his words, " the price of one is doubled, 
 of the other halved — one is multiplied by two, one divided 
 by two " ; just as, in the seventeenth-century problem, the 
 true value was multiplied in the one estimate by ten and in 
 the other it was divided by ten ; wherefore Jevons, like 
 Galileo, considered that " on the average " the prices of 
 these articles had remained as they were before,^" their 
 opposite deviations really being equal and compensating 
 each other. Only Jevons did not state this last proposition 
 about the geometric equality of the opposite deviations, did 
 not generalise and formulate it, did not reach the bottom of 
 the subject. And as far as he did go, he did merely for one 
 set of deviations, but hastily and without demonstration, 
 what Galileo had more fully done for another. We may 
 therefore say, that the first advocate of the geometric mean 
 or average for the measurement of deviations, when they are 
 proportional, was a greater genius even than Jevons. It is 
 another instance, also, of astronomy leading the way for 
 statistics. And it is curious to note a further parallelism in 
 the fact that Jevons's position was disputed by Laspejnres 
 as Galileo's had been disputed by Nozzohni ; and Laspejnres 
 converted the question into one of gain and loss, just as 
 Nozzohni had done. We shall have occasion later to revert 
 to this nineteenth-century controversy. 
 
 «» First in the pamphlet A serious Fall, etc., in Investigations, 
 p. 23. Later in his Theory of Political Economy, ist ed., pp. 81, 
 83, he defined exchange- value as a " ratio of exchange." 
 
 '» Investigations, pp. 23-4. 
 
78 THE PROBLEM OF ESTIMATION 
 
 Jevons did not confine his recommendation of the geome- 
 tric average to index- numbers. " In almost all the calcula- 
 tions of statistics and commerce," he wrote, " the geornetric 
 mean ought, strictly speaking, to be used." ''^ This is an 
 exaggeration, but the amount of truth in it has been too 
 httle heeded by statisticians. At most they admit that the 
 geometric average has an advantage in smoothing off extra- 
 vagant fluctuations, and therefore some of them advise its 
 use when it differs much from the arithmetic.''^ This is, 
 as it were, a concession made by the arithmetic to the 
 geometric average, whereas in truth the concession should 
 be the other way: the arithmetic average may be used 
 where it does not much differ from the geometric. That is, 
 this concession can be made to the arithmetic average, on 
 account of its convenience, in subjects where the geometric 
 is the theoretically correct average ; but where the arithmetic 
 average is the theoretically correct one, no concession on its 
 part need ever be made to the geometric average, for the 
 geometric average, being the more difficult, should never be 
 used except where theory demands it. Thus the procedure 
 of these economists who concede the use of the geometric 
 average for extravagant cases, is really an unconscious 
 tribute to the theoretical correctness of the geometric 
 average. A similarly unconscious recognition of it is made 
 by some economists, who, of late, have advocated the 
 drawing of logarithmic diagrams, and for this purpose the 
 use of logarithmically ruled paper. '^ For, when two or more 
 companion curves are drawn in this way, the eye naturally 
 makes a resume of them by imagining a line halfway between, 
 or even this is actually inserted by the operator ; but the 
 arithmetic mean or average (represented by this line) 
 between logarithms gives the geometric mean or average 
 between the natural figures.'^ Using logarithms also, 
 
 '^ Principles of Science, 3rd ed., p. 361 ; cf. Investigations, 
 p. 128 n. 
 
 '^ E.g., Bowley, who calls it a " tentative rule," Elements of 
 Statistics, pp. 128-9 '> followed by Zizek, Statistical Averages, 
 p. 197. 
 
 " E.g., Bowley, op. cit., pp. 188-96 ; I. Fisher, The " Ratio " 
 Chart for flatting Statistics, Quarterly Publications of the 
 American Statistical Association, June, 1917. The latter's 
 bibliography shows that Jevons (see Investigations, p. 128) was 
 the first to use and recommend this kind of charting. It fitted 
 in with his recommendation of the geometric average. 
 
 '* Note that in this connection Bowley says : " For the pur- 
 
DEVIATIONS AND VARIATIONS 79 
 
 Fechner has ccaiverted the Gaussian law, originally applied 
 to the arithmetic average, into a law appHcable to the 
 geometric average of deviations. He did so for the reason 
 that the deviations on large objects are absolutely greater 
 than the deviations on small objects, although they are not 
 necessarily so proportionally, as, to use an instance he cited, 
 the deviations of the sizes of men compared with those of 
 fleas. For the measurement of all deviations, therefore, 
 including the average stature of men, which we have seen 
 to be correctly averaged arithmetically, he thought the 
 geometric average to be the strictly and theoretically 
 (" prinzipiell streng," strictly according to theory) correct 
 one, although he would concede the use of the arithmetic 
 average when the deviations are comparatively small, since 
 then it is inappreciably different from the geometric.'^ But 
 the reason he gave is not enough. For the use of the geo- 
 metric average it needs to be shown, not only that the 
 deviations of large objects are greater than those of small 
 ones, but that the deviations above the normal position 
 are or can be greater than the deviations below it. This 
 Fechner did not take into account, and so he applied his 
 method to such objects as the dimensions of visiting cards 
 and of pictures. He merely experimented with the geometric 
 average, without any correct theoretical guidance, and with- 
 out any satisfactory theoretical or practical results. 
 
 Credit for first perceiving the resemblance of prices to 
 estimates and thereby getting an insight into the true nature 
 of their variations, belongs to Profesor Edgeworth. After 
 adopting, in 1887, in the passage already quoted, Galton's 
 re-discovery of the geometric nature of estimation, and 
 referring to Fechner's and Weber's law, he continued : " This 
 law of prizing may well extend to prices " ; and he added : 
 " There exists a simple reason why prices are apt to deviate 
 more in excess than in defect, namely that a price may rise 
 to any amount, but cannot sink below zero." '^ In this he 
 
 poses of price index-numbers it is ratios which are important 
 and which the diagram should represent," p. 190. 
 
 " KoUektivniasslehre, pp. 82-3, and so pp. 24-5, 77-82, 91, 95, 
 307, 339-64. The opinions of this posthumously published work 
 he had reached twenty years before, for their outline is found in 
 his paper Ueber den Ausgangswerth der kleinsten Abweichungs- 
 summe of 1878, pp. 14-15. 
 
 " Memorandum to the first Report of the British Association 
 Committee, p. 283. He remarked that Venn had just called this 
 kind of rising and falling " a one-ended phenomenon." It may 
 
8o THE PROBLEM OF ESTIMATION 
 
 adduced a better reason than Jevons had given for the 
 geometric average in connection with prices, or rather gave 
 it more explicitly. But he went on and invoked another. 
 " It appears," he wrote, " that prices group themselves about 
 a mean, not according to a symmetrical curve like that which 
 corresponds to the arithmetic mean, but according to an 
 unsymmetrical curve like that which corresponds to the 
 geometric mean." " And he added some " empirical 
 evidence " in support of this statement.'® This last is not 
 so good a reason as the one first quoted, which Professor 
 Edgeworth did not bring into immediate connection with the 
 geometric average. The primary reason for the propriety 
 of the geometric averaging of price-variations is because 
 prices have a lower and no upper limit to their movements. 
 All that can subsequently be discovered about their multiple 
 dispersion may be interesting, but it is superfluous. The 
 criterion of dispersion we have seen to be needed only for 
 dubious cases. This is not a dubious case. The geometric 
 average is here determined by a perfectly good criterion. 
 To repeat an illustration already used : Professor Edgeworth, 
 we can have no doubt, would not ascribe the reason for our 
 employing the geometric average in averaging the annual 
 rates of increase of a country's population over a number of 
 
 be noted that Quetelet had noticed (on very insufficient data) 
 that prices deviate more above than below their mean, Lettres 
 stir la Theorie des Prohabilitis, pp. 72-3 ; but he inferred from this 
 only that the causes at work for raising prices are more energetic 
 than those for lowering them, p. 183. 
 
 " Memorandum to the First Report, ibid. Inhis Memorandum 
 to the second Report, p. 208, he claims that, so far as he knew, 
 he was the first to advance this theory of prices ; and he referred 
 to it again as his discovery in New Methods of Measuring Varia- 
 tions in General Prices, in the Journal of the Royal Statistical 
 Society, June, 1888, pp. 351, 356. 
 
 " Memorandum to the first Report, pp. 284-6. Professor 
 Wesley C. Mitchell has likewise given an analysis of over 5,000 
 price-variations, which supports it, in Index Numbers of Wholesale 
 Prices, in the Bulletin of U.S. Labor Statistics, No. 173, 1915, 
 pp. 11-24 ' where his table No. 2 presents a specimen of almost 
 perfect geometric dispersion around the median. But very 
 curiously Professor Mitchell calls it, p. 18 " a distribution accord- 
 ing perfectly with the so-called ' normal law of error,' " like errors 
 of observation, etc., as if it exhibited arithmetic dispersion, while 
 on pp. 20-1 he recognises that it is not " perfectly symmetrical " 
 although it is " regular." On p. 82 he, too, notes that prices can 
 rise without limit, but have a limit to their fall. 
 
DEVIATIONS AND VARIATIONS 8i 
 
 years to the fact, if fact it be, that such annual rates group 
 themselves around their average according to an unsymme- 
 trical curve which is the regular curve representative of 
 geometric dispersion. And if the ulterior study of dispersion 
 is not needed in this case, it is not needed in the case of price- 
 variations. Because of not resting at the simple, true, and 
 sufficient reason. Professor Edgeworth has unfortunately 
 been led astray into studies of dispersion involving the 
 calculus of probabilities, which, however interesting in 
 themselves and useful for their own purposes, are not per- 
 tinent to the study of averaging price-variations for the 
 purpose of measuring variations in the exchange-value of 
 money ; but which Professor Edgeworth has come to consider 
 basic to this study. There will be need of going into this 
 question more fully later. 
 
 Weighting. 
 
 Prices not only have some resemblance to estimates, and 
 through them to observations, but they have some difference. 
 They are properly variations, while estimates and observa- 
 tions are deviations. For this reason their dispersion around 
 a centre is of no importance. The problem, as has been 
 shown, is not to get from them the original centre : it is to 
 get an average of their departures from a common starting 
 point, or more simply, an average of their variations. This 
 average may form a new centre for new departures — or rather 
 a new level for new risings and fallings. 
 
 Among the various problems we have reviewed, one having 
 partial analogy with the problem of axiometry is the 
 problem of getting the true position when the estimates and 
 their errors are both known. For here what corresponds to 
 the estimates are the prices, and what corresponds to the 
 errors are the deviations or variations of the prices. Prices 
 and their deviations are here the two factors given, and 
 sought is the value around which the deviations compensate 
 each other and balance, just as in that other problem the 
 position sought is one around which the erroneous estimates, 
 properly weighted, equal each other and balance. But 
 between the present problem and that problem a great 
 difference comes in with the question of weighting. There 
 the weighting is given with the other two elements, since it is 
 inversely according to the errors. Here no such weighting 
 can be employed ; for the variations of the prices have 
 influence on the weights only as they affect another set of 
 
82 THE PROBLEM OF ESTIMATION 
 
 factors, which exist here but have no existence there. These 
 other factors are the fuU values of the articles whose varying 
 prices are being averaged. The full value of any kind of 
 article itself depends upon the price of the article and the 
 quantity of the article (or the number of its units so priced) 
 that come under consideration. This element of quantity, 
 and the full value which it produces with the price-element, 
 have nothing analogous to them in the problem of the errors 
 of estimates or observations ; and so they very seriously 
 differentiate the problem of averaging price-variations from 
 the problem of averaging errors of estimates and observa- 
 tions — and, it may be added, from the problem of averaging 
 estimates or observations in any of its forms. 
 
 But the principle of weighting is the same at bottom in all 
 kinds and instances of averaging. In the words of Professor 
 Merriman, " weights are merely numbers denoting repeti- 
 tion." '^ Certain items are to be averaged : several of them 
 may be the same quantity or magnitude ; then, instead of 
 treating such equal items_ as so many distinct items, we 
 combine them into one item several times repeated, and the 
 coefficient representing its repetition is its weight.^" Thus 
 in the problem of finding the true position from two known 
 estimates and their known errors, the estimate with the 
 smaller error has to be repeated in order that its error may 
 equal and so compensate the larger error of the other estimate. 
 In averaging estimates or observations of which the errors 
 are unknown, in order to obtain the nearest approach to the 
 true position, the estimates or observations must be treated 
 as single items (and then if two are alike, they are combined 
 into one item twice repeated, and so on), provided they have 
 been made with equal care and therefore can be considered 
 as equally good. But if some have been made with greater 
 care, or with finer instruments, we may consider them as 
 being twice or thrice as good as the others, and so we may 
 treat the former as equivalent to two or three of the latter, 
 and weight them accordingly.®^ The question of weighting, 
 then, is a question as to the nature of the single items that 
 are to be averaged.®^ In price- variations what are the single 
 items ? 
 
 It might seem at first sight as if simply every price- 
 quotation were a single item, and since every commodity 
 
 "• The Method of Least Squares, § 51, cf. § 39. 
 
 " Cf. M. G. E.-V., pp. 87-8. 
 
 *^ Cf. Venn, Logic of Chance, p. 346. 
 
 «2 Cf. M. G. E.-V., pp. 89-90. 
 
DEVIATIONS AND VARIATIONS 83 
 
 (any kind of commodity) has one price-quotation attached 
 to it, it would seem as if price-variations of every kind of 
 commodity were the single item in question. This is the 
 way the question struck the first inquirers into price-varia- 
 tions, wherefore they used simple averaging with even 
 weighting. But a price-quotation is the quotation of the 
 price of a generic name for many articles ; and one such 
 generic name covers a few articles, and another covers many. 
 The few and the many articles themselves consist of so many 
 pounds or bushels or other denominations of measures ; 
 but these as such have nothing to do in the economic world 
 of values — that is, they do not themselves appear in that 
 world, though they act, so to speak, as substances underlying 
 it. The generic name of a commodity, covering so many 
 pounds or bushels at a certain price, covers so many value- 
 units of those singly-named articles, and these value-units 
 evidently are the single articles variously repeated. A 
 single price-quotation, therefore, may be the quotation of 
 the price of a hundred, a thousand, or a million dollar's 
 worths, or pound's worths, of the articles that make up the 
 commodity named. Its weight in the averaging, therefore, 
 ought to be according to these money-unit's worths.®* Or, to 
 reach the same result more briefly : the weights of com- 
 modities are directly according to their importance ; and the 
 importance of every commodity is proportional to what may 
 be called its full value, or its quantity multiplied by its price.®* 
 This may be rendered still plainer by an example. Sup- 
 pose wheat is classified into two grades and their prices 
 quoted separately, and barley is quoted only as one commo- 
 dity. Suppose, further, the full value of wheat is, grade A 
 five millions, grade B five millions, and of barley five millions 
 
 8' Cf. M. G. E.-V., p. 93, cf. pp. 80-1. 
 
 ^* A warning may be needed against misuse of the word 
 " importance " in this connection. It is used simply as a con- 
 venient term for summing up so many money-units' worth. 
 There is no reference to utility, whether total or final. Utility 
 has no more to do with measuring price-variations than with 
 measuring temperature. Utility may be among the causes of 
 exchange-value ; but knowledge of causes is not necessary for 
 measuring effects. Objective ratios of exchange can be measured 
 without such knowledge, even by observers who have no notion 
 of utility. Nor is there any reference to the fact that a more 
 important commodity " absorbs more currency." It refers only 
 to the fact that in a more important commodity there is a greater 
 number of price-variations. Cf. M. G. E.-V., pp. 8-9, 94. 
 
84 THE PROBLEM OF ESTIMATION 
 
 (money units of any sort being understood). Evidently we 
 have three equal items. Now suppose the distinction 
 between the two grades of wheat had not been made and the 
 prices of wheat had all been thrown together. Then we 
 should have had, wheat ten millions, barley five millions. 
 Evidently the difference between the two cases is only 
 nominal : two items have been merged into one, with only 
 one price-quotation, giving the appearance of one item ; but 
 in truth we stUl have two itertis of wheat, or ^hree items all 
 told. Nor is the matter affected by the fact, or not, whether 
 wheat admits of being classified into two grades. Let all 
 wheat be aUke : yet five millions of wheat is a single item 
 over against five millions of banley, and another five millions 
 of wheat is either another item, or it is the same item 
 repeated. Therefore we should weight wheat as two 
 compared with barley as one. For weights are proportional 
 figures, and we naturally reduce them to their least common 
 denominator. 
 
 When the early investigators simply took each quotation 
 of a price-variation as one item, we may suspect they were 
 influenced also by the consideration of convenience. There 
 is prevalent a stupid distinction between weighted and 
 unweighted or simple averages, as if the former contained 
 weights and the latter did not. The latter contain weights 
 just as much as the former, only they are equal or even 
 weights. ^^ You cannot avoid the subject of weighting by 
 adopting the simple average, for it has pursued you there. 
 You must have either a right or wrong system of weighting, 
 or rather a better or a worse. If you make an earnest 
 attempt to adopt a good system, something can generally 
 be said for it ; but if you adopt even weighting, nothing can 
 be said for it — except the lazy man's recommendation of 
 superior easiness. It is, however, argued : all systems of 
 weighting give nearly the same results in practice ; therefore 
 adopt the easiest, which is even weighting. It is Hke the 
 argument also made for the arithmetic average compared 
 with the others. All depends on the word " nearly." If 
 index-numbers are ever to be put to any serious use, the 
 closest possible approximation to the truth wiU be necessary. 
 For it will be necessary to make the measurements from 
 year to year, or even oftener, and to combine the results in a 
 series ; and if a method be adopted that gives a small error 
 always or mostly on the same side of the truth, its errors will 
 
 »' Cf. M. G. E.-V., p. 82. 
 
DEVIATIONS AND VARIATIONS 85 
 
 accumulate from period to period. If, to avoid this danger, 
 the ordinary system be kept in use of comparing every 
 subsequent period with the same period as a base, this 
 system with a faulty average or weighting, not excepting 
 even weighting, produces a varying haphazard weighting, 
 with variously resulting errors, in all comparisons between 
 the subsequent periods, increasing in irregidarity as the 
 series advances,^* — and it is principally for comparison with 
 the immediately preceding period that these measurements 
 are desired.^' 
 
 Besides, we need first of all to examine the matter in 
 theory, by which practice may afterwards be regulated. 
 Theory demands precision, and can make no compromise 
 with sloth. " A life of action," says Macaulay, " if it is to 
 be useful, must be a life of compromise. But speculation 
 admits of no compromise." ^^ Now, in theory the considera- 
 tion of weighting is absolutely necessary. And it must be 
 treated in close connection with the averages that are being 
 investigated.^® For subconsciously various kinds of weight- 
 ing obtrude themselves in company with various kinds of 
 averages. The subject may be illustrated by the controversy 
 between Jevons and Laspeyres. 
 
 Jevons asserted that because the price of one article had 
 risen 100 per cent, and the price of another had fallen 50 per 
 cent., it would be " totally erroneous " to say the average 
 variation of the two prices was a rise of 25 per cent., since 
 the geometric mean between them indicates no variation 
 at all, and the geometric mean of the ratios is the right one 
 to use.®" To this Laspeyres objected that as it later requires 
 200 money-units to purchase the one article which 100 would 
 have purchased before, and fifty to purchase another which 
 100 would have purchased before, it now requires 250 to 
 purchase what 200 would have purchased before, which is 
 the same as requiring 125 to purchase what 100 would have 
 purchased before, wherefore the prices of these two articles 
 together have risen from .100 to 125, as indicated by the 
 arithmetic mean, or by 25 per cent.*^ Obvious is the 
 
 »» Cf. M. G. E.-V., pp. 189, 206-7. 
 
 " Cf. Mitchell, Index-Numbers of Wholesale Prices, pp. 36-7 ; 
 also Fisher, Purchasing Power of Money, pp. 203, 423-4. 
 
 *' Essay on Mahon's History. 
 
 8» Cf. M. G. £.-7., p. 275. 
 
 '" Investigations, pp. 23-4. 
 
 •^ Hamburger Waarenpreise, 1850-1863, in JahrbUcher fUr 
 Nationaloekonomie und Statistik, 1864, p. 97. 
 
86 THE PROBLEM OF ESTIMATION 
 
 resemblance of Laspeyres's procedure to Nozzolini's, in imme- 
 diately converting the problem into one of gain and loss ; 
 while Jevons, like Galileo, treated it as a question of ratios. 
 But Jevons for a moment went over into Laspeyres's camp 
 to dispute with him on his own ground. He urged that we 
 cotdd with equal reason " suppose a certain uniform quantity 
 of gold money to be expended in equal portions in the pur- 
 chase of certain commodities, and that we ought to take the 
 average quantity purchased each year " ; which " might be 
 ascertained by taking the harmonic mean " of the prices. ^^ 
 This suggestion was actually adopted by Messedaglia. ®' 
 Professor N. G. Pierson reviewed this debate, and because 
 of the inconsistencies it involved, various results being one 
 as good as another, concluded that the whole subject is 
 insoluble.^* Now, originally Jevons had no right, as Galileo 
 had had, to use the simple geometric mean, unless he added 
 the assumption that the two articles were equally important, 
 and unless he gave the proper definition of what is to be 
 meant by their equal importance. Laspeyres, on his part, 
 did make a further assumption, namely that the same 
 quantities of the two articles are purchased at the two 
 periods, notwithstanding the changes in their prices. This 
 assumption he had a perfect right to make, as such ; only 
 he should have drawn attention to it, and have noted its 
 arbitrariness, which permitted it to be countered by Jevons's 
 later equally arbitrary assumption of another sort. Thus 
 Laspeyres's problem was different from Jevons's original 
 problem, because Jevons had not made the assumption 
 which Laspeyres made ; and it was different from Jevons's 
 later suggested problem, because that rested on another 
 different assumption. They argued at cross purposes ; and 
 they did so because they did not introduce the subject of 
 weighting except in this secondary clandestine manner, and 
 they never argued as to which kind of weighting is the more 
 correct. Galileo, we have seen, advanced some incomplete 
 suppositions, and then remarked that no solution could be 
 made from them, because some essential data were not 
 supplied. Jevons set before himself a problem with just 
 such insufficient data, and reasoned about them just as 
 if he had given himself all the data needed ; and when 
 
 '2 Investigations, pp. 120-1. 
 "^ II Calcolo dei Valori Medii, p. 39. 
 
 "^ Further Considerations on Index-Numbers, Economic Journal, 
 March, 1896, pp. 129-30. 
 
DEVIATIONS AND VARIATIONS 87 
 
 Laspeyres supplied some data, he supplied some others, and 
 neither examined what the additional data ought to be in 
 order to represent the ordinary state of things. The two 
 commodities under consideration were really two classes con- 
 taining different numbers of individuals, whose prices, them- 
 selves averaged at each period, varied between the periods ; so 
 that the price- variations they were averaging were the price- 
 variations, not of two commodities, but of an unknown (or 
 unposited) number of individuals distributed over two classes. 
 Yet neither of the disputants (nor Messedaglia and Pierson 
 after them) made the slightest attempt to ascertain or to 
 define what and how many were those individual things 
 whose price-variations they were averaging. It was as if 
 two anthropologists should hear that the average height of 
 one group of men were so and so, and the average height of 
 another group of men were so and so, and then, without 
 trying to find out how many individuals were in each group, 
 should dispute over the proper kind of average for averaging 
 these averaged heights. It is pitiable to think that men of 
 scientific attainment should argue in this way.^^ And yet 
 others have continued to take sides in this dispute in the same 
 old incomplete way ; and his opponents have criticised 
 Jevons rather for using the geometric average than for 
 neglecting to inquire into the proper weighting. In all his 
 writings only two passages can be found in which, in this 
 connection, Jevons referred to weighting, and then he showed 
 indifference to the subject. He even entertained the idea 
 of weighting his commodities inversely according to their 
 price- variations ! ^® In his practical work he was bothered 
 by the high prices of cotton in 1862, and although the 
 geometric average was recommended as especially suitable 
 for such extravagant cases, he actually doctored those prices. 
 Had he noticed that the quantity of cotton had been even 
 more extremely reduced, those high prices, properly weighted, 
 would have given him no cause of uneasiness. It may be 
 remarked that Laspeyres added at the end of his own argu- 
 ment : " What is true of two commodities, is true also of any 
 
 »= Cf. M. G. E.-V., pp. 266-8, 275. 
 
 »« Investigations, pp. 21, 57. Jevons was one of the first, if 
 not the first, to mention the use of "ponderation " in the subject 
 of axiometry ; and therefore he may well be excused for not 
 being precise with regard to it, for making a wild suggestion, and 
 for refusing to " attempt to decide " the questions involved. 
 But we should advance beyond his first step. 
 
88 THE PROBLEM OF ESTIMATION 
 
 number of commodities." He happened to be right ; for 
 this is true of the arithmetic average with which he was 
 dealing, as also of the harmonic. But neither he nor Jevons 
 perceived that this statement, as it stands in all its general- 
 ness, is not true ; for it is not true of the geometric average. 
 Jevons never saw, what we shall soon see, that he had no 
 right to extend to the geometric average of many price- 
 variations, as he tacitly did, what he might have proved of 
 the geometric mean in the case of two commodities ; and 
 Laspeyres never took him to task for this oversight. 
 
 In measuring variations of the exchange-value of money 
 we pass from one period to another. Now, at any one period 
 the importance of every commodity is measured by its fuU 
 value, which is calculated by the quantity of the commodity 
 that has been dealt with during that period multiplied by its 
 price. If we are measuring the variation of the money of a 
 country (not of any one class of its population), we should in 
 strict theory take into account all commodities (along with 
 some other things, possibly, such as house-rent, but not 
 wages), and the complete quantities of them that have 
 appeared in the markets throughout the whole country. 
 This is evidently impossible, and samples of the most simple 
 and useful commodities have to be selected, and their quan- 
 tities have to be estimated more or less accurately. Some 
 nice questions arise as to whether only what is consumed 
 in the country, or only what is produced in it, or both 
 together are to be counted ; and also there are difficulties as 
 to the single price-quotation that is to be given at each period 
 to each commodity, since this, too, must be an average. ^^ If, 
 
 " Throughout the country during the period a commodity is 
 not sold at one price, nor even at one wholesale price in its prin- 
 cipal market. Various quantities of it are sold at different 
 prices, and the full value is obtained by adding all the sums 
 spent (at the same stage in its advance towards the consumer), 
 and the average price is found by dividing the total sum (or the 
 full value) by the total quantities. This result is the arithmetic 
 average of the prices weighted according to the quantities sold 
 at the prices. Between producer and consumer some goods pass 
 through few and some through many hands. It would seem as 
 if only one sale (say the first or the last) ought to be counted ; 
 but this has been disputed. As it is impossible to record all sales, 
 estimates have to be relied on ; which leaves room for error, and 
 for the calculation of error by the doctrine of probabilities. This 
 doctrine shows that, when the investigation has been conducted 
 with care and without bias, because of the likelihood of the errors 
 
DEVIATIONS AND VARIATIONS 89 
 
 further, instead of its being the money of the country that 
 is being measured, it is the money of some class (say labourers), 
 then a different selection of articles, with different quantities, 
 and even with different prices (retail instead of wholesale), 
 has to be made. Into these details it is impossible here to 
 enter, and it is not necessary ; for they have no effect upon 
 the problem of averaging price-variations, which is the 
 problem before us. This problem presupposes that the 
 prices and the quantities are given, ^® and it then proceeds 
 to operate upon them. This problem is a large one in itself, 
 and quite deserving of being isolated from the problems 
 involved in the providing of its data. It is a mathematical 
 problem, needing to be solved in theory first, while those are 
 mostly practical questions, for practical men to decide. The 
 theoretician may leave those problems to others. The 
 others should take the solution of the problem of averaging 
 from the theoretician, modifying it to suit their practic^ 
 purposes. Being utterly distinct, this problem and the 
 other problems should never be mixed. 
 
 We then assume that (i) commodities, (2) their quantities, 
 and (3) their prices are given, wherewith (4) their full values 
 are also given, whereby their relative importance is measured 
 and their weights in the averaging determined. The commo- 
 dities, as classes of things, must be the same at both periods. 
 If a commodity exists at one period and not at another, it 
 must be omitted from any comparison of these two periods ; 
 for its prices have not varied, it has merely appeared or 
 disappeared.^® The quantities, however, as well as the 
 prices, are apt to vary between the two periods, involving 
 also variations in the full values. But we are not averaging 
 the variations of the quantities, nor of the fioll values : we 
 are averaging only the variations of the prices. Yet the 
 variations of the quantities, along with the variations of the 
 prices, by affecting the full values cause variations in these ; 
 and the variations of the fuU values cause variations in the 
 weights wherewith the price-variations are to be averaged. 
 
 being on opposite sides and therefore neutralising one another, 
 the effect of the individual errors upon the final result is likely 
 to be small. 
 
 " This is the position taken by Professor Edgeworth at the 
 beginning of his Memorandum to the first Report of the British 
 Association Committee, where, p. 254, he wrote : " It is supposed 
 that the prices of commodities, and also the quantities pur - 
 chased, at two epochs, are given." 
 
 " Cf. M. G. E.-V., pp. 112-13. 
 
go THE PROBLEM OF ESTIMATION 
 
 Thus, after aU, the variations of a commodity's price, in so 
 far as it affects the commodity's full value, affects the weight 
 to be assigned to that commodity's price-variation, and so 
 restores a partial resemblance to the weights in the problem 
 of estimates above cited, where the weights are entirely 
 determined by the deviations, or errors, of the estimates, 
 but inversely. Here, however, the influence, as far as it goes, 
 is neither inverse nor direct, being itself affected by another 
 factor. For here there is another factor, the quantities, 
 which is entirely absent from the other problem. 
 
 With respect to quantities, the present problem has more 
 analogy with the problem of estimates set up by Dr. Bowley. 
 For in Dr. Bowley 's problem estimates of two or more 
 different things were involved, and the quantities, or sizes, 
 of these things had influence. But here still another 
 difference subsists. 
 
 Commodities are to be weighted according to their im- 
 portance, or their full values. But the problem of axiometry 
 always involves at least two periods. There is a first period, 
 and there is a second period which is compared with it. 
 Price- variations have taken place between the two, and these 
 are to be averaged to get the amount of their variation as a 
 whole. But the weights of the commodities at the second 
 period are apt to be different from their weights at the first 
 period. Which weights, then, are the right ones — those of 
 the first period ? or those of the second ? or should there be 
 a combination of the two sets ? There is no reason for 
 preferring either the first or the second. Then the combina- 
 tion of both would seem to be the proper answer. And this 
 combination itself involves an averaging of the weights of the 
 two periods. 
 
 A hint of this difficulty appeared in Dr. Bowley 's problem. 
 And there we saw that a wrong system of weighting with a 
 wrong average may give the right result. Possibly the same 
 may take place here. Perhaps, then, a wrong average with 
 a wrong weighting may serve us, even though the right 
 average with the right weighting may fail us. It is aU the 
 plainer how close is the connection between the systems of 
 weighting and the averages.^"" 
 
 i"" In their dispute above reviewed, Jevons simply treated 
 the two commodities as equally important. Laspeyres posited 
 that they were equally important at the first period only. Then 
 Jevons in his reply posited that they were equally important at 
 both periods. In this way they were able to use different evenly 
 
DEVIATIONS AND VARIATIONS 91 
 
 Index-numbers. 
 
 In averaging the price- variations of varying quantities and 
 full values of commodities, the whole ground is covered by 
 three possible suppositions. (I.) The quantities may be the 
 same in both periods, and the full values may vary with the 
 prices. (II.) In spite of variations of prices, the full values 
 may be the same in both periods, because of inverse varia- 
 tions of the quantities. (III.) Both the prices and the quan- 
 tities may vary irregularly, and therefore also the full values 
 vary. The first two suppositions are very improbable, as the 
 third is the one which almost always takes place ; for the 
 usual course of things is that, the exchange- value of money 
 remaining nearly stable, in most commodities a rise of price 
 generally goes with a diminution of quantity, and a fall of 
 price with an augmentation of quantity, or reversely (for 
 here, that is, for our purpose, it is indifferent which is cause 
 and which effect). These opposite movements may tend to 
 keep the fuU values more stable than either of the factors. 
 But there always is much irregularity. Now, the first two 
 suppositions contain separately the elements into which the 
 third can be analysed ; and therefore, as logical method 
 enjoins going from the simple to the complex, we may 
 briefly review the matter in this order, since the first two 
 suppositions, though not probable in practice, are possible 
 in theory. 
 
 Throughout the following let us represent quantities by q, 
 prices by p, and price-levels by P. The quantities and 
 prices of particular commodities may be distinguished by 
 accents, and those of different periods by subscript numerals. 
 Full values, then, which may be represented by v, are = qp. 
 
 (I.) The quantities being the same in both periods, there is 
 no need of subscripts for them. The case is formulated thus : 
 
 -Pa _ q'Pi + q"p^' + q"'p^" + ...ton terms 
 ■Pi q'Pi + q"fx' + q"'Pi" + ...ion terms 
 
 (I) 
 
 weighted averages. But neither of them inquired whether the 
 weighting they chose fitted the average they used. They may 
 have hit upon a right combination, or not, but they put forth no 
 effort to prove what was right or wrong in the matter. They 
 did not posit all the possible, nor even the most Ukely, com- 
 binations of weighting. They refrained from any systematic 
 inquiry into the subject. 
 
92 THE PROBLEM OF ESTIMATION 
 
 It is evident that the price-level has varied according to the 
 sums of the full values (and the exchange-value of money 
 has varied inversely). A whole people (or a sample portion 
 of it) has purchased the same quantities at different prices. 
 This is the same as saying the average person has bought his 
 portions of the goods, the same in both periods, at different 
 prices. Obviously if he has bought them all with the same 
 expenditure, their average price has not changed ; if he has 
 bought them with a greater expenditure, their average price 
 has risen ; and if he has bought them with a lesser expendi- 
 ture, their average price has fallen ; and the rise or fall is 
 proportional to the greater or lesser expenditure.i"^ The 
 case here is one of gain and loss. Suppose the total expendi- 
 tures are the same ; this means that what you gain by the 
 faU of one price you lose by the equal rise (arithmetically 
 reckoned) of another, or by a distribution of it over several 
 others. Or again, if the price of one article has risen 50 per 
 cent, (in the usual way of reckoning percentage), it can be 
 compensated by another falling 50 per cent., or by two others 
 falling 25 per cent, each, or the one 30 per cent, and the other 
 20 per cent., etc. It is NozzoUni's interpretation of the 
 problem of estimation over again. But how about the 
 average of the price- variations ? Is this necessarily and only 
 the arithmetic average, as that example would seem to 
 require ? No ; for the weighting that has to be employed in 
 the averaging brings in diversity. 
 
 The equation of formula b above (p. 31) exactly to two 
 other formulae and approximately to a third, can be imitated 
 here. The formula just given is equal to this, 
 
 /// 
 
 rf""^ q'Pi\ q"Pi" + rk'" + ■■■ 
 
 which is the formula for the arithmetic average of the price- 
 variations with weighting according to the full values of the 
 first period. It is also equal to this, 
 
 1"! If the prices all vary alike, say by r, every p^ would be 
 rpi, and every v^ would be rv^, and the variation of the price- 
 level would be r. If the quantities all vary alike, say by s, 
 every q^ would be sqi, and the variation of the price-level could 
 be obtained by dividing the result by s. 
 
DEVIATIONS AND VARIATIONS 93 
 
 P2_ q'-p2' +J"W + q"'P2"'^+ . . . 
 
 p^X- qpi -\- pj,X q p2. + )~Tr, Xq Pi + • • ■ 
 
 2d, 
 
 S.^fl (3) 
 
 Pi 
 
 which is the formula for the harmonic average of the price- 
 variations with weighting according to the full values of the 
 second period}^^ It is also found by trial to be, mostly, 
 approximately equal to this, 
 
 Pg _ Sj ypipij (p^Y VPiFi (p^'\ i" '^P^W' ( pi" \ i'" \'Pi"'t'i"' 
 
 Pi~ y \p7) '\pZ'j '^pr^ ■•■' ^4^ 
 
 which is the formula for the geometric average of the price- 
 variations with weighting according to the full values over 
 both the periods}'^ Thus Galileo's advocacy of the geometric 
 average fares here almost as well as Nozzolini's advocaey of 
 the arithmetic. For Nozzohni could advocate the arithmetic 
 average here only with a weighting for which no reason can 
 be given, and which is no better than another weighting 
 which, used with the harmonic average, gives the same result, 
 so that he must share the honours of the arithmetic average 
 with the harmonic average ; whereas the geometric average, 
 itself apparently the right one, is used with the right weight- 
 ing, so that everything about it seems to be in its favour. 
 Yet it fails, since it does not give with perfect accuracy the 
 result known to be the right one. We have the paradox, 
 that wrong averages with wrong weightings give the right 
 result, while the right average with the right weighting does 
 not. 
 
 ^"2 If the ^I's are all the same, formula 2 reduces to 5?!ip 
 
 Pi^v^' 
 If the p^'s are all the same, formula 3 reduces to ^' ' . 
 
 ^»' The full values of the two periods are compared, not by 
 their arithmetic, but by their geometric mean. The reason for 
 this is given in M. G. E.-V., pp. 105-10, 118-21, 387-8. For the 
 proof of this formula's approximation to the others, see above, 
 note 25. Further of. M. G. E.-V., p. 350. The first two iden- 
 tifications are repeated by Professor Fisher in The Purchasing 
 Power of Money, pp. 365, 397, cf. 393-5 ; but he apparently has 
 no use for the simiUtude of the third additional formula (4). 
 And the first alone is repeated by Professor Mitchell, Index- 
 Numbers of Wholesale Prices, pp. 92-3 and n. 
 
v^ 
 
 94 THE PROBLEM OF ESTIMATION 
 
 The geometric average, however, does give the right resiilt 
 with perfect accuracy in one stiU more narrowed case, when, 
 in fact, it is reduced to being the geometric mean between 
 two equally important and hence evenly weighted price- 
 variations. If we deal with only two commodities, or with 
 two groups of commodities, and if their full values over the 
 two periods together are equal (geometrically measured, that 
 is by their products), then it is demonstrable that the geome- 
 tric mean of their two price- variations with even weighting 
 is the same as the arithmetic average of them with the weight- 
 ing of the first period, and as the harmonic average of them 
 with the weighting of the second period. This may be 
 represented simply as follows, 
 
 fi ■ Pi' q'Px + q"i>i" 
 
 on condition that q'p-^ . q'p^ = q"-p{' . q"p4'?-^^ And so, 
 after all, the geometric mean with the right weighting, where 
 it i* apphcable, does give the right result. It is only the 
 geometric average, that is at fault. 
 
 The reason for this agreement in the case of the geometric 
 mean, depends on the well-known proposition that between 
 only two terms the geometric mean is the geometric mean 
 between their arithmetic and their harmonic means. And 
 the reason why the geometric average, in formula (4), applied 
 to more than two equally important price-variations (and 
 even to two that are not equally important ^''*), is only 
 
 103a.. Xheproof is given in M. G. E.-V., pp. 519-20, cf. 351-2. 
 In this case the arithmetic and harmonic means are not subject 
 to the objections brought against them in the case of estimations. 
 Those objections apply to these averages only when used with 
 the weighting proper to the geometric average. When used 
 with their own proper weightings, they give the right result. 
 ' ^"^ For then we no longer have two single items, since one of 
 the named items is composed of more than one real item. If 
 the two articles are only slightly divergent in importance over 
 the two periods (the prices diverging but slightly), the geometric 
 average with its proper weighting will still be very close to the 
 two others with their proper weighting ; but it will diverge con- 
 siderably from them if the one article becomes three or four times 
 more important than the other, and its discrepancy may even 
 be very great with a still greater divergence of the relative 
 weights : cf. M. G. E.-V., pp. 365-7. Something analogous to 
 this is stated by Professor Fisher, Purchasing Power of Money, 
 pp. 412-13 n., although what he there says is applied to the 
 geometric average wrongly, because evenly, weighted. 
 
DEVIATIONS AND VARIATIONS 95 
 
 approximately true (in ordinary cases), depends on the fact 
 that between more than two terms the geometric average 
 is not exactly (except for special reasons), but only approxi- 
 mately, the geometric mean between their arithmetic and 
 harmonic averages, as any one may see who will take the 
 trouble to look into the matter, although this fact is often 
 ignored.^"^ And any one can find that the geometric average 
 of all the terms is sometimes above and sometimes below 
 the geometric mean of the other two averages of them ; 
 and the same he will find to hold of formula (4) compared 
 with either of the three preceding formulas. And we can 
 find this general rule, that if the prices of the preponderating 
 commodities (those with the greater weights) rise above the 
 average price- variation, formula (4) gives a result above the 
 other three, and if they fall below that average, it gives a 
 result below the others.^"® Even so, that is, if it happens 
 (which is not likely) that the prices of all the preponderating 
 commodities vary in the same direction and the prices of 
 all the less important commodities vary in the opposite 
 direction (compared with the variation of the price-level), 
 the discrepancy of formula (4) from the others is found to be 
 small, unless the opposite price- variations are excessive ; 
 and of course in the more usual cases of a mixture of the 
 variations and the weights, such as occur in practice, some 
 of the preponderating commodities rising and others falling 
 in price, and some of the less important commodities falling 
 and others rising, and none or few varying greatly, the discre- 
 pancy will be still smaller, and may even be reduced to 
 nothing. And in a series of measurements extending over 
 many periods, as the errors will sometimes be in excess and 
 sometimes in defect, there will not be cumulation, but there 
 will be neutraHsation, of the errors, wherefore it is probable 
 that the total error will increase in amount very slowly. 
 
 i»5 It has been ignored even after the proof of it was first given 
 in M. G. E.-V., p. 517. Thus as late as 1913 Zizek and his 
 American translator assert that " of the three means of the same 
 set of values [no limitation being made to two values] the g eometric 
 is always the geometric mean of the other two," the arithmetic 
 and the harmonic. Statistical Averages, p. 132 n., and again 
 pp. 196-7 n. In the last passage Zizek illustrates the proposition 
 by an instance of two figures. Had he tried it on three figures, 
 he would have seen his mistake. 
 
 !»' M. G. E.-V., p. 365. Unfortunately the rule, there given 
 in italics, by a double slip inverts the true statement. Cf. 
 pp. 321, 521. 
 
96 THE PROBLEM OF ESTIMATION 
 
 (II.) The full values remaining the same at both periods, 
 this supposition means that the community has the same 
 total income in both the periods, and lays it out in the same 
 sums to purchase the various commodities, whatever be the 
 changes of their prices.^"' If, then, the price of a commo- 
 dity falls, the community buys more of it ; if the price of a 
 commodity rises, the community buys less of it — and so does 
 the average person in the community. Hence we can 
 examine the purchasing power of money directly by the total 
 quantities which the same total incomes of the community 
 can in this way purchase in the two periods. But a compli- 
 cation enters here, in that the comparison cannot be merely 
 according to the quantities purchasable, since these have 
 different preciousness, but it must be according to the 
 exchange- values of these quantities ; and it cannot be 
 according to the exchange-values of these quantities at 
 either of the periods alone, but it must be according to their 
 exchange-values over both the periods together (of course 
 geometrically reckoned). ^"^ Therefore the ordinary units 
 of quantity will not serve, and we must construct units, 
 specially for each comparison, that have the same exchange- 
 value over both the periods. When this is done, we can 
 measure variations in the purchasing power of money by 
 averaging the quantities purchasable of such units in very 
 much the same way as before, getting three exactly equal 
 formulae and another approximately equal to the three. 
 But here things are inverted, and the arithmetic average of 
 the quantity- variations must use weighting according to the 
 units purchased in the second period, and the harmonic 
 average must use weighting according to the units purchased 
 in the first period ; but of course the geometric average wiU 
 continue to use weighting according to the geometric mean 
 of the units purchased in both the periods. The formulae 
 thus directly giving the variation of the purchasing power 
 (or exchange-value) of money, can now easily be turned 
 back into formulae of the usual sort giving the variation of the 
 price-level. The first of the three equal formulae (for the 
 arithmetic and harmonic equivalents may be omitted) has 
 either of these two equivalent forms, 
 
 "' If the total income increases or decreases, and the sums 
 spent on every commodity are varied in the same proportion, 
 the common variation may be divided out, leaving the same 
 results as in the text. 
 
 108 por the principle see M. G. E.-V., p. 304 ; for the method 
 and for the formulae that follow in the text here, see pp. 305-10. 
 
DEVIATIONS AND VARIATIONS 97 
 
 Px i^'^fJxR + qz'^-Px'-p^' + ?u"'v?7^' + . . . 
 
 or 
 
 P^ - P^'^^qTql' + P^'^qT^' + fr^q^qz" + ■•■ 
 
 Pi PiWq,'ql+ P^"Vq,"q," + P^V^Tq^' + • ■ • 
 
 And the approximately correct geometric average has this 
 formula, 
 p^ ^ 2 viilW ip^'\ vii^ /^g"\ vi;?^;;^ /'?^a""\ ^'"■"''"'"' („\ 
 
 This geometric average errs here just as the corresponding 
 geometric average, in formula (4), errs in the preceding 
 supposition, no more and no less, not cumulatively, but 
 neutralisingly.io* And, again applied to two commodities 
 equally important over both the periods, so that the simple 
 
 geometric mean of their price-Variations [yj^ ■ ^rr,) can 
 
 be used, this method gives the exact truth.^^" 
 
 But unfortunately, except in the last narrow case of only 
 two equally important commodities (or two such groups of 
 commodities with similar price- variations in each group), 
 formvilse (5) and (6) are open to an objection to which 
 formula (i) is not exposed — an objection which shows that 
 they are not perfectly true, as that formula is. It would 
 seem to be evident that if prices or quantities or both vary 
 over a number of periods and then return to exactly the same 
 state at a later period as they were in at the first (being the 
 same quantities and the same prices of every commodity), 
 then the measurement of the variations of the price-level 
 from period to period, when combined and strung out in a 
 series, should at the last period indicate the same level as at 
 the first."^ This is only a negative test ; for many different 
 erroneous methods — ^in fact, the very worst method of all, 
 
 "• M. G. E.-V., pp. 321, 323. 
 
 "» M. G. E.-V., p. 330. 
 
 ^" E.g., if the comparison of the second period with the first 
 showed a rise of 20 per cent., and the comparison of the third 
 with the second showed a fall of 5 per cent., this is a fall from 
 
gS THE iPROBLEM OF ESTIMATION 
 
 Dutot's — satisfy it ^^^ ; but if a method does not satisfy this 
 test, there is something wrong about it. This test is precisely 
 analogous to the test which surveyors use in their triangu- 
 lations ; for they measure round many hues and angles until 
 they come back to their starting point, and the two ends of 
 their measurements ought to fit, and when they do not (as 
 they never do), they apply the method of least squares to 
 adjust the error over all the intervening measurements.^^^ 
 So here, after three or more measurements of price- variations 
 over three or more periods after the starting period, if we 
 assume as the next period the conditions of the first period, 
 this last measurement, when combined with the others in a 
 series, ought to indicate no variation, and if it does indicate 
 some variation, it has in so many measurements committed 
 error to that extent. This test, which because of its return- 
 ing to the starting point may be called the circular test, not 
 only indicates that a method is erroneous (unfortunately it 
 cannot indicate that it is correct), but also indicates the 
 amount of its error, and its direction ; and when various 
 methods are tried by it in application to the same data, it 
 indicates fairly well their comparative erroneousness. It is 
 not serviceable for testing all methods ; for those which 
 satisfy it must be tested in other ways. But it is especially 
 useful for testing the methods which have otherwise proved 
 themselves nearly correct."* 
 
 Now, tried by this test, formula (i), with its two variants, 
 shows no error at any later period, however extended be the 
 series ; and, being otherwise demonstrated, it may be 
 regarded as absolutely true, on its purely theoretical condi- 
 tion, extended of course to require constancy of the quan- 
 tities over all the periods. But on the condition laid down 
 in the present second supposition likewise extended to aU the 
 
 120 of yItji so that the series is loo, 120, 114. To form the series 
 100, 120, 114, 100, the comparison of the fourth ( = the first) with 
 the third would have to show a fall of 12 -3 per cent. This, of 
 course, is the inverse of comparing the third with the first, which 
 should show a rise of 14 per cent. 
 
 1" M. G. E.-V., pp. 203, 205. This test is a particular appli- 
 cation of Professor Westergaard's test. The negative nature of 
 Professor Westergaard's test has not always been recognised. 
 
 ^" This last operation suggests a hint for axiometry. 
 
 ^'* This circular test is frequently used for this purpose in 
 M. G. E.-V., pp. 324-41, 368-70, 389-92, 416-18, and especially 
 423-33. When the quantities are not supposed the same at the 
 final period, a complication arises : see pp. 399-402. 
 
DEVIATIONS AND VARIATIONS 99 
 
 periods, the duplicate fornmlse (5) and (6) satisfy this test 
 only in the case of three periods, but not always in the case 
 of four and more periods. The reason is that, in the measure- 
 ment from the second to the third period the units have been 
 changed, and if the next measurement is a comparison with 
 the initial state of things, it, too, has another set of units. 
 Only if we construct units that are equivalent over the whole 
 series of periods compared, wUl this formula, adapted to such 
 an adjustment, satisfy this test, and then only for that set 
 of periods ; wherefore it does not admit of extension into the 
 future.^^^ But, again, it is proved by trial, with the use of 
 this test, that the aberration of these equivalent formulae is 
 even smaller than that of formula (7), being for ordinary 
 price-variations and quantity-variations, such as are likely 
 to occur in experience, very small, and sometimes above and 
 sometimes below, wherefore their errors are not cumulative, 
 but tend to neutralise one another.^^* 
 
 All the formulae so far given, whatever their merits and 
 defects under the conditions supposed, have no practical 
 uses, since the suppositions for which they are invented 
 never occur actually. But, being simpler, they contribute 
 to elucidate the remaining more complex cases, which alone 
 represent the actual economic world. But before going on, 
 it should be emphasised that aU that has been advanced, 
 except as to the amounts of error determined by trial, is 
 demonstrative, and especially that in the first supposition 
 formula (i) and its variants, and even formula (4) when 
 restricted to its narrowed condition, are demonstratively 
 true — as true as anything in Euclid. The problem of 
 measuring the variation of the price-level, and inversely of 
 the exchange- value of money, is solved, as far as an economic 
 world represented by that supposition is concerned.^^' 
 
 (III.) In the actual world both prices and quantities vary, 
 so that it contains features from both the preceding imaginary 
 economic worlds. Now, neither the arithmetic nor the har- 
 monic average of price-variations as obtained in the first 
 supposition is transferable to this third case, because if we 
 use the former with the weighting of the first period and the 
 latter with the weighting of the second period, we no longer 
 get identical results ; and of course it would be absurd to com- 
 
 i» iif. G. fi.-F., pp. 334-6 ; cf- 398-9- 
 
 "« M. G. E.-V., p. 341. 
 
 11' M. G. E.-V., pp. 401-2. That, on this supposition, for- 
 mula (i) is the 'correct method, has been recognised by many 
 writers on the subject. Refer'uices are given, p. 544. 
 
100 THE PROBLEM OF ESTIMATION 
 
 pare merely the sums of the full values of the two periods."® 
 But the geometric average is transferable from those sup- 
 positions to this case, because it requires the use of the weight- 
 ing of both periods. The form of it to be used is that given 
 in formula (7) ; fo r in form ula (4) the q really stands for 
 
 •v/5'15'2, and now '^qxliPifi is = V'^i^'z. ^.nd formula (7) is 
 merely a more comprehensive, or distinguishing form of 
 formula (4). Now, if our third supposition is confined to a 
 world with only two commodities, or of two groups of 
 commodities in each of which the prices vary alike, and if, 
 further, these two have the same importance over both the 
 periods compared, then the simple geometric mean of these 
 price-variations yields the absolutely true measurement of 
 the variation of the price-level.^^* But our misfortune is 
 that the geometric average applied to more than two equally 
 
 ^^^ Suppose we draw the arithmetic average of the price- 
 variations with the weighting of the first period : this gives us 
 
 -^ = V^/-^^^^ =_^2£i, which is the method recommended 
 
 by Laspeyres. Or suppose we use it with the weighting of the 
 
 second period : thus ^ = Xpir"^" = \pi / , which is a 
 
 ' ^P,q2 ^v, 
 
 method recommended by Palgrave. An arithmetic mean 
 between these has never been recommended. Evidently none 
 of these have any validity. Nor would the harmonic average 
 with any such weightings. Yet the harmonic average with the 
 
 weighting of the second period gives -^ = 2^^ which has been 
 
 recommended by Paasche ; and with the weighting of the first 
 
 period it gives p- = — -,■ ■ , which has been recommended by 
 
 nobody. 
 
 "» As do also formulae 6 (already given) and 8, 9, and 10 (still 
 to be given). See M. G. E.-V., pp. 402-3, 413, 422. We may 
 now perceive the solution of the Jevons-Laspeyres dispute. It 
 the two commodities are equally important over both the periods 
 together, Jevons was right. If they were as Laspeyres supposed 
 them to be, Laspeyres was right. But neither was fully right 
 unless he expressly confined his solution to the particular sup- 
 position about the weights (or the quantities). This was done 
 by neither of them. Cf. M. G. E.-V., p. 354 
 
DEVIATIONS AND VARIATIONS loi 
 
 important commodities is only approximative even under 
 the first two suppositions, and it can be no more than 
 approximative for the present case. The expectation that 
 we could pass from the geometric mean to the geometric 
 average has not been fidfiUed. Like Moses, we have just 
 caught a glimpse of the promised land, and are then kept out 
 of it. But we have come near it, and we may abide near it. 
 The geometric average is approximative in the present 
 general case just as much, and in the same way — ^not cumula- 
 tively, but neutralisingly — as it was in those special cases. 
 
 Other approximations may also be made. The geometric 
 method is not the only one that can be taken over from the 
 earlier suppositions to this third. In the second supposition 
 formula (6) can be used without the condition that 5' 1:^1 is = 
 q>^2, that is, it can be used generally, or under our third 
 supposition.^^" And now it yields results, as tested by our 
 circular test, as good, but no better, than it yielded there. 
 This formula (6), being analysed as formula b above was 
 analysed, may be shown to be (i) the arithmetic average of 
 the price-variations with weighting according to the prices 
 of the first period multiplied by the geometric mean between 
 the quantities of the two periods ; (2) the harmonic average 
 of the price- variations with weighting according to the prices 
 of the second period multiplied by the same geometric means. 
 By analogy we see that it will also be approximately (3) the 
 geometric average of the price-variations with weighting 
 according to the geometric means between the full values of the 
 two periods. It is, then, approximately equal to formula (7) . 
 This is the inverse of what we knew before ; for we have 
 known that formula (7) is approximately equal to it. In 
 the second supposition formula (6) is better than formula (7). 
 Therefore in general we may expect formula (6) to be better 
 than formula (7) — better than the geometric method. 
 
 Another approximative method can be obtained by com- 
 bining the best results from the two elementary suppositions, 
 that is, combining formula (i), in which the quantities of the 
 two periods must be distinguished, with formula (5), which 
 cannot be taken over alone as could formula (6), into this : 
 
 P2 ^ q2 'p2+q2"P," + --- giV^7^ + gi"V^^W' + ... 
 ^1 qi'Pi + qi"Pi" + • • • 'q2'^Pip2 + q2"^Pi"p2" + ■•• 
 
 _tv2 Mx'^P\p2 . 
 
 2f 1 ■ ^qzVpipi ' 
 "° a. M.G.E.-V., p. 360. This method is there called Scrope's 
 
 (8) 
 
102 THE PROBLEM OF ESTIMATION 
 
 in which it is evident that if the quantities are the same in 
 both the periods, it reduces to formula (i), and if the full 
 values are the same in both the periods, it reduces to formula 
 (5). There is good reasoning leading up to this combination, 
 too long to repeat here ; ^^ and yet this formula does not 
 satisfy the circular test any better than does either the 
 properly weighted geometric average or formula (6), though 
 it meets that test about as well as they do.^^^ 
 
 Again : we have remarked that in the present extensive 
 supposition the arithmetic average with the weighting of the 
 first period and the harmonic average with the weighting of 
 the second period yield different results. The fact is, as trial 
 shows, they yield results which, when tested by the circular 
 test, are found to lie on opposite sides of the truth, 'and 
 apparently equally above and below it proportionally. This 
 suggests taking the geometric mean between them, which 
 works out as follows. 
 
 This method still fails to satisfy the circular test ; but perhaps 
 it satisfies it best of all.^^^ Note that it involves the arith- 
 metic average, the harmonic average, the weightings of the 
 first and second periods, and the geometric mean. It is, too, 
 neither more nor less than the geometric mean between two 
 forms of formula (i) applied to the quantities of the two 
 periods one at a time. It seems to contain everything that 
 could be desired. 
 
 Nos. 6, 7, 8 and 9, then, are four methods which seem 
 to have much in their favour. They all keep very close 
 together, and each comes very near to satisfying the circular 
 test ; and they err sometimes above and sometimes below 
 the truth, so that their errors are not cumulative over a series 
 of successive periods, but tend to neutralise one another.^^* 
 
 emended method, pp. 373, 543. It would have been better if this 
 method, in all its forms, had been called Lowe's method. 
 
 "1 See M. G. E.-V., pp. 374-82. 
 
 122 This method is analysed and compared with Drobisch's, 
 Lehr's, and Nicholson's, in M. G. E.-V., pp. 383-96, and with 
 formulse (6) arid (7), pp. 396-407. 
 
 "3 This method is not formulated in M. G. E.-V. ; but in 
 note 16, on p. 429, it is tested and found wanting, but it happens 
 to meet this particular test better than do any of the preceding 
 methods. It was unduly overlooked in that work. 
 
 "* For remarks on the comparative merits of three of them. 
 
DEVIATIONS AND VARIATIONS 103 
 
 We have seen reason to believe formula (6) better than 
 formula (7). Perhaps formula (9) is the best of the rest, 
 but between it and Nos. 6 and 8 it would be difficult to decide 
 with assurance. The circular test may provide the criterion. 
 Only it wiU require many laborious operations before a 
 decision can be made. Who wiU undertake the labour ? 
 
 It may be questioned whether any method can be found 
 that will satisfy the circxolar test perfectly, at the same time 
 showing other ear-marks of correctness ; for, as we have 
 seen, the test can easily be satisfied by many wrong methods. 
 The problem of axiometry may after all be mathematicadly 
 insoluble with perfect precision, like the problem of squaring 
 the circle. ^^* Mathematicians can work out the value of the 
 symbol w to any number of decimals ; but practical workers 
 rarely employ more than four or five. Surveyors, in working 
 around a territory and coming back to their base, never strike 
 it with perfect accuracy ; and even after adjusting their 
 measurements they are not sure of their results to every 
 millimetre. If, then, our best methods, after proceeding 
 over many periods, when worked back to their starting point, 
 do not fit into it with perfect exactness, we need not be dis- 
 couraged. Their precision is sufficient for practical purposes. 
 
 Indeed, their precision is greater than is needed, when we 
 consider the lack of precision that exists in the data that are 
 presented to these methods to operate upon. Therefore 
 even less precise methods may be adopted in practice, if more 
 convenient and yet fairly agreeing with these theoretically 
 best methods. 
 
 see M. G. E.-V., p. 407. Unfortunately the fourth, formula (9), 
 was not there discussed. 
 
 ^^^ Cf. M. G. E.-V., p. 408. There in a note some hints are 
 offered for possible improvement of formula (7). Perhaps the 
 reason why that formula fails is because in the compound unit 
 constructed that is to have the same exchange-value over both 
 the periods together, there is error due to the fact that the full 
 values at the two periods are reckoned in moneys of different 
 exchange- value, and it is impossible to allow for that fact till the 
 measurement of the variation of the exchange-value of money is 
 completed, whereas that compound unit is needed for the making 
 of that very measurement. Now, one of the exchange-values of 
 money must be smaller than the other, and therefore its greater 
 smallness should be allowed for. The arithmetic mean does not 
 allow for it. The geometric mean allows for it too much. There- 
 fore some intermediate mean is needed that will allow for it 
 correctly. 
 
i04 THE PROBLEM OF ESTIMATION 
 
 AU these four theoretically best methods contain a geome- 
 tric feature, involving the extraction of square roots. These 
 operations, even with the aid of logarithms, are laborious. 
 Two methods, substituting an arithmetic for the geometric 
 mean, have been proposed. The first, independently sug- 
 gested by Professor Marshall and by Professor Edgeworth, 
 has been actually recommended by the latter and by the 
 British Association Committee.^^* The second has been 
 merely suggested, likewise independently, by Drobisch and 
 Sidgwick,"'^ and has been used by Dr. Bowley, as will be seen 
 presently. They are 
 
 and 
 
 (II) 
 
 The first of these arithmeticises formula (6), and the second 
 arithmeticises formula (9). Which is the better, theory does 
 not say, although if (9) is better than (6), (11) is probably 
 better than (10). To decide between them appeal must 
 again be made to the laborious task of testing them many 
 times by the circular test.^^^ Sufficient trial of them has 
 already been made to show that even in unfavourable 
 examples they come pretty close to the theoretically best 
 methods. This congruence, along with their practical 
 convenience, is the proper, and the only proper, reason for 
 recommending them ; and on this ground, formula (10), 
 being easier, may be preferred to formula (11). Other 
 methods, many of them in use to-day, are absolutely bad 
 ar.d pernicious. In many of them the errors are so great 
 
 i^« First Report, pp. 249-50 ; Edgeworth's Memorandum 
 thereto, p. 266. Cf. M. G. E.-V., p. 543. 
 
 ^^' Both suggestions were merely incidental. Sidgwick made 
 his in a footnote, in which he added the remark that such a mean 
 is wanting in " practical significance," Political Economy, p. 68 n. 
 Likewise Wicksell has said that the use of this mean has only a 
 " conventional meaning," and therefore he rejected it, Geldzins 
 und Giiterpreise, Jena, 1898, pp. 8-9. 
 
 "8 The single use of this test in M. G. E.-V., pp. 427-8, shows 
 a slight advantage in formula (11). Formula (10) is examined in 
 that work at length, pp. 409-23. There it is shown that this 
 method and Lehr's method apparently err almost equally on 
 opposite sides of the truth, suggestive of drawing a mean between 
 them, though this would be too laborious for practice. 
 
DEVIATIONS AND VARIATIONS 105 
 
 and so cumiilative, that they cannot be used in the proper 
 way, in the " chain " system ; but must be measured from 
 a common base, and then they give rise to all sorts of hap- 
 hazard weighting, involving unknowable errors, when later 
 periods (the more recent periods) are compared with one 
 another. AU these should be thrown on the scrap heap.^^' 
 The two before us are sufficiently easy ; and any one who will 
 shirk the trouble of working them, should abandon the 
 subject. These two, then, may be declared unequivocally 
 the best practical methods ; and it is only necessary for the 
 practical workers to come together and choose between them. 
 Let us illustrate the matter by an example. In the 
 chapter on index-numbers in his Elements of Statistics 
 Dr. Bowley uses an example with three commodities, one 
 of which is made to vary rather violently in price and 
 quantity, as follows : 
 
 First year. 
 
 s. d. 
 6 quarterns bread at 6d,. 3 o 
 4 lbs. meat at yd. . 2 4 
 
 j lb. tea at 3s. . .16 
 
 6 10 
 
 Second year. 
 
 s. d. 
 7 quarterns at ^d. . 2 11 
 5 lbs. at 8^. . .34 
 
 I J lbs. at IS. 4<f. . .20 
 
 8 3 
 
 On these data Dr. Bowley performs five operations, rejects 
 three as resting "on no sound hypothesis," and retains the 
 first two because " they may be regarded " as giving " inferior 
 and superior limits of the index-number, which may be 
 estimated as their arithmetic mean (80-5) as a first approxi- 
 mation " (pp. 226-7). This final performance is in accord 
 
 128 There is, however, one practical method which avoids those 
 errors and the consequent condemnation. This is the method 
 used by Lowe and recommended by Scrope, and by many others 
 since, notably by G. H. Knibbs, who calls it " the aggregate 
 expenditure method," Prices, Indexes, and Cost of Living in 
 Australia, 1912, pp. 11-14. Professor Mitchell, who adopts the 
 same name for it, describes it thus : " By this method the cost 
 of an unvarying bill of goods is calculated at the varying prices 
 prevailing during different years," Index-Numbers and Wholesale 
 Prices, p. 160. Its formula is No. i above, but with the q's 
 representing no real quantities, but only general estimates of the 
 average quantities that have been used during certain years. 
 Professor Mitchell recommends it because its results can be 
 shifted from one base to another. This is analogous to saying it 
 satisfies the circular test. But we have seen that this test does 
 not prove the correctness of a method. The truthfulness of a 
 
io6 THE PROBLEM OF ESTIMATION 
 
 with formula (ii) (multiplied by loo to bring the restilt into 
 comparison with loo as the starting point) — and perhaps 
 Dr. Bowley is the first to put that formula into operation. 
 But because of these " may he's " Dr. Bowley's readers can 
 hardly be impressed with the validity of this result, and still 
 less when they remember that he has behttled the use of 
 weighting (and never once mentioned double weighting, 
 which is here employed) except in cases of great variations 
 and few items, such as this is, and for such cases has recom- 
 mended the geometric average. Moreover, he himself casts 
 doubt on the result by speaking of it as " a first approxima- 
 tion," though he gives no hint of any subsequent procedure.^^" 
 We are not surprised, then, that in his latest utterance on the 
 subject, in a paper on The Measurement of Changes in the 
 Cost of Living in the Journal of the Royal Statistical Society, 
 May, 1919 (pp. 348-9), Dr. Bowley says little in favour of 
 this method (which there is given as Method III.), but thinks 
 it " appears valid " for certain purposes. This is not the 
 way to recommend a good method. 
 
 Now, on applying to these data the several theoretically 
 best methods, worked out to the second decimal for distinct- 
 ness, and giving Dr. Bowley's also in this form, along with 
 the other practical method, we get the following results : 
 
 No. 6, giving index-numbers So-ga, indicating fall of ig-oS per cent, 
 
 7 .. .. 81-07 " .. 18-93 
 
 8 „ „ 8o-8o „ „ 19-20 
 
 9 „ „ 80-32 . „ „ 19-68 
 
 10 „ „ 79-34 „ „ 20-66 
 
 11 Dr. Bowley's „ 80-47 „ „ 19-53 
 
 If formula (9) be the best of the theoretically best methods, 
 we see that Dr. Bowley's " first approximation," the practical 
 formula (11), is better, here at least, than some of the 
 theoretically best. Here again also it looks better than 
 No. 10. Then let us recommend it firmly, and not with 
 mere "may be's."^*^ 
 
 To return to theory : would anything be gained by drawing 
 an average of the results yielded by several methods ? 
 
 method should not be sacrificed to obtain this and some other 
 merely apparent advantages. But in past epochs, when the 
 annual variations of quantities cannot now be ascertained, this 
 is by all odds the best practical method to use. 
 
 1^° Jevons likewise had spoken of his simple geometric average 
 as " a first approximation," without making any second. Inves- 
 tigations, p. 122. 
 
 1" M. G. E.-V. was published three months after the first 
 
DEVIATIONS AND VARIATIONS 107 
 
 Hardly, as they have different merits. All that we can do is 
 to choose the best, after testing all the candidates ; for to 
 average the others with the best, would only vitiate the 
 result. This is said in spite of Professor Edgeworth to the 
 contrary, who has advocated such averaging of various 
 methods in his first Memorandum (pp. 266, 297). Only if 
 two methods can be shown to err nearly equally on opposite 
 sides of the truth, can a mean between them give a closer 
 approach to the truth, and then this mean constitutes a new 
 method, as in formulae (9) and (11). ^^^ Professor Edgeworth 
 apparently conceives of all the methods as so many chance 
 shots at a mark, and, having no knowledge of any way of 
 testing them, he relies on the theory of probabilities that they 
 are likely to err equally on opposite sides, so that their 
 average is more likely to approximate to the truth. But 
 this procedure rests on the supposition that the observations 
 are made with equal care, or else they must be weighted — 
 and no astronomer or surveyor would include very carelessly 
 made observations. But the methods which Professor 
 Edgeworth would average are not made with equal care : 
 some of them have been invented without any care whatever, 
 being mere first guesses. They would, then, need to be 
 weighted. But nobody can reckon the comparative careful- 
 ness with which the different methods have been invented. 
 Therefore the only way to compare methods is to try them 
 on cases where the true result is known, whereby we can 
 measure their comparative erroneousness and their compara- 
 tive accuracy. And then we should choose the best, and let 
 the others go. 
 
 Probabilities and the Median. 
 
 Making reference, reported in the Journal of the Royal 
 Statistical Society, May, 1919 (p. 366), to The Measurement 
 of General Exchange-Value, Professor Edgeworth criticised 
 that work for " attempting to dispense with the need of 
 Probabilities." If that work has any merit, it is because of 
 that attempt. That work aimed at precision, at the attain- 
 ment of absolutely correct results as far as within our power — 
 not at " the judicious compromise and happy ambiguity " of 
 another writer, which Professor Edgeworth in a previous 
 
 edition of Dr. Bowley's work. Consequently the means of testing 
 the method he so slightingly recommended were in existence 
 when Dr. Bowley revised his later editions. 
 1'* And compare what is said in note 128. 
 
io8 THE PROBLEM OF ESTIMATION 
 
 review of that work held up for imitation.^^' Everything 
 in it about the theory of averaging price-variations is 
 demonstrative, except the calciilations of the amounts of 
 error of methods proved not to be correct. But, for the 
 unexpected breakdown of the geometric average in not 
 fulfilling the promise of the geometric mean, that work would 
 have left no room for the use of probabilities in the theory 
 of the subject, but only in its application to practice. 
 
 In that work the best methods of averaging price-varia- 
 tions were put to various tests — among them the circular 
 test, never before used for measuring their comparative 
 errors, and unfortunately never since made use of ; and they 
 were thus tested on hypothetical examples, and those 
 examples were mostly, as in the usual practice with other 
 writers, cases of two commodities varying in price (and often 
 in quantity) oppositely. Two commodities were chosen 
 because they are the simplest, and because they are always 
 sufficient to disprove a method, though not always sufficient 
 to prove one. They were chosen also because it is natural, 
 in balancing opposite forces or influences, first to use two, 
 and only later to extend the investigation to the case of 
 many. And those two prices were generally supposed to 
 vary extravagantly. This was in order that they should 
 serve, like Bacon's instantice ostensivce, to magnify any error 
 that might lurk in the method under examination, and thus 
 lay it open to more ready detection ; and especially this was 
 done when the erroneousness of different methods was being 
 compared. Nobody knows better than Professor Edgeworth 
 (for he has drawn attention to it ^'*) that different kinds of 
 averages applied to the same many slightly divergent figures 
 yield nearly the same results. Therefore little information 
 can be elicited from applying the different kinds of averages 
 to ordinary cases of price- variations ; and while we have 
 them in our power, as we have in hypotheses, it is the part 
 of wisdom to make them strongly divergent, that the averages 
 may be clearly distinguished. 
 
 This is not Professor Edgeworth's method, and it seems to 
 have displeased him. So much so that he has lost his 
 judicial balance ; for at the same time he made the above 
 criticism he added another, to the effect that of the work in 
 question " a great part was devoted to the investigation of a 
 formula proper to the case, in which there were only two 
 
 123 In the Economic Journal, September, 1901, p. 410. 
 "* In the Economic Journal, March, 1896, pp. 136-7. 
 
DEVIATIONS AND VARIATIONS 109 
 
 prices ! " ^^ This is simply a misstatement of fact, all the 
 more astonishing as that work was the first to point out the 
 danger of relying on only two items in deahng with the 
 geometric average.i^* But that work did make much use 
 of two prices of the sort mentioned, in negative tests mostly, 
 or else in comparative tests. This is the point of offence to 
 Professor Edgeworth. In the Economic Journal of March, 
 1896, Professor N. G. Pierson made an attack on index- 
 numbers, in which he used extravagant price-variations of 
 two groups of commodities described as " equally impor- 
 tant," their full values being posited as equal sometimes at 
 the first period, sometimes at the second, and sometimes over 
 both together. Then he applied to these price-variations 
 the same system of index-numbers — the arithmetic average 
 with even weighting. Naturally he got very discordant 
 results. Thereupon, without making any attempt to solve 
 the difficulty, except by suggesting another worse system, 
 the error of which he discovered, he hastily concluded that 
 " all attempts to calculate and represent average movements 
 of prices, either by index-numbers or otherwise, ought to be 
 abandoned " (p. 131). In the same number of that journal 
 Professor Edgeworth took up the cudgels in A Defence of 
 Index-Numbers. How did he conduct the defence ? Did he 
 point out that Professor Pierson, while using always the 
 simple arithmetic average, had really obtained the "equal 
 importance " of his groups in three different ways, only one 
 of which could be proper for that average, though the others 
 might be proper for the other averages ? And did he show 
 that each of the three simple averages applied to the suppo- 
 sition fitted for it would yield the correct result, and that the 
 same would be given by the other averages (or by the geome- 
 tric at least approximately) when applied to the same 
 suppositions, each with its proper weighting ? No, he knew 
 nothing of all this. He perceived, indeed, and pointed out, 
 that different weightings were involved. He put his finger 
 on the source- of the trouble. But he did not know the 
 remedy. So he brushed aside the various suppositions, 
 calling them " odd " "in view of the sporadic dispersion [of 
 price-variations] which very generally prevails in this 
 world " (p. 133). In their place he advised us to deal only 
 
 ^^^ This and the preceding are only repetitions of criticisms 
 made in the review already referred to, several pages of which 
 were devoted to the discussion of a problem in probabilities 
 entirely irrevelant to the main subject. 
 
 "» Cf. above, note 68. 
 
no THE PROBLEM OF ESTIMATION 
 
 with " concrete, sporadically dispersed, price-ratios," mean- 
 ing many and not very divergent price-variations, as these 
 would show only " insignificant " discrepancies ; and to 
 make our experiments, not on " artificially simplified 
 examples," but " in the spirit of Probabilities," " not 
 consciously selecting cases which will not work well " (p. 135). 
 Such conscious selection of extravagant examples, simplified 
 by confinement to two articles, but with their discrepancies 
 magnified for better discrimination, we have seen purposely 
 made by Galileo in his controversy with Nozzolini ; and we 
 have seen Nozzolini throw them out as improbable absurdities 
 unfit for consideration by rational dialecticians, but really 
 because they were cases that would not work well according 
 to his view of the subject. It may be submitted that 
 Professor Edgeworth follows Nozzolini rather than Galileo 
 in his method of treating the subject. He woiold purposely 
 confine attention to examples which minimise discrepancies 
 and so hide the need of getting rid of them. If this is " the 
 spirit of Probabilities," it is not the spirit of science. 
 
 But Professor Edgeworth has two special reasons for 
 objecting to the use of only two prices. The one is that they 
 do not admit the use of the calculus of probabilities ; the 
 other, that they do not admit the use of another pet of his, 
 the median. Of the latter, later. As for the former, the 
 fact that an example of two price-variations excludes the 
 calculus of probabilities, merely proves that the calculus of 
 probabilities should not be invoked — until it is needed. But 
 Professor Edgeworth has made up his mind that " the 
 problem now before us, in its data, methods and result, is 
 germane to the Calculus of Probabilities." ^*' So convinced 
 is he of this that, as he tells us in another paper, if a case 
 should turn up in experience of two large groups of commo- 
 dities simultaneously rising and falling in price, he would 
 deny to the idea of a variation in the exchange-value of 
 money any determinate meaning. His words are : " That 
 in such a case the exchange- value of money has varied by so 
 much, would appear to me a somewhat indefinite proposition 
 — ^its subject deficient in logical clearness, and its predicate 
 in numerical precision." Think of it ! Because his demand 
 for " Probabilities " has no application to this case, he denies 
 to it significance ! He denies, too, that the problem in this 
 case admits of solution, for he adds : " On such a supposition 
 
 1^' Ibid., p. 132. He asserts that this " is generally implied," 
 but does not say by whom. 
 
DEVIATIONS AND VARIATIONS in 
 
 the objections which have been urged by a distinguished 
 economist against index-numbers, that the results are widely 
 different according as different species of averages are 
 employed, would seem to me a fatal objection." ^^ That is, 
 Professor Pierson's unweighted and trivial objections are to 
 be allowed, if such things happen ! Most certainly Professor 
 Edgeworth has not answered Professor Pierson — ^has not 
 known how to answer him, and does not wish to know. The 
 balancing of opposite price- variations — how the rise of one 
 price may compensate, or come short of or exceed in com- 
 pensating, the faU of another — ^remains beyond his ken. 
 Yet what about the doubly weighted method indicated by 
 formula (lo), which Professor Edgeworth has joined others 
 in recommending ? That method is perfectly applicable to 
 this case, provided data about the quantities be also supplied, 
 and it would give a result insignificantly discrepant from the 
 results given by the theoretically best methods. Would 
 Professor Edgeworth throw it overboard also, merely because 
 some other wrongly weighted averages might give widely 
 different results ? Again, Professor Edgeworth is one of 
 those who hold that formula (i) is correct on the supposition 
 that the quantities have not varied.^^^ Would he say that 
 this is no longer true, or that it has no meaning, if only two 
 prices vary ? And how about Dr. Bowley's example, above 
 cited ? That has three commodities, to be sure ; but when 
 does the number used begin to have meaning ? In that 
 example the price-variations are, at least, not sporadic, so 
 that Professor Edgeworth would still have to discard it. 
 Prof essor _ Edgeworth would have done better to confine 
 himself, as before, to denying the probabiUty of such things 
 happening. 1*" But he has admitted that it can happen by 
 taking it into consideration ; and now it seems strange if he 
 cannot understand a problem which so many other econo- 
 mists — Jevons, Laspeyres, Messedaglia, etc. — have worked 
 at, though unsuccessfully. Just because another has 
 succeeded where they have failed and he himself has not 
 tried — ^is that a reason for him to continue looking at the 
 subject through a mist ? The assertion may be added : if 
 the case he supposed should happen, and if in addition it 
 
 ^"^ In the Economic Journal, September, 1901, p. 408. Page4io 
 he says the widely different results may amount " as likely as not 
 to 25 per cent." 
 
 ^'' Memorandum to the first Report, pp. 264, 272, 293. 
 
 1'° This, in fact, he does again do, a little further on in the same 
 paper, Economic Journal, September, 1901, p. 410. 
 
112 THE PROBLEM OF ESTIMATION 
 
 happened that the two groups should change in price in 
 inverse ratio, and if they were of equal importance over both 
 the periods together, then, no matter how many other 
 commodities remained unchanged in price, the exchange- 
 value of money has remained constant. And this proposi- 
 tion has been proved and demonstrated, and is as certain (to 
 repeat), and (we may add) as plain, as anj^hing in EucUd. 
 If Professor Edgeworth can disprove that proposition, let 
 him do so. If he cannot, what value has his statement that 
 he cannot see the meaning in it ? "^ 
 
 But here a concession may be made. Even if an abso- 
 lutely correct method of averaging price-variations were 
 discovered, or one as correct as the use of the arithmetic 
 average in reducing observations (and have we not reached 
 one as good as that ?), yet the application of it to practice 
 would give occasion for error, just as even the application of 
 an absolutely exact yard stick to measuring the length of a 
 line would give rise to error. Now, The Measurement of 
 General Exchange-Value might have been a little more 
 explicit by making some allusion to these errors of appli- 
 
 1" Professor Edgeworth seems to desire to retain what Mill 
 called " the necessary indefiniteness of the idea of general 
 exchange value," and says " we will never reach an exactly true 
 method [of measuring it] . . . until we are able to handle and 
 weigh final utility, or ... ' esteem value,' as we do material 
 commodities " {ibid., p. 409). On the contrary, we shall never 
 reach an exactly true method even of understanding exchange- 
 value until we cease to mix with it the idea of esteem-value. 
 Any one who accepts Mill's denial of the possibility of measuring 
 general exchange-value (a denial due to ignoring the balancing 
 of opposite variations) ought, like Mill himself, to refrain from 
 working over price index-numbers. Professor Edgeworth further 
 agrees with Professor Marshall that, in the world as it is, " an 
 absolutely perfect standard [of exchange value] is ' unthink- 
 able ' " (p. 410). There is, of course, some inconsistency in saying 
 that something one is thinking about is " unthinkable " ; but 
 the meaning seems to be that it is unthinkable, that is, it cannot 
 be rightly thought that such a thing exists or can exist — that is, 
 again, it must be thought or believed that the thing does not or 
 cannot exist. But such a statement needs proof. Well, 
 M. G. E.-V. has gone perhaps the furthest of all in showing that 
 an absolutely perfect standard of exchange-value, in the complex 
 economic world, does not exist, just as (see above, p. 103) there 
 is no absolutely perfect value of tt, or just as an absolutely perfect 
 value of TT is " unthinkable," since we simply cannot think what 
 it is. See in that work p. 408. 
 
DEVIATIONS AND VARIATIONS 113 
 
 cation according to the theory of probabiHties. It might 
 have expatiated more on the inaccuracy of observations 
 or measurements, and on the probabihty of the increase or 
 decrease of errors. The theory of probabiHties shows that 
 if in measuring a given magnitude four times as many 
 observations be used as at first, the accuracy of the result, 
 as indicated by its " probable error," is doubled, that is, it 
 increases with the square root of the number of observa- 
 tions. Again, although a surveyor in measuring 100 miles, 
 involving 100 times more partial measurements, pro- 
 bably commits 100 times more errors and altogether 100 
 times greater amount (or arithmetic sum) of errors, than 
 in measuring a mile, yet, because his errors probably fall 
 pretty equaUy on opposite sides of the truth, some of his 
 partial measurements being too great and some too small, 
 and because in the greater length greater opportunity is 
 offered for the compensation of opposite errors, the pro- 
 bability is found to be that in measuring the longer line 
 the balance of his errors, his final aggregate error (the 
 algebraic sum of all the errors) is only ten times greater, or 
 relatively to the lengths measured it is ten times less, or 
 again, in other words, his accuracy is ten times greater, than 
 in measuring the shorter line ; that is, in general, the 
 absolute error increases, the relative error decreases, and the 
 accuracy increases, with the square root of the increase in the 
 magnitude measured.^*'* Now, these principles may be 
 taken over bodily into the mensuration of price-variations, 
 with the proper mutanda mutata of course, the numbers of 
 commodities being substituted for the numbers of observa- 
 tions, and the relative numbers of periods for the relative 
 lengths of the lines. Here it will appear that the probability 
 is (i) that, even if we employ a perfectly correct method, the 
 final errors which we shall inevitably commit in practice, 
 being by the nature of the case relative, wiU decrease, and 
 our accuracy increase, with the square root of the increase 
 in the numbers of the commodities operated on ; and (2) that 
 as the measurements advance over a course of years, each 
 being compared with the preceding in a new measurement, 
 and the whole being strung out in one line, the errors to 
 which even the perfectly correct method is exposed in 
 practice will increase from the starting period (unless 
 
 ^*2 Ct. Professor Edgeworth, in the Journal of the Royal 
 Statistical Society, June and September, 1888, pp. 365, 3,67, 
 602, 627. 
 
114 THE PROBLEM OF ESTIMATION 
 
 adjusted by direct comparison with it) with the square root 
 of the number of years traversed. The first of these propo- 
 sitions may be extended to all methods ; the second to some, 
 but not to all. The second can be extended only to the 
 methods of which the errors are shown by trial to fall pretty 
 equally on both sides, of the truth. It cannot be applied 
 to those of which the errors faU always or mostly on one side ; 
 for when such methods are used in the " chain " system just 
 referred to, their errors wiU grow in almost geometrical 
 progression with the arithmetical advance in periods.^*^ 
 That work omitted to elaborate these precise propositions. 
 They are here supplied. Perhaps Professor Edgeworth will 
 be satisfied with this filling up of the gap left by that omis- 
 sion. It does not appear, however, that very much is 
 gained. Common sense, especially as it may rely on the 
 study of probabilities made by others, would give the correct 
 injunction to employ as many commodities in our averaging 
 as is possible ; which injunction was given in that work 
 (p. 77) on the authority of Professor Edgeworth himself. 
 Common sense also teaches that compensating errors do 
 not increase as do errors that do not compensate. That 
 work made considerable use of common-sense probability 
 on the occasions when it was needed,^** although it did not 
 employ the calculus of probabilities. 
 
 And continuing to use such common-sense probabiUty, we 
 may argue in this wise. The arithmetic average is no better 
 than the harmonic, and the weighting of the first period is 
 no better than the weighting of the second ; therefore let us 
 use all these things together, and the result wih have the 
 greatest probability in its favour. Now, precisely all this 
 is done, when the quantities are the same in both periods, 
 by the single formula (i), which we have found to be true 
 on that supposition. And it might further be argued : the 
 geometric average lies between the arithmetic and the 
 harmonic, therefore it with the weighting of both the periods 
 (likewise geometrically averaged) wiU most probably give 
 the true result, and therefore agree with the preceding. And 
 this, too, we have found to be approximately true on the 
 same condition, and exactly true if the condition is still 
 
 1*' More specifically, methods are found that augment or 
 diminish both rises and falls. If in a cycle prices first rise and 
 then fall back to their former positions, such methods may be 
 right at the end, but in the middle they will exaggerate or imder- 
 rate the rise. 
 
 "* Pp. 223 n., 321-4, 332, 341, 365, 394, 404, 406, 422, 423. 
 
DEVIATIONS AND VARIATIONS 115 
 
 further narrowed to two commodities equally important 
 over both the periods. Furthermore, if we drop the restric- 
 tion to quantities the same at both periods, we may still use 
 combinations involving all the averages and weightings 
 mentioned, and these mU then be the most probable. Such 
 combining is done in formulae (6), (7), (8) and (9), and with 
 most perfect mixture in the last, which seems to stand the 
 circular test best, and if it does, confirms the probability. 
 And formalae (10) and (11) are only slight departures from 
 these, a more convenient average being substituted for a less 
 convenient in parts of them ; which substitution, when the 
 formulae are applied, as they usually are, to a fairly large 
 number of commodities varying not very greatly in prices 
 and quantities, wiU produce little discrepancy in the results, 
 so that they too have probabiHty in their favour. One of 
 these is the very formula recommended by Professor Edge- 
 worth in the second section of his Memorandum to the first 
 Report of the British Association Committee, as also by that 
 Committee itself. Yet Professor Edgeworth did not make 
 this argument from probability for that method — in fact 
 neither he nor that Committee gave any argument in support 
 of that method.^*^ If he himself did not use probability for 
 deriving and defending the method he recommended, why 
 should he require others to do so ? Or if he had used it, would 
 he have improved upon the arguments which have been 
 given for it in the preceding section, and which were first 
 given in The Measurement of General Exchange-Value, 
 although Professor Edgeworth failed to see them there.^*® 
 
 But Professor Edgeworth has made use of the theory of 
 probabilities, not in connection with that practical method, 
 but in connection with the geometric average, and through 
 it with the median. For Professor Edgeworth not only 
 follows Laspeyres in advocating an arithmetic average, but, 
 quite independently, he follows Jevons in advocating the 
 geometric average — he follows both Galileo and Nozzolini. 
 He advocates the geometric average for a special purpose 
 
 1" The only thing like an argument appears in his Memorandum 
 to the Third Report, p. 150, where he speaks of the " twin 
 methods " using the quantities of the two periods separately, 
 and adds, " there is no reason to think this method [one of the 
 twins] would be less accurate than its converse " — the other 
 twin ; . which at least suggests combining them. 
 
 ^*' Had Professor Edgeworth noticed what is written in favour 
 of his own method in that work (p. 423), he would have perceived 
 that that work strengthened his position. 
 
ii6 THE PROBLEM OF ESTIMATION 
 
 different from the special purpose for which he advocated 
 fornmla (lo). The speciahty of purposes — or of qucBsita, as 
 he calls them — is a peculiar feature in Professor Edgeworth's 
 treatment of the subject before us. In one of his papers, 
 making a reference to Dr. Venn's Logic of Chance, Professor 
 Edgeworth, perhaps influenced also by a remark of Jevons, 
 writes : " The answer to the question what is the Mean of 
 a given set of magnitudes cannot in general be found, unless 
 there is given also the object for the sake of which a mean 
 value is required." "' In another he again writes : " It is 
 with the index-numbers [of prices] as with conduct ; in order 
 to form a just judgment, we must always look to the under- 
 lying idea and purpose. ' ' ^*^ Apparently Professor Edgeworth 
 views these two statements as equivalent. They are very 
 different. The former is true. The latter is not true. The 
 truth of the former has already been explained. The error 
 of the latter consists in its putting a variety of purposes 
 within a purpose. In averaging price- variations the purpose 
 or object is given : it is to measure variations in the exchange- 
 value or purchasing power of money. This is precisely the 
 purpose which Professor Edgeworth declares to be the 
 object of his Memorandum to the first Report of the British 
 Association Committee ; which Committee itself had been 
 appointed " for the purpose of investigating the best methods 
 of ascertaining and measuring variations in the value of the 
 
 1*' New Methods, in the Journal of the Royal Statistical Society, 
 June, 1888, p. 347. Jevons had previously in the Journal of the 
 Royal Statistical Society, June, 1865, written : " It is probable 
 that each of these [the three averages] is right for its own pur- 
 poses, when these are more clearly understood in theory," 
 Investigations, p. 121. As it stands, Jevons's statement is per- 
 fectly true — and we have seen different purposes, or uses, for 
 which each of these averages is proper. Jevons, however, seems 
 to have had the present subject only in mind. Even so, if we 
 substitute " on certain conditions appropriate to each " for " for 
 its own purposes," his statement is true : see M. G. E.-V., p. 355, 
 and cf. above, note 119. 
 
 ^*^ Recent Writings on Index-Numlers, Economic Journal, 
 March, 1894, p. 163. It was perhaps Professor Edgeworth who 
 inserted a similar statement near the opening of the first Report 
 of the British Association Committee, making them say that the 
 theoretical part of their work was " to distinguish analytically 
 the different purposes which may be subserved by constructing 
 a measure of a change in the value of money, and then to show 
 what formula, what particular mode of combining the statistical 
 data, is appropriate to each case." 
 
DEVIATIONS AND VARIATIONS 117 
 
 monetary standard." ^*' " There are many methods — not 
 one method — of measuring and ascertaining variations in the 
 value of money," says Professor Edge worth in the same 
 Memorandum (p. 259). There may be many methods, but 
 only one true or best method — the one which comes nearest 
 to giving the true variation of the exchange- value of money. 
 To get at this true variation in the exchange-value of money 
 is the sole object or purpose of averaging the price- variations, 
 just as to get at the true length of a line by averaging the 
 observations of it is the sole object or purpose of averaging 
 those observations. There is no other object or purpose — 
 that is, no other main or final purpose.^^" The single object 
 or purpose has been stated. It is specific.^^^ No further 
 varietal subdivision of it is needed, nor is it possible, so far 
 as the method of averaging the price- variations is concerned, 
 including the method of weighting them. 
 
 The only differences that can come into our subject are 
 differences as to the money itself of which the variations in 
 exchange- value are being measured. ^^'^ Is it the money of 
 England or the money of France ? Is it the money of a whole 
 country or the money which passes through the pockets of 
 one class of its population ? Obviously different data must be 
 collected in each of these cases — different quantities always, 
 
 "' So runs the official title of that Committee. Professor 
 Edgeworth opens his Memorandum thus : " The object of this 
 paper is to define the meaning, and measure the magnitude, of 
 variations in the value of money. " 
 
 "" There may be some subsidiary purpose sometimes in averag- 
 ing price-variations, as to study their dispersion. But this is 
 only in aid of the metrical purpose, and cannot be put on a level 
 with it or be substituted for it. It is conducted on the same 
 price-variations and by the use of the same method of averaging. 
 There may be such a subsidiary purpose in averaging observa- 
 tions of length, etc. ; but nobody has supposed that this calls 
 for a different method of nieasuring length. 
 
 151 Dr. Venn has thus repeated his doctrine : an average, 
 being a fictitious value, cannot take the place of actual values 
 " for purposes in general, but only for this or that specific pur- 
 pose," On the Nature and Uses of Averages, in the Journal of the 
 Royal Statistical Society, September, 1891, p. 430, again p. 432. 
 
 1°^ There are other kinds of value, such as the esteem-value 
 and the cost-value, of money, that may be measured ; and 
 naturally the methods of measuring them, whatever they be, 
 are different from the method of measuring the exchange- value 
 of money. But these are not in question. Who cannot dis- 
 tinguish between these, ought not to dabble with economics. 
 
ii8 THE PROBLEM OF ESTIMATION 
 
 different prices sometimes. But the method of averaging the 
 data is the same in all of the cases. We do not require 
 different methods for averaging prices in England and in 
 France ; and the method of averaging the prices of family 
 budgets should be the same as the method of averaging the 
 prices of the national budget. Only the collection of the data 
 is different in these different cases, different being the quan- 
 tities and sometimes the prices that go into the budgets of 
 employer families from those which go into the budgets of 
 employee families, different those which make up the national 
 budget of England from those which make up the national 
 budget of France. Of course for political economists the 
 exchange- value of the money of their country — of the whole 
 country, not of any one class in it — what Professor Mitchell 
 calls " the general-purpose index-number " ^^' — ^is the prin- 
 cipal object of interest. " Let debtors and creditors," well 
 says Professor Edgeworth, " regulate their private affairs by 
 a special index-number, if they like. That is not the affair 
 of statesmen and financiers. But the currency is within the 
 province of government." ^^ In the collection of the 
 commodities and of their quantities and prices many varieties 
 of detail have to be considered (see above, pp. 88-9) — and 
 these may be spoken of as different methods, if you care to 
 do so ; but they are different methods of collecting the data. 
 Totally distinct is the method of averaging the price-varia- 
 tions of these data once collected. This is the method under 
 consideration. Of this method there is no subdivision. It 
 is simply the method proper for averaging what is presented 
 to it, for the one common purpose of measuring variation in 
 exchange- value of the money brought into relation with the 
 
 1" Index-Numbers and Wholesale Prices, p. 26. Professor 
 Mitchell complains that we " cannot conceivably devise a single 
 series that will serve all uses equally well " — that is, all par- 
 ticular or private uses. There is nothing peculiar about this in 
 political economy. Every tax, every tariff, affects individuals 
 and classes differently. The political purpose is to look after the 
 general interest. 
 
 1" Memorandum to the Third Report, p. 135. 
 
 154a jjj fact the measuring proper is done in collecting the prices 
 and quantities at the different periods ; which provides the 
 particular price-variations and their weights. These are the 
 analogues of the measurements made by surveyors and the 
 observations made by astronomers. The inversions of the price- 
 variations provide the variations of the particular exchange- values 
 
DEVIATIONS AND VARIATIONS 119 
 
 In no other subject of mensuration is there any differentia- 
 tion of methods to be used according to different ulterior 
 purposes held in view. How, indeed, in any subject could 
 there be two different equally correct methods of measuring 
 any magnitude ? If the different methods give different 
 results, how could they both be correct ? It does not help 
 matters to say that the one is correct for one purpose and the 
 other for another. Should a civil engineer employ two 
 different methods of measuring a line (or rather of reducing 
 his measurements to a final result), and on obtaining different 
 results should he say the one was correct for laying out a 
 railway and the other was correct for lajdng out a highway, 
 he would probably get few customers. Of course different 
 degrees of precision may be required for different purposes ; 
 but this acknowledges that only the most precise method is 
 the truest. When the magnitude to be measured is the 
 variation in the exchange-value of money (of the same 
 money, naturally, at the same time and place, and for the 
 same people), there is no more room for two different but 
 equally correct results. In other kinds of mensuration there 
 may be variety of different instruments fashioned to fit 
 different objects that are to be measured : these different 
 objects correspond to the different collections of data in our 
 subject. There may even be sometimes a secondary more 
 convenient method of roughly making some measurements, 
 always tested by the one true method, as when we measure 
 elevations by the barometer : such a method corresponds 
 to the practical methods in our subject, substituted for the 
 one best method, to which they should approximately 
 conform. Otherwise the purposes we have in view when we 
 measure the length of something (to repeat this instance) 
 have nothing to do with the method of measuring it : what 
 we are going to do subsequently with the measurements 
 made does not enter into the making of them. 
 
 And so it is with the measurement of variations in the 
 exchange-value of money. To measure variation in the 
 exchange- value of money is the object or purpose of averaging 
 
 of money. The averaging of these data yields the final measure- 
 ment, but is not a measurement itself, although it is of course 
 an essential element in the final measurement. In axiometry, 
 where the averaging yields the inverse variation of the general 
 exchange- value of money, the theoretical problem is concerned 
 with the averaging. And there can be two or more ways of 
 averaging (as regards the best method) for different purposes or 
 qucBsUa, no more in axiometry than in astronomy or in surveying. 
 
120 THE PROBLEM OF ESTIMATION 
 
 price- variations. If there is any other object or purpose 
 in averaging them, then that is something else, and does not 
 belong to our subject. To seek within the operation of 
 measuring variation in the exchange-value of money by 
 averaging price-variations for an undefined number of 
 different methods according to arbitrarily chosen vilterior 
 objects or purposes called qucesita, is an illusory refinement. '^^ 
 In fact, a couple of Professor Edgeworth's eight qucesita 
 involve merely varieties of ways of forming the collection of 
 data,^^^ and four others do not belong in our problem at all.^" 
 
 "^ In a notice of M. G. E.-V. in the Yale Review, May, 1902, 
 following Professor Edgeworth's lead, Professor Fisher criticised 
 that work for seeking only one method, instead of taking different 
 puiposes into view ; and illustrated the matter by a very curious 
 description of two methods of measuring the height of a car, when 
 really different heights of its whole and of a part of it were 
 measured. But in his own Purchasing Power of Money, pub- 
 lished nine years later, in the sections devoted to the mensuration 
 of that power, he, too, sought only one method of averaging (with 
 another as substitute), and said that the elements which enter 
 into the construction of index-numbers must be chosen according 
 to the puipose for which the index-number is desired (pp. 204-5, 
 232 ; and p. 392) ; giving an instance of such a purpose, he said 
 " the same formula " may be employed (as in the other case), 
 but the terms of this formula " have different meanings," that 
 is, apply to different prices and quantities. Professor Fisher's 
 last position is the correct one. He also criticised that work for 
 not using the theory of probability ; and in his own work he did 
 not use it. He further criticised that work for devoting all its 
 efforts to measuring variation in the exchange- value of money, 
 instead of extending them to measure variation also in the 
 esteem-value of money ; and he confined the treatment of 
 mensuration in his own work to the measru-ement of variation in 
 the purchasing power ( = exchange- value) of money. 
 
 1" The second differs from the first (" the consumption stan- 
 dard " of a whole country) by employing private budgets, and 
 the eighth differs from it by employing all the quantities sold 
 and resold in all the mercantile and financial transactions of a 
 country. 
 
 ^" The third, fourth, and fifth are concerned with the ques- 
 tions whether the standard should itself vary with the national 
 affluence, income, or capital ; and the sixth with the definition 
 of the appreciation (then going on, now depreciation) which 
 monetary reformers (then the bimetallists) wished to rectify by 
 infusing more money into (now abstracting some from) the 
 circulation. All these are questions which the author of 
 
DEVIATIONS AND VARIATIONS 121 
 
 Only the first and the seventh require two distinct methods 
 of averaging price-variations. These two are one too 
 many.^^^ 
 
 In his first qucBsitum Professor Edgeworth recommends 
 formula (10). In the seventh he recommends the geometric 
 average. AU the first six qucssita are expressly made without 
 any hypothesis as to the causes of the changes in prices. 
 The seventh drops this denial, and introduces a clause about 
 the " changes [of prices ?] affecting the supply of money," 
 while the eighth contains the seemingly opposite " hypothesis 
 that a common cause has produced a general variation of 
 prices." ^^* Incidentally, we might have supposed that 
 particular causes produce variations in the prices (the true 
 prices, relatively to variations in the exchange-value of 
 money ^®'') of particular commodities, while a general cause 
 with reference to commodities, but a particular cause with 
 reference to money, is what produces the general variation 
 in prices and the particular inverse variation in the general 
 
 M. G. E.-V. has discussed in a separate work, The Fundamental 
 Problem in Monetary Science. 
 
 168 Professor Edgeworth even has a third ; for in the Economic 
 Journal, June, 1918, p. 197, he agrees with Professor Mitchell in 
 recommending " the aggregate expenditure method " as " the 
 best form" for the latter' s "general purpose series." (Above, 
 in note 129, that method has also been recommended for use in 
 past times, as the best practical method feasible for such occa- 
 sions. It is there not recommended for any different purpose, 
 the single purpose always recognised being that of making the 
 best measurement we can of variations in the exchange- value of 
 money.) Exactly how Professor Edgeworth intended to relate 
 this new " best " method with formula (10) already recommended 
 as the best, is not stated. As it is only briefly mentioned in 
 passing, we may ignore it. 
 
 ^" Apparently suggested by Jevons, who, for continuing to 
 use the simple geometric average, gave three equally poor addi^ 
 tional reasons, the third of which is : " It [the simple geometric 
 average] seems likely to give in the most accurate manner such 
 general change in prices as is due to a change on the part of gold. 
 For any change in gold will affect all prices in an equal ratio," 
 Investigations, p. 121. If by " change in gold " he meant change 
 in the exchange- value of money, the last statement is true 
 (cf. ihid., p. 128). But if he meant change in the quantity of 
 money (which is the way Professor Edgeworth seems to take it) 
 the statement is wrong, as it ignores what Del Mar felicitously 
 named " the precession of prices." 
 
 "» Cf. M. G. E.-V.. pp. 466, 475. 
 
122 THE PROBLEM OF ESTIMATION 
 
 exchange- value of money — the bias above referred to (p. 72) 
 — whenever such variations occur. However this be, causes 
 that have produced a magnitude which we wish to measure, 
 do not properly enter into the measurement of the magnitude. 
 Professor Edgeworth recognises this by sa5nng near the 
 beginning of his first Memorandum that " the business of this 
 Committee is to measure a fact, not to speculate about its 
 causes or consequences" (p. 258). Unfortunately he 
 immediately abandons this restraint by restricting it ; for he 
 subjoins "it is only in the simpler kinds of measurement 
 that the metretic art can be entirely divorced from theory 
 about its subject-matter." And so now in the seventh 
 queesitum, though he had not done so in the first, he brings 
 into consideration the quantity theory that, other things 
 being equal, prices vary inversely with the quantity of money 
 (p. 280). This theory would seem to most of us to call 
 especially for the consideration of the quantities of the 
 commodities the prices of which are varying. Not so to 
 Professor Edgeworth. To him it seems that the price- 
 variations of particular commodities may be affected by the 
 " other things," but the general price- variation of aU commo- 
 dities is a "residual phenomenon,"^*"* affected by the 
 quantity of money. The general price- variation is obtained 
 by an average of the particular price- variations, and accord- 
 ing to Professor Edgeworth it is to be obtained from them 
 alone, simpliciter and per se. Thus he says : "it would be a 
 significant operation " — significant, apparently, of the quan- 
 tity of money (or of what ?) — " to take the average of all the 
 price-variations, irrespective of the quantities of the corre- 
 sponding commodities " (p. 281}. This gets rid at least of the 
 usual kind of weighting, and suggests either even weighting 
 or some other kind. Then, " the problem before us may be 
 thus defined. Given a number of observations consisting 
 each of the ratio between the new price and the old price of 
 an article, to find the mean of the observations." Finally, 
 " the problem as thus conceived belongs to that higher 
 branch of the calculus of probabilities which may be called 
 the doctrine of error " (p. 282). Here are several questions — 
 about the average, the weighting, the distinction from the 
 first queesitum and its formula, and the invocation of the 
 higher probabihties. It is difficult to disentangle these 
 various things. So let us get down at once to the last, and 
 investigate its meaning and validity. 
 
 160a fjjg Journal of the Royal Statistical Society, June, 1888, 
 P- 353 ; Economic Journal, March, 1896, pp. 132-3. 
 
DEVIATIONS AND VARIATIONS 123 
 
 Wovild it not have been better to reverse the last statement 
 quoted, and say that the calculus of probabilities belongs to 
 the economic problem — if we choose to apply it thereto ? 
 If we choose, we can apply it to anthropological measure- 
 ments, as Quetelet did. But Quetelet had his single method 
 of making his measurements before he subjected — not his 
 method, which he never did, but — ^his measurements to the 
 calculus of probabilities. And he did this only because he 
 found an analogy, before unsuspected, between his measure- 
 ments and the chances of black and white baUs coming out 
 of an urn in certain combinations. Now, Professor Edge- 
 worth, having made a collection of " price-observations " 
 (which phrase means what is generally meant, not by 
 " prices," but by " price- variations," ^*^), arranged them in 
 order, as Quetelet had done with his chest measurements 
 of Scottish soldiers, and found that — ^they did not follow " the 
 typical law of error " (of observations), but fell into an 
 unsymmetrical, though regular, curve, as he had suspected 
 they would because of their similarity with estimates.^*^ 
 He therefore concluded that they should be averaged geome- 
 trically, like estimates ; and as estimates are averaged 
 simply, or with even weighting, and because of some mis- 
 applied analogy with gravity and the motion of the solar 
 system through the sidereal system, he inferred that price- 
 variations ought also to be averaged simply, " after the 
 manner of Jevons " (pp. 281-2, 288). 
 
 AU this is very pretty. The analogy with estimates sets 
 up a rival method, by the evenly weighted geometric average, 
 over against the unevenly weighted formula (10), which he 
 has already recommended. We might expect, then, that 
 Professor Edgeworth would examine and compare them, to 
 see whether the geometric average, with proper weighting, 
 would agree with the other method, and if not, how much 
 they would differ ; and then weigh them in the balance and 
 determine which is the better. No, he adopts them both ! 
 Only the one is for one qucssitum, and the other is for another ! 
 And meanwhile what is the calculus of probabilities doing ? 
 As regards the method of measuring variations in the 
 exchange- value of money by averaging price- variations, it is 
 
 '" The phrase occurs again in the second Memorandum, p. 206. 
 The price- variations, however, have probably been put in the form 
 of price-deviations from a common starting point, which is 
 omitted, leaving only the deviated prices to be operated on. 
 
 i«2 Here, in the first Memorandum, pp. 282-3, occur the 
 passages quoted above, pp. 79-80. 
 
124 THE PROBLEM OF ESTIMATION 
 
 doing nothing. It had nothing to do with Quetelet's method 
 of averaging chest measurements. It has nothing to do with 
 Professor Edgeworth's method of averaging price- variations. 
 Professor Edgeworth tries his best to bring this calculus 
 into the inquiry about the axiometric problem, but without 
 success except to confuse the subject. All that this calculus 
 can properly do is to come in after the method has been 
 adopted and the measurements made. Then it may be 
 applied to the dispersion of the price- variations around their 
 average. And if Professor Edgeworth would confine it to 
 this use, there could be no objection to his doing so. He is 
 at liberty to experiment all he pleases with the dispersion 
 of price-variations, to investigate their modes and medians 
 and quartiles, etc. Perhaps some interesting developments 
 may come from this application of the calculus, — but we 
 still wait for them. In the hands of Quetelet some interest- 
 ing results did materialise. In the hands of Professor 
 Edgeworth none have appeared.^®* 
 
 And the reason is clear. Professor Edgeworth has mis- 
 taken the analogy. Because Quetelet got what he considered 
 an objective fact from his measurements — ^his observations, 
 not on the stars, but on men, — ^which he called a type, and 
 conceived of as having objective existence (which others have 
 since controverted). Professor Edgeworth thinks that in 
 averaging " price-observations " in the simple geometric 
 way he is getting something more than he got by the complex 
 formula (lo). Formula (lo) yields only an " average," not 
 a " mean " (in the Quetelet-Herschel-Jevons phraseology), — 
 only an ideal thing, a mere general average variation of 
 prices (although its reciprocal is a variation in the exchange- 
 value of money, which might seem objective enough) ; but 
 
 1*' He makes a suggestion, however, and leaves it at that. 
 Quetelet, we have seen, noticed that if a population were made up 
 of two races, the dispersion of their height measurements (and of 
 others too) might show two modes (which would make the curve 
 representing such dispersion look like the silhouette of a Bactrian 
 camel's back). And we have seen that Bertillon got some 
 positive result from this study of the dispersion of stature in the 
 Department of Doubs. This question has also exercised Professor 
 Edgeworth. Perhaps price-variations are not of one type, but 
 of two types — perhaps their curve is not " monocephalous," but 
 " bicephalous " (not one-humped, but two-humped). " This is 
 a question of fact," which he leaves for others to decide. New 
 Methods, in the Journal of the Royal Statistical Society, June, 1888, 
 p. 368. 
 
DEVIATIONS AND VARIATIONS 125 
 
 the simple geometric formula, because it conforms to the 
 nature of price-variations as dispersing themselves geome- 
 trically, reveals, he thinks, a real thing, a cause. This latter 
 operation he condescendingly calls " secondary " to the 
 other as " the primary." But he conceives it as a close 
 second. Its " position of high collateral dignity," he says, 
 " is all the more deserved in that the secondary measure 
 enjoys an objective or external character, which cannot, 
 according to my view of the subject, be accorded to the 
 primary qucesitum." ^** That it is viewed as yielding some- 
 thing objective is expressed also in his speaking of it as 
 serving the " purpose of ascertaining a type." ^*^ More 
 fully he once refers to " that variety of index-number which 
 purports to determine a real quantity, a cause or charac- 
 teristic, such as ' scarcity of gold,' in some more objective 
 sense than a mere fall of prices on an average. The qucesitum 
 in this case may be likened to a physical quantity which is to 
 be ascertained from a set of measurements. The method 
 accordingly presents certain peculiarities derived from the 
 theory of errors-of-observation." ^®* In other words. Pro- 
 fessor Edgeworth thinks that in averaging (geometrically) 
 not prices, but price- variations, we are performing an opera- 
 tion analogous to the operation of averaging observations. 
 The latter operation is conducted for the purpose of getting 
 the true magnitude — " a real thing." Therefore the former 
 also aims to get a true and real magnitude — ^which we may 
 grant, seeing that it is the true magnitude of the variation 
 of the price-level and inversely of the exchange-value of 
 money which is aimed at (at least when we perform the 
 averaging properly, or fairly well, as by formula (10) ), 
 although this is not his meaning ; for his meaning refers to 
 some other " real thing," discoverable only by his geometric 
 average — a type or a cause, never yet described in clear 
 lineaments, but a something which he once adumbrated as 
 hovering " between genuine being and not being," " a 
 muffled or masked representation of some real entity " ^*' — 
 for such nebulous phrases are his delight. Now, this analogy 
 
 "* In the Economic Journal, September, 1901, p. 415, cf. 
 pp. 409, 413. 
 
 ^°' In the Journal of the Royal Statistical Society, June, 1888, 
 p. 360, again p. 363 ; and pp. 352, 367, the average is said to 
 represent " a real thing," an " objective thing." 
 
 "" In the Economic Journal, March, 1894, p. 163. 
 
 "' In the Journal of the Royal Statistical Society, June, 1888, 
 P- 352. 
 
iz6 THE PROBLEM OF ESTIMATION 
 
 is false. We have already seen the true state of the case 
 (above, p. 71). In averaging price- variations for the 
 piurpose of ascertaining and measuring variation in the 
 exchange-value of money, we are doing something slightly 
 analogous to what astronomers and surveyors do when they 
 average their errors for the purpose of ascertaining and 
 measuring the " probable error " of their observations or 
 measurements. This " probable error " is far from being 
 an objective reality, and certainly it is not a type or cause 
 of anything, though it may be an effect of something (of our 
 carelessness or imperfection). Or, better stUl, it has con- 
 siderable resemblance to the operation of getting the centre 
 of shots deflected from the bull's-eye by a common cause, the 
 wind ; which new centre never was, and never can be, aimed 
 at, and is not a real thing existing anywhere " in heaven or 
 earth." Hence the failure of Professor Edgeworth's search 
 after an objective reality in the sense of an independently 
 existing entity. He has missed Quetelet's trail. And what 
 good, now, the appeal to the theory of probabilities (which 
 he so identifies with the theory of averages as to call it by 
 the latter's name ^^^) can do in our problem of axiometry, 
 still remains to be seen ; and until the good it serves is shown, 
 we may be excused from bothering ourselves with it. 
 
 Possibly, however, an objective thing may be disclosed by 
 the geometric average of price-variations, even though the 
 analogy is false which is offered as a base for it. In fact, it 
 may be repeated that an objective thing — a variation in the 
 general exchange-value of money — ^is very approximately 
 shown by the inverse of the geometric average of price- 
 variations when properly weighted (and perhaps equally well 
 by formula (10) ). But Professor Edgeworth is now search- 
 ing for some other objective thing, to be disclosed by the 
 geometric average not weighted in the way we have seen to 
 be its proper weighting. That other objective thing, con- 
 ceived as existing apart from the price-variations, he has 
 generally described very vaguely, but sometimes he has given 
 it a body and a name, referring to it as " a scarcity of gold," 
 which is only a particular way of describing the proper 
 quantity of money " in relation to the work it has to do " ; '*' 
 
 ^«' In the Economic Journal, September, 1901, p. 412. 
 
 169 ^gj^ Methods, in the Journal of the Royal Statistical Society, 
 June, 1888, p. 355. Obviously scarcity of money may exist 
 independently of the averaging of prices, just as the height of a 
 star above the horizon exists independently of the astronomer's 
 averaging of his observations of it ; whereas general exchange- 
 
DEVIATIONS AND VARIATIONS 127 
 
 for he was writing in a period of monetary appreciation. If 
 it is an3rthing so definite as this, we can only wonder why 
 Professor Edgeworth on other occasions made so much of a 
 mystery of it.^'" Yet if we pin him down to this definite 
 description of the " real thing " to be disclosed by his 
 geometric average, it becomes merely a question of fact 
 whether his geometric average does disclose such a thing. 
 This question must be simply negatived. Here of course is 
 where the quantity theory comes into play ; for if prices 
 vary inversely according to the quantity of money, we have 
 only to measure the variation of prices to determine the 
 inverse variation in the quantity of money. But Professor 
 Edgeworth knows that the quantity theory rests on the 
 condition of " other things being the same." He therefore 
 tells us that a " prior operation " is necessary to " reduce the 
 problem to the simple case of general prices varying, other 
 things being the same," which extraordinary operation he 
 leaves " to the currency doctors." ^'^ As he abandons to 
 others the most difficult part of the job, he has not solved it. 
 Moreover, in a similar connection he truly says that " varia- 
 tion in the effect indicates, but does not measure, variation 
 in the cause." "^ Therefore if scarcity of gold is the cause 
 of falling prices, its ardount is not measured by the amount 
 of their fall. Finally, if the purpose of averaging price- 
 variations geometrically were to measure variation in the 
 relative quantity of money, its purpose would be different 
 from that of measuring variation in the exchange-value of 
 money, and would not belong to axiometry. 
 
 value of money cannot exist independently of the particular 
 exchange-values of money, of which it is composed, and which 
 are inversely indicated by prices. But this does not prevent the 
 general exchange- value of money from being as objective as are 
 its particular exchange-values. See M. G. E.-V., pp. 10, 12-13. 
 Of course the general exchange- value of money that is measured 
 is as general only as are the quantities of the commodities taken 
 into consideration compared with the totality of commodities. 
 So the general nm of the stature of a race, rendered definite by 
 an average, can be measured only from samples, as already 
 remarked. 
 
 "" Perhaps it is because the " scarcity of gold," being a cause 
 rather than an effect, belongs as the object to the eighth qucesitum, 
 leaving the " real thing " which is the object of the seventh 
 qucesitum entirely undefined. 
 
 "1 Loc. cit. 
 
 1" Ibid., p. 332. 
 
128 THE PROBLEM OF ESTIMATION 
 
 Still, the geometric average has been advanced as one 
 belonging to the subject before us, and we must consider the 
 question of the use in our problem of the geometric average 
 either with even weighting or weighted as Professor Edge- 
 worth would weight it. Here Professor Edgeworth has been 
 misled, not so much by an analogy wrong from the beginning, 
 as by one which he either does not follow or would still f oUow 
 beyond the point where it ceases. The model he has set up 
 for imitation is what should be done with estimates, which 
 itself is modelled upon what is done with observations. Now, 
 astronomers and surveyors average their single observations 
 of the same magnitude with even weighting, provided they 
 have made them with equal care. But they have two ways 
 of weighting their observations of the same thing. The one 
 is when two or more observations turn out alike : then they 
 treat them as one observation and weight this with a coeffi- 
 cient corresponding to the number of the repetitions. As 
 already quoted, weights are " merely numbers denoting 
 repetition." The other is when two or more sets of observa- 
 tions of the same thing — the different sets made perhaps by 
 different observers, or with different instruments — are 
 themselves to be combined and averaged : then the procedure 
 is the following. In each set its " probable error " is 
 calculated, which shows inversely its precision. If the 
 probable errors are equal, the sets (supposed to contain the 
 same number of observations) are averaged with even 
 weighting. If the probable errors are unequal, the sets are 
 averaged with weights inversely proportional to the squares 
 of their probable errors ; for when one set is, for instance, 
 twice as precise as another, it is as if it contained four times 
 as many observations made with the same care as those in 
 the other, wherefore it must be weighted as four to the other 
 as one.^'* The principle is the same in both cases. Let us 
 take the latter model first. If Professor Edgeworth should 
 desire, for instance, to combine two sets of measurements 
 of the same money over the same two periods, say Mr. Sauer- 
 beck's and The Economist's index-numbers, he might 
 calculate, if he could, their probable errors, and weight them 
 accordingly. As a fact, we rarely care to make any such 
 combination of two sets of index-numbers. We generally 
 prefer to accept the one we belive the better and let the other 
 go. Remains the first method of weighting observations. 
 Now, this model has been imitated above in the section 
 
 ^" Cf. above, pp. 67, 113. 
 
DEVIATIONS AND VARIATIONS 129 
 
 devoted to weighting, where we found that in our subject 
 the individual can only be a certain quantity of value, 
 wherefore commodities are to be weighted according to the 
 repetitions of this quantity in their full values (with extra 
 complication caused by the question of periods). And this 
 kind of weighting has been employed in all the subsequent 
 formulae for averaging price- variations, including formula (10), 
 as recommended by Professor Edgeworth, though it may be 
 questioned whether he understood the fuU nature of that 
 formula. Professor Edgeworth, however, in the present 
 part of his work, takes for a single or individual price-varia- 
 tion, corresponding to a single observation of an astronomer 
 or surveyor, the price-variation formed by the price- 
 quotations at two periods of every single commodity that 
 is entered in his list. But a single commodity is only a name 
 for a greater or less number of objects, or of money-unit's 
 worths of objects, so that it is slieer nominalism to treat 
 every name of a commodity as an individual thing. Yet in 
 his use of the geometric average in imitation of the use of the 
 arithmetic average by Herschel or by Quetelet, Professor 
 Edgeworth would give no more weight to the price- variations 
 of the multitudinous money-unit's worths of wheat than to 
 the scanty money-unit's worths of cloves. How far he 
 wanders from true imitation of his model, should be evident. 
 And it should be evident that he has no right to depart from 
 it in the formula for his seventh qucBsitum any more than in 
 the formula for his first qucBsitum, where he kept to the 
 model. Or rather, he has no right to lay down a qucssitum 
 " irrespective of the quantities of commodities," since this 
 has no rational meaning. 
 
 But Professor Edgeworth is not altogether satisfied with 
 even weighting. The oft-repeated condemnation of weight- 
 ing wheat and cloves alike was perhaps too much for him. 
 He makes concessions, and speaks of allowing some different 
 weights to the price-quotations. Only they must stiU be 
 different from the weights used in formula (lo) ; they must 
 not be regulated by the importance of the commodities to 
 the consumers (or to the nation) — not by the numbers of 
 money-unit's worths repeated in them. Why this continual 
 refusal to weight the geometric average as he weighted 
 formula (10) ? Because astronomers and surveyors allow 
 two similar observations to count as two only if they are 
 independent of each other ; but a double quantity of wheat 
 (over against barley) has similar prices not independent of 
 each other, since the same conMnodity always commands the 
 
130 THE PROBLEM OF ESTIMATION 
 
 same price in the same market. i'* Here is where Professor 
 Edgeworth dings to an analogy no longer existing. For as 
 a matter of fact the price of barley is also not independent 
 of that of wheat. And in wheat itself a factor of its price is 
 its material quantity. A new element here comes in, which 
 has no existence in the problem of averaging observations. 
 This new element differentiates our present problem from 
 that problem, and henceforth the analogy ceases ; for to 
 leave the new element out is simply to shut one's eyes to a 
 factor which exists all the same. But there is one principle 
 which underlies all averaging, which is that the individuals 
 averaged must be weighted according to their repetitions. 
 Now, at least the two quantities which make up the double 
 quantity of wheat are independent of each other, and to that 
 extent the full values which make up the double fuU value 
 of wheat (compared with that of barley) are independent of 
 each other. This is as far as the alleged total independence 
 of observations can be carried. And there is no need of 
 carrying it further. There is a good reason why observations 
 must be independent ; for their individuality is conditioned 
 by their independence. But in prices this is not so : the 
 individuality of the price-variation of one quantity of wheat 
 exists notwithstanding that the price-variation itself must 
 go hand in hand with the price-variation of some other 
 quantity of wheat. Thus independence of the items, though 
 necessary in observations, is not necessary in prices. We 
 average many things that are not totally independent. The 
 heights of men of the same race, for instance, are not totally 
 independent, since they are dependent upon the physical 
 conditions to which their ancestors have been subjected. 
 For that matter, the observations of astronomers and sur- 
 veyors are not totally independent, since they depend upon 
 three common factors — the object itself, the instruments, and 
 the observer. They are independent only as separately made. 
 Independence as well as individuality differ in different 
 objects. But individuality, not independence, is the uni- 
 versal element. Professor Edgeworth has inverted the true 
 relation, giving to independence the precedence which is due 
 to individuality. 
 
 But having rejected weighting according to full values or 
 importance, and being not altogether satisfied with even 
 weighting, Professor Edgeworth looks around for another 
 system of weighting, and finds it in the dispersion of the 
 
 "* Cf. Memorandum to the First Report, p. 287. 
 
DEVIATIONS AND VARIATIONS 131 
 
 prices. The quantities of the commodities are stiU to be 
 excluded : the weighting is still to be irrespective of them. 
 Thus in his first Memorandum (pp. 286, 287-8) he speaks 
 of weighting the price-variations inversely according to the 
 width of their dispersion, that is, according to their own 
 amounts, so that commodities whose prices fluctuate little 
 shall have greater weight than those whose prices fluctuate 
 much. Herein he adopts one of the careless suggestions 
 thrown out by Jevons (see above, p. 87). The width of a 
 commodity's price-variation is considered a gauge of that 
 commodity's importance, not to its consumers, but to " the 
 calculators of probabilities " ! I's -j-jiig jg g^id to be on the 
 principle that according to the theory of errors of observation 
 (and of estimates) " in the combination of the given observa- 
 tions less weight should be attached to observations belonging 
 to a class which are subject to a wider deviation from the 
 mean." "^ But that rule of astronomers and surveyors 
 refers to the combination of different sets of observations 
 of the same magnitude, not to the averaging of different 
 observations in the same set, and so is an entirely different 
 affair, misused as a model here. Perhaps, however, this is 
 not his meaning ; for then his weighting would have had to 
 be inversely according to the squares of the price-variations, 
 treated as " probable errors." Perhaps he, and Jevons 
 before him, had reference to a problem of true errors hke that 
 which we have discussed above, and in which we found, 
 in effect, that the weighting had to be inversely according 
 to the relative amounts of the errors (p. 62) ; which may 
 here be assimilated to the relative amounts of the fluctua- 
 tions of the prices. But that, too, was another considerably 
 different problem. Price-variations, put in the artificial 
 form of price-deviations, have some resemblance to errors ; 
 but they are not errors, and have differences too. And these 
 differences are especially influential in this very subject of 
 weighting, as we have already seen (above, pp. 81-2). The 
 mere fact that a certain weighting is proper in one problem, 
 is not a sufficient reason for introducing it into another. 
 In that problem of true errors we saw plainly the reason for 
 giving a greater weight to the smaller error. Professor 
 Edgeworth here says that " if more weight attaches to a 
 change of price in one article rather than in another, it is 
 ... on account of its importance ... as affording an 
 
 1" Memorandum to the Third Report, p. 157. 
 
 i'« Repeated in the Economic Journal, June, 1918, p. 188. 
 
132 THE PROBLEM OF ESTIMATION 
 
 observation which is peculiarly likely to be correct." ^^'^ 
 This is hardly so. The data of a commodity whose price 
 has varied 5 per cent, from one period to another are not 
 more likely to be correct than the data of a commodity 
 whose price has varied 10 per cent. Certainly they are not 
 likely to be twice as correct in the former as in the latter. 
 The consideration of likelihood or probability brings the 
 analogy back to that of probable errors ; and now our 
 astronomers and surveyors do not attach more weight to 
 observations near the average than to observations further 
 from it, notwithstanding that the former are more likely to be 
 correct than the latter. Professor Edgeworth both foUows 
 and does not follow his models. It is difficult to make out 
 his meaning with regard to his new kind of weighting ; and 
 it is needless to do so, as he reverts in the end to even 
 weighting.^' ^ It is mostly with the median that he would 
 use his new kind of weighting. To the median, then, let us 
 turn. 
 
 Professor Edgeworth has noticed in this connection what 
 he calls " the curious fact, that the median seems to keep 
 closer to the geometric than [to] the arithmetic " average.^'^ 
 There is nothing curious about it (provided the data be 
 numerous enough), if the data are those which arrange 
 themselves according to the geometric law of dispersion. 
 The median then follows the geometric average, as it foUows 
 the arithmetic average when the data arrange themselves 
 iaccording to the arithmetic law of dispersion (c/. above, 
 p. 43). Observations arrange themselves in the latter way, 
 estimates in the former ; and prices have been found to 
 arrange themselves in the former way too. And now, for the 
 same special qucBsitum for which he recommended the 
 geometric average, he recommends the median as following 
 it.^^° He uses even an opposite reason for it, because of its 
 departure from the geometric average on some occasions, 
 when the latter is unduly influenced (and the arithmetic 
 average still more) by some abnormally high price- variations, 
 as of cotton in 1862, which troubled Jevons ; for then he 
 thinks the median, being unaffected by their extravagation, 
 would have given a better result.^®^ But we have seen that 
 
 ^" Third Memorandum, loc. cit. 
 1" First Memorandum, p. 288. 
 1" Memorandum to the Second Report, p. 208. 
 "» Ibid., pp. 208-9. 
 
 ^'1 New Methods, in the Journal of the Royal Statistical Society, 
 June, 1888, pp. 358, 360, 362. 
 
DEVIATIONS AND VARIATIONS 133 
 
 that trouble was due, not to the geometric average in itself, 
 but to its use with wrong weighting (above, p. 87). This, 
 however, is a minor detail ^^^ for extraordinary occasions. 
 For ordinary occasions the principal argument is the median's 
 fellowship with the other average. There is even a double- 
 headed argument for the median. For if it turns out after 
 all that he was mistaken and price-variations do behave 
 according to " the normal probability curve," that is, the 
 arithmetic, then the median follows the arithmetic average ; 
 and if he was right and price- variations behave according to 
 the revised (logarithmetic) probability curve, the geometric, 
 then the median follows the geometric average ; and so it is 
 the safest, as it is close to being right in either case."* This 
 is the argument of one who is not sure of his position. But 
 perhaps his idea rather is, that prices do disperse themselves 
 in both these ways on different occasions, and therefore the 
 median is the best, as it automatically adjusts itself to either 
 behaviour of the prices.^^* The median, then, is " the 
 method " — " the most comprehensive combination," the 
 best " compromise between all the modes and purposes." ^^^ 
 Apparently it is recommended also to replace formula (10) : 
 its following the arithmetic average as well as the geometric 
 justifies its ousting them both."* The argument is lax, 
 because the median follows the one or the other with any 
 highly probable closeness only in cases when a great many 
 items are taken into consideration. Otherwise it follows the 
 others erratically, as Professor Mitchell, who has much dealt 
 
 ^'2 Professor Fisher, who likewise recommends the median on 
 top of another method already recommended as the theoretically 
 best, urges another detail in its favour : that " it has the advan- 
 tage of easily exhibiting the tendency to dispersion of prices," 
 Purchasing Power of Money, p. 231, again p. 427. If there is any 
 need of such an exhibition, there is nothing to prevent any 
 statistician from arranging price-variations in their order of 
 magnitude and studjdng their dispersion, whatever be the 
 method he adopts for measuring variation in the exchange-value 
 of money. Why not keep distinct things separate ? 
 
 1'^ Cf. Memorandam to the First Report, p. 291. 
 
 1'* Cf. New Methods, in the Journal of the Royal Statistical 
 Society, June, 1888, p. 362. 
 
 1'' First Memorandum, p. 296. 
 
 186 But we have seen in note 158 that Professor Edgeworth 
 admits the median is not so good an average as another for at 
 least one purpose not clearly defined. What his exact opinion 
 is, cannot be fathomed. 
 
134 THE PROBLEM OF ESTIMATION 
 
 with it, has found by experience, as might have been expected 
 from theory, since " its precise position is often dependent 
 on the relative price of a single commodity, which stands in 
 the middle of a scale of relative prices." ^^'' Yet Professor 
 Edgeworth would allow its use applied to only forty-five 
 commodities, that is, to forty-five price-variations.^^^ With 
 such small collections, close agreement of the median with 
 either the geometric or the arithmetic average can be expected 
 only in the long run. " But we must remember " — ^it is 
 Professor Edgeworth himself who says this — " that what is 
 true only in the long run is apt not to be serviceable on a 
 particular occasion." ^^^ What opinion, indeed, would 
 people have of an astronomer or a surveyor who should use 
 the median ? " To take the median for the sake of avoiding 
 computation," says an authority on the subject, " can only 
 be justified when the observations are rough ones, and then 
 the median itself is liable to differ considerably from the 
 arithmetical mean. The uncertainty of the probable error 
 of the median is greater than that of the arithmetical mean, 
 217 observations being necessary in the former case to give 
 the same [as little] uncertainty as 100 observations give in 
 the latter case." ^^'* What is here said of the median in 
 comparison with the arithmetic average in connection with 
 observations, applies similarly, though with somewhat 
 differing proportions, to the median in comparison with the 
 geometric average in connection with estimates and with 
 price-variations. To be sure, in the field of price-variations 
 (and still more of estimates) the data collected are only rough 
 approximations themselves ; yet this is not a good reason 
 for avoiding the trouble of computation, with risk of raising 
 the uncertainty by nearly half. Professor Edgeworth him- 
 self admits that the probable error of the median is slightly 
 greater than that of the arithmetic average ; ^^^ and for 
 measuring the general trend of prices he has maintained that 
 the arithmetic average is less accurate than the geometric. 
 Then why all this advocacy of an average admitted to be the 
 least accurate ? 
 
 1*' Mitchell, Gold Prices under the Greenback Standard, 1908, 
 p. 58, cf. p. 33 n. ; of. Index-Numbers and Wholesale Prices, 
 pp. 85, 90. 
 
 '*' Memorandum to the Second Report, p. 209. 
 
 i*» New Methods, in the Journal of the Royal Statistical Society, 
 June, 1888, p. 350 n. 
 
 189a Merriman, Method of Least Squares, § 167. 
 
 ^"' In the Economic Journal, June, 1918, pp. 193-4. 
 
DEVIATIONS AND VARIATIONS 135 
 
 But here again the consistency of his argument is broken 
 by his allowing some uneven weighting to the median. Or 
 is it only when the median is supposed to take the place of 
 formula (10) that uneven weighting is to be allowed ? But 
 then the automatic working of the median, which was 
 supposed itself to choose its own guide, is interfered with. 
 This does not seem to be his position. But what the weight- 
 ing is to be, it is difficult to make out. It must not be 
 according to the importance of the commodities, or to their 
 relative " masses of value " ; for, " from the present point 
 of view, in order to determine a type by the Calculus of 
 Probabilities, the attribute which gives importance to an 
 article is not its quantity, but the independence of its 
 fluctuation. From this point of view pepper may be as good 
 as cotton," ^^^ that is, apparently, cotton need have no more 
 weight than pepper. Yet " upon this ground," he " recom- 
 mended assigning rather larger weights to the more massive 
 commodities," that is, more to cotton than to pepper ! 
 " For technical reasons I advise that each price- variation 
 should count for as many, should in effect be repeated as 
 often, as the square root of the number of units in the 
 quantity of the corresponding commodity" (ibid.). What 
 the units are, is not specified. Nor are the technical reasons 
 given ; but we may suspect one to be the familiar law that 
 closeness of practice to theory, or the diminution of error, 
 is increased proportionally to the square root of the number 
 of items employed. The idea may be that the quantities 
 of the more massive commodities, being more numerous or 
 more often repeated, yield more accurate results in this 
 proportion. But we have seen that they should then be 
 weighted according to the squares of their probable errors. 
 This would bring their weights back into direct proportion 
 with their relative massiveness or importance or fuU values 
 (for we cannot believe he referred to physical masses), 
 though the question of the periods is left untouched. StiU, 
 if weighting is to be allowed to the median, it might equally 
 well be allowed to the geometric average ; and if we used 
 the weighting of both periods, we should be using formula (7), 
 which Professor Edgeworth never dreamt of. At all events, 
 no superiority of the median over the geometric average 
 would now appear : rather, the median would sink back 
 into its proper inferior position as a mere more or less 
 
 i»i New Methods, in the Journal of the Royal Statistical Society, 
 June, 1888, p. 363- 
 
136 THE PROBLEM OF ESTIMATION 
 
 faithful attendant. The median 'can be advocated only 
 because it is more convenient to consult than is the average 
 it is supposed to attend. But a weighted median is again a 
 difficult average to work with, and the margin of convenience 
 is much reduced. As for the geometric average, this, with 
 its proper weighting (as in formula (7) ), does not need to be 
 confined to any such special qucBsUum as Professor Edge- 
 worth would confine it. Professor Edgeworth did not know 
 that, applied to the data collected for his first qucBsitum — the 
 practical part of his subject, the case for which he recom- 
 mended formula (10), — ^the geometric average with its 
 proper weighting even in extravagant examples closely 
 approximates to (and in practice so closely as to be almost 
 indistinguishable from) that other formula. Had he known 
 this, he would have had no occasion to distinguish the cases. 
 Instead of following both Galileo and Nozzolini, both Jevons 
 and Laspe3n:es, when they differed, he would have seen that 
 in this our subject, because of the peculiarity here of the 
 weighting, the positions of these opponents, when properly 
 la,id off, come so close together that their difference is 
 negUgible.^^^ 
 
 One root of trouble in all this matter is the unwillingness of 
 Professor Edgeworth and his followers to give due attention 
 to weighting even in theory, because they think it of little 
 importance in practice. Even in practice, however, it is 
 important. Professor Edgeworth in his Memorandum to the 
 Second Report of the British Association Committee studied 
 the discrepancies of averages differently weighted, and came 
 to the conclusion that " other things being the same, the 
 inaccuracy of the price-returns affects the result more than 
 inaccuracy of the weights" (pp. 197-8). As "a practical 
 conclusion " he therefore advised : " Take more care about 
 the prices than the weights " (p. 200), Yet this is not a good 
 reason for not giving all the care that we can to weights ; and 
 
 192 Professor Edgeworth not only did not know this and some 
 other things when he invented his theory, but he continues in the 
 same ignorance since the publication of M. G. E.-V., which he 
 reviewed and criticised and presumably read. So enamoured is 
 he of his own probabilistic way of lodiing at the subject, that 
 he refuses to be instructed, and shuts his eyes to the truth, just 
 because it has come from another mode of treating the subject 
 more analytical (dialectical he called it) and systematic than his 
 own. Yet where plain algebra is sufficient for dealing with a 
 jproblem, it is simple pedantry to lug in the so-called higher 
 mathematics. 
 
DEVIATIONS AND VARIATIONS 137 
 
 it is only perverted when Dr. Bowley says " Do not strain 
 after exactness in weighting." "^ ^ consequence has been 
 that, although in their Reports the British Association 
 Committee always recommended the use of uneven weighting, 
 they did so half-heartedly, and damned it by adding that, 
 notwithstanding their recommendation of it, it was " in one 
 aspect [whatever that refers to] almost an unnecessary 
 practice to secure accuracy." "* What is meant by " accu- 
 racy " here may be judged by the fact that in the annexed 
 Memorandum their secretary. Professor Edgeworth, had 
 characterised as small such discrepancies as those of 110-4 
 and 115, 94-6 and 92-4, 119 and 116, etc^'^ If a government, 
 whose business it is to regulate the currency, were to use 
 index-numbers hable to such errors, which might accumulate 
 from year to year (for the Committee rightly recommended 
 what has since been called the " chain " system of making 
 index-numbers for every year compared with its prede- 
 cessor 1^®), that government .would do better to leave the 
 currency alone. Evidently the statistical problem has been 
 injuriously affected by such want of precision ; and few 
 statisticians have paid attention to the recommendation of 
 the Committee, despite the fact that the formula they 
 recommended is a very good one, and perhaps the best 
 (unless formula (11) is better) for practical purposes. 
 
 Is it not time, then, that the Report of the Committee 
 should cease to be spoilt by the imperfect theoretical treat- 
 
 ^"^ Elements of Statistics, p. 118. More recently he has written : 
 " It does not seem at all certain that any improvement is made 
 by weighting," in the Journal of the Royal Statistical Society, 
 May, 1919, p. 346. Compare with the above the practical advice 
 given in M. G. E.-V., p. 180 n. 
 
 1" Second Report, p. 185. 
 
 195 Memorandum to the Second Report, pp. 202, 206. In 
 the Economic Journal, June, 1918, p. 185, Professor Edgeworth 
 disputes Professor Mitchell's statement that the " chain " 
 system is more trustworthy than the " fixed-base " systeni, by 
 attempting to show on an example that the difference is slight. 
 The one system indicates a rise from 111-5 to 113, and the other 
 one from 126 to 130. The latter is the same as a rise from in -5 
 to 115. The difference between 113 and 115, or between rises of 
 1-35 per cent, and 3-13 per cent., is treated by Professor Edge- 
 worth as a " remarkable consilience . . . greater than was to be 
 expected " ! Evidently he would have been content with an 
 even greater discrepancy. 
 
 "« First Report, pp. 250, 252. 
 
138 THE PROBLEM OF ESTIMATION 
 
 ment with which it was accompanied ? Professor Edgeworth 
 has won a high place of authority in this subject of averaging 
 price-variations. But it may be contended that the recog- 
 nition of his authority has stood in the way of progress. A 
 five-fold charge may be brought against him. He has led 
 others astray (i) by underrating the importance of weighting 
 as well in theory as in practice, (2) by complicating the 
 subject with a variety of qucesita artificially constructed and 
 apparently unlimited in number, (3) by ignoring the balanc- 
 ing of opposite price-variations, (4) by insisting on introduc- 
 ing the theory of probabilities in places where it does not 
 belong, and (5) by winding up Jilways with the median 
 because of the mere probability that it follows close upon the 
 average for which he would substitute it. 
 
 Would it not be well if statisticians and economists should 
 again come together and decide authoritatively on the 
 proper method of constructing index-numbers ? And as 
 regards the formula to be adopted, why should they choose 
 any other than the one recommended by the Committee, 
 unless indeed it be formula (11) ? If this had been done a few 
 years ago, it might have prevented such a regrettable relapse 
 as that of Professor Fisher, who has tried to revive Paasche's 
 use of the quantities of the second period, in a form which 
 really is the harmonic average of the price-variations with 
 the weighting of that period.^*' It might also have prevented 
 the still more regrettable, since practically misleading, 
 relapse of the Labour Department in their Labour Gazette 
 
 1" Cf. above, note 118 ; The Purchasing Power of Money, 
 pp. 201-3, 218, 231-2, 421. Professor Fisher used four tests 
 (each with a duplicate), of which the first is incomplete, the second 
 unnecessary, the third questionable, and the fourth is Wester- 
 gaard's test. All these tests can be satisfied by very bad methods. 
 But the circular form of the last test — the most important of aU, 
 since it alone measures the amount of error in a method — he did 
 not use. He therefore never measured whether a method failed 
 to meet his tests only slightly or very badly. He apparently did 
 not notice the criticism of Paasche's method in M. G. E.-V., 
 pp. 193-4, 541-2, nor the bad failure of that method to meet the 
 circular test in an example on p. 428. But in his recently pub- 
 lished Stabilising the Dollar, he has dropped that method (and the 
 median too), and recommended, pp. 3-4, 85-6, 206-7, Lowe's 
 "tabular standard" method referred to above in note 129. 
 This, really a multiple standard, though not the best method, is 
 of course a great improvement on the single gold standard which 
 it is intended to supplant. Biit cf. p. 494 of the first work. 
 
DEVIATIONS AND VARIATIONS 139 
 
 into revival of Laspeyres's equally faulty use of the quanti- 
 ties and weighting of the first period, with arithmetic averag- 
 ing. The subject is one that is growing in importance. 
 Labourers are more and more clamouring for regulation of 
 their wages by index-numbers. The use of them has been 
 recommended to bankers for the regulation of their issues. 
 Professor Fisher himself is advocating their use for stabilising 
 the monetary standard. The day for sHp-shod methods, it 
 would seem, is passing. 
 
INDEX 
 
 Aristotle, 3, 13 
 
 Arithmetic average, behaves like 
 the arithmetic mean, 76 ; fre- 
 quent use of, 38, 69 ; adopted 
 as the most convenient, 13-14, 
 30, 77 ; may in practice con- 
 tinue to be used for this 
 reason, 78, 104, since it 
 diverges Uttle from the theore- 
 tically correct average, 13, 79, 
 104-105 ; correctly used in 
 observations, 37, 46 ; pro- 
 perties of, 46-47, 49, 50 n. ; 
 formula of, the simple, 31 ; 
 the weighted, 32. See A verage, 
 Errors. 
 
 Arithmetic mean, 2, 3, 5, 8-9, 
 13, 14 n. ; pecuHarity of, 16. 
 
 Arithmetic terms, behaviour of, 
 16, 27 ; curve of, 40, 48 n. 
 
 Averages, the criteria of, 18-365 
 43-44. 45-50. 80 ; proper 
 where the corresponding means 
 are proper, 75-76 ; of methods 
 in general, not desirable, 107, 
 128 
 
 Axiometry, 69 ; problem of, 
 perhaps not solvable abso- 
 lutely, 103 and n., 112 n. 
 
 Back-and-forth argument, the, 
 
 6-7, 11-12 n., 23, 74-75 
 Bacon, 108 
 Baynes, J., 25 n. 
 BernoulU, Daniel, 23, 24 n., 25, 
 
 39 
 Bertillon, A., 29 n., 30-31, 76, 
 
 124 n. 
 Bowley, A. L., 53-56, 62, 66, 76, 
 
 78 n., 90, 104, 105-107, III, 
 
 137 and n. 
 British Association Committee, 
 
 the, 115, 116 and n., 122, 137- 
 
 138 
 BufEon, 23, 24 n., 25 
 
 Butchers, wagers of, 4-5, 13, 14, 
 15-16, 30 
 
 Cajori, 24 n. 
 
 Capacity, measurement of busi- 
 ness, 6, 21 
 
 Castelli, i, 2, 5, 6, 7, 10, 75 
 
 Cauchy, 33 n. 
 
 Chain system, the, of index- 
 numbers (84-85), 105, 137 
 and n. 
 
 Circular test, the, 98, 102, 103, 
 104, 108 
 
 Compensation of opposite changes, 
 72, 92, III, 112 n. 
 
 Criteria of average. See Averages, 
 
 Data, need of sufficient, for 
 solving a problem, 11, 86-87 
 
 Death-rate, 29, 34 n. 
 
 Del Mar, 121 n. 
 
 De Morgan, 34-35 n., 38 n. 
 
 Deviations, distinguished from 
 variations, 68 
 
 Dispersion, the arithmetic and 
 the geometric, 28, 42, 69 ; the 
 harmonic, 28 n. ; use of, 35, 
 80 ; study of, 117 n. 
 
 Drobisch, 102 n., 104 
 
 Dutot, 98 
 
 Economist, The, 128 
 
 Edgeworth, F. Y., 39, 42 n., 79- 
 81, 89 n., 104, 107, 107-138 
 
 Encke, 76 n. 
 
 Error, law of, 45 
 
 Errors, as deviations, 68 ; 
 measurement of, 51-67 
 
 Errors of estimation, measure- 
 ment of, 12 ; defined as a 
 ratio, 12, cf. 41-42 ; unlimited 
 above, hmited below, 16, 21 ; 
 
142 
 
 INDEX 
 
 thereby difEerentiated from 
 errors of observation, 36-37, 
 43 ; geometric, 52-53. 
 
 Errors of observation, compared 
 •with chances, 45 ; arithmetic, 
 52-53 
 
 Estimate, treated as unratified 
 purchase, 3, 8 ; difference 
 between, and purchase, 9 ; 
 two meanings of, 36-37 
 
 Estimation, the controversy 
 over, 1-18 ; problem of, not 
 confiined to values, 9-10. See 
 Errors. 
 
 Euchd, 99, 112 
 
 Extravagant cases, use of, by 
 GaUleo, 2, lo-ii ; by Castilh, 
 5 ; disallowed by Nozzohni, 
 14-15; by Edgeworth, no; 
 propriety of, 108. 
 
 Fechner, 24 n., 27, 33 and n., 
 
 38, 50 n., 79 
 Fisher, I., 33 n., 73 n., 78 n., 85, 
 
 93 n., 94 n., 120 n., 133 n., 138 
 
 and n., 139 
 Franklin, 25 n. 
 
 Gain and loss, question of, 6, 8, 
 21, 22, 77, 86, 92 ; not neces- 
 sary in problem of estimation, 
 9-10 ; connection of, with 
 arithmetic average, 19 
 
 Gahleo, i, 2-3, 6-7, 9-13, 15 n., 
 16, 17, 18 and n., 21, 25, 37-38, 
 39 n.. 40- 51- 74. 75. 77. 86. 93, 
 no, 115, 136 
 
 Galton, 28 and n., 38 and n., 39, 
 
 40, 79 
 Gauss, 45, 47, 48 and n. ; law of, 
 
 79 
 Geometric average, formula of, 
 simple, 31, weighted, 32 ; 
 use of, in estimates, 37-39 ; 
 connection of, with ratios and 
 variations, 29 ; recommended 
 in general by Jevons, 78 ; the 
 simple, advocated for index- 
 numbers by Jevons, 77, 121 n., 
 by Edgeworth, 115, 121-129 ; 
 difficulty of, 57-61 ; failure of, 
 where the geometric mean is 
 correct, 61, 76, 93^95. 97> 
 
 loo-ioi, 108 ; cases where it 
 
 works, 61-65. See Averages. 
 Geometric mean, 2-3, 7, 8, 10, 
 
 13, 14 n. ; pecuUarity of, 16, 
 
 39 
 Geometric terms, behaviour of, 
 
 16, 27 ; curve of, 41, 48 n. 
 
 See Dispersion. 
 Guesses, 37 n., 39 n. 
 
 Harmonic average, 75 ; behaves 
 
 like the harmonic mean, 76 ; 
 
 of prices, 86 ; formula of, 
 
 weighted, 32, 93 
 Harmonic errors, 53 n. 
 Harmonic mean, 2, 5, 9, 20-21 n. 
 Harmonic terms, behaviour of, 
 
 28 n. ; curve of, 42 n. 
 Herschel, Sir J., 48, 124, 129 
 
 Importance, warning concern- 
 ing, 83 n. 
 Index, cephahc, 29-30, 33 
 Index-numbers, 78, 84, 91-107, 
 115, 118, 139 
 
 Jacoby, H., 49 n. 
 
 James, W., 28 n. 
 
 Jevons, 24 n., 48 n., 70, 74 n., yf, 
 78 and n., 80, 85-88, 90 n., 
 100 n., 106 n.. Ill, 115, 116 
 and n., 121 n., 123, 124, 131, 
 136 
 
 Justice, commutative and dis- 
 tributive, 3-4, 10, 12 
 
 Knibbs, G. H., 105 n. 
 
 Labour Department, The, 138 
 Lacroix, 24 n. 
 Laplace, 24 n., 25 
 Laspe5Tes, 77, 85-88, 90 n., 
 
 100 n.. Ill, 115, 136, 138 
 Least squares, method of, 35 n., 
 
 46-50 
 Legendre, 47, 49 n. 
 Lehr, 102 n., 104 n. 
 Lesson-marks, 13-14 n., 31 n., 
 
 39n- 
 Libri, 18 n. 
 
INDEX 
 
 143 
 
 Logarithmic diagrams, 78 and n. 
 Lowe, loi n., 105 n., 138 n. 
 
 McAlister, 40, 68 
 
 Macaulay, 85 
 
 Mark, aiming at a, 16, 71-72 
 
 Marshall, A., 104, 112 n. 
 
 Measurements, requirements in, 
 4, 12 
 
 Median, the, 43 ; use of, 44, 
 50 n. ; a questionable employ- 
 ment of, 44 n. ; a property of, 
 46, 47, 50 n. ; recommended 
 by Edgeworth, 115, 132-136; 
 by Fisher, 133 n., cf. 138 n. 
 
 Merriman, M., 39 n., 49 n., 67 n., 
 82, 134 
 
 MessedagUa, 39 n., 86, 87, 11 1 
 
 Mill, J. S., 112 n. 
 
 MitcheU, W. C, 80 n., 85 n., 93 n., 
 105 n., 118 n., 121 n., 133-134, 
 1371. 
 
 Mode, the, 27, 30, 35-36, 42, 43, 
 44, 68 
 
 Money, kinds of, of which the 
 exchange-value is measured, 
 89, 117 
 
 Nicholson, J. S., 102 n. 
 
 NozzoUni, i, 2, 3-4, 5-6, 7-9, 10, 
 11-12 n., 13-17, 18, 21, 22, 30, 
 39 n., 40, 52, 75, jj, 86, 92, 
 93, no, 115, 136 
 
 Observations, as measurements, 
 36. See Errors, Weighting. 
 
 Paasche, 100 n., 138 and n. 
 
 Palgrave, 100 n. 
 
 Percentage, how reckoned, 7, 
 II n. ; comparisons of, 5, 6 
 
 Persons, W. M., 31 n., 95 n. 
 
 Pierson, N. G., 86, 87, 109, in 
 
 Practice, inexactitude permitted 
 in, 13-14. See Theory. 
 
 Preciousness, 70 
 
 Prices, as ratios, 70 ; as varia- 
 tions, 69, 81 ; like estimates, 
 73-75, 79-80, 123, also different, 
 81-82, 90, 131 ; averaging of, 
 70-73 ; negative, 22 n., 74 n., 
 level of, desired to be stable. 
 
 69 ; strict meaning of price 
 level, 70 
 Probabilities, calculus of, 42- 
 
 43 n., 81, no, 122-125 '• spirit 
 of, no; common-sense, 114- 
 
 "5 
 Probable error, measurement of, 
 
 44 n., 113, 132 
 
 Quaesita, used by Edgeworth, 
 
 116-125, 136, 138 
 Quantities of articles priced, 82 ; 
 
 lurk beneath fuU values, 83, 
 
 89-90 
 Quetelet, 24 n., 26, 28, 29 n., 
 
 123, 124 and n., 126, 129 
 
 Ratio, difierent from fraction, 
 
 29-30 
 RomUly, Sir S., 25 n. 
 
 Samples, only possible in some 
 investigations, 26, 88, 127 n. 
 
 Sauerbeck, 128 
 
 Scrope, loi n., 105 n. 
 
 Sidgwick, 104 and n. 
 
 Single items, question of, in 
 price- variations, 82-84, 129- 
 130 
 
 Theory, guide to practice, 34, 
 85 ; should precede, 89 
 
 Todhunter, 19 and n., 23-24 n. 
 
 Tooke, Horn, 25 n. 
 
 Two objects or instances, em- 
 ployment of, 76, 87-88, 108, 
 109-112 
 
 Unweighted averages, none 
 such, 84 
 
 Value, esteem, 23-24, 117 n. 
 Values, full, 82, 83, 88 n., 89, 91 
 Value-units, 83, 96 
 Variations, connection of, with 
 
 the geometric average, 29 ; 
 
 distinguished from deviations, 
 
 68 
 
144 
 
 INDEX 
 
 Venn, J., 24 n., 48 n., 76 n., 
 79 n., 82 n., 116, 117 n. 
 
 Walras, 33 n. 
 
 Weber, law of, 24 n., 38, 39 and n., 
 
 79 
 Weighting, unrecogmsed, 32-33, 
 34 n., 55 ; wrong, with wrong 
 average may give right result, 
 
 33-34. 56. 90. 93 ; in the 
 measurement of errors, 56-57 ; 
 proper, in averaging estimates, 
 62 ; in comparing different 
 sets of estimates or observa- 
 tions, 67 ; of prices, 81-90 ; 
 first considered by Jevons, 
 
 87 n. ; complication of two 
 periods, 88-90, 90 ff. ; hap- 
 hazard, 71 n., 85, 105 ; prin- 
 ciple of, 82, 128 ; necessity of, 
 in theory, 85 ; not unimpor- 
 tant in practice, 84-85 ; con- 
 nected with the kinds of 
 averages, 85, 92-93, 94 n., 96, 
 loi ; neglect of, 136-137 
 
 Westergaard, test of, 97-98, 98 n., 
 138 n. 
 
 Whewell, 47 
 
 Wicksell, 104 n. 
 
 ZiZEK, 31 n., 78 n., 95 n. 
 
 r. S. KING Sc SON, LTD., OSCHAHD HOUSE, WESTMINSTER. 
 
^r