price of every article to influence the result" (71). In making this objection, the author seems to have in view two groups in pari materia such that in passing from one to the other we find no change in the median, while there are changes among the other observations other than those determining the median, which changes affect the more sensitive arithmetic mean. Upon this it may be remarked that if there is this difference in the behaviour of the two averages, it is not to the credit of the arithmetic mean. The slight advantage which we have already allowed to the arithmetic mean would not be enhanced by this circumstance. Supposing that slight advantage corrected by basing the median on a greater number of observations, then the sensitiveness attributed to the arithmetic mean would be rather a defect than an advantage.1

But does the difference exist? Does the median, oftener than the arithmetic mean, does it even often, remain unchanged from one group to another? This may be doubted, if the data are finely graduated, or if graduation of the median by adjustment is practised. The median seems, indeed, but only seems, to be irresponsive in certain circumstances of perhaps frequent occurrence in the statistics of prices-which we shall indicate by continuing the parable of the indoor race. Suppose that in the first five minutes several of the numerous players-late or dilatory— do not make a start, and that their positions at the end of the period are registered as being at the starting-point. Accordingly, at the end of a short period a good number of observations would be heaped up at the starting-point; the median would appear unmoved. But, of course, the position of those players who have not moved-whose position is not the result of steps determined by tossing coins-cannot be used to ascertain the asymmetry of the coins. For that purpose it would be proper to omit those dead-head observations, or to prolong the game until the slow players should come in. But for other purposes, of perhaps greater interest to the players, as relevant to the betting, it might be proper to take account of those nullities.

1 The arithmetic mean in this respect might be compared with the method of examination by summing arithmetic marks practised at some public competitions as contrasted with examinations at one at least of our Universities where general unanalysable impressions have a due weight. The former method, no doubt, more frequently brings out candidates as unequal, but the distinction does not correspond to a real difference.

2 Above, p. 403, note 2.

* Above, p. 391. Note that the spike-shaped " mode "there noticed is formed by prices which have not moved at all in the period under consideration; to be distinguished from those which have moved less than one mill.

Here, probably, is to be found the reason of the difference between Professor Mitchell and ourselves as to the worth of the median. We have been all along seeking to extricate from fallible observations a mean apt to represent the "general trend of prices" (9). That is the sort of index-number to which we submit that the median may be appropriate. But Professor Mitchell in his criticism of this average has presumably often in view some of the more directly practical purposes which have been distinguished, such as par excellence the determination of changes in the cost of living. For these purposes we at once admit that the median is not so appropriate as the combination of the kind which Professor Mitchell calls an "aggregate." 1 We entirely agree with him that "the best form for general purpose series is a weighted aggregate of actual prices."

1 The term 66 aggregate" is felicitous as suggesting approach to that type which, as above explained (p. 396), is furthest removed from an index-number, the term least connected with the Calculus of Probabilities; infelicitous so far as it masks the affinity, not to say identity, between the proposed construction and the weighted arithmetic means used by Giffen, Palgrave, and the older statisticians (as to whom see British Association Memoranda, 1887, p. 264, and 1889, p. 139 et seq.). The words of Sidgwick there quoted: Summing up the amounts of money paid for the things consumed at the old and the new prices respectively. . ." (Political Economy, Book I. ch. ii. section iii.), are appropriate to aggregates.

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EVALUATION OF THE METALLIC CURRENCY

[THE attempts to evaluate the amount of coin circulating in a country which form the subject of this paper, published in the ECONOMIC JOURNAL, 1891 and 1892, were conducted partly on the lines of Newmarch's method (discussed by the present writer at the meeting of the British Association for 1888), partly on the fresh lines struck out by Jevons. To the second class belong De Foville's calculation based on three French enquêtes, noticed here. Mr. F. C. Harrison's computation of the rupee circulation, which occupies a great part of the paper, is an improvement on the method of Jevons. By bringing to bear on the calculation the evidence afforded by the examination of samples pertaining to several successive years, he has obtained a result which seems to have almost the certainty of physical science.

The reader may like a reference to the Journal of the Royal Statistical Society (Vol. LXXXIII, 1920, p. 609 et seq.), where the subject is further discussed in connection with Mr. Shirras' excellent Paper on the effects of the war on gold and silver.]

I. Among recent attempts to evaluate the amount of coins circulating in the country a prominent place is due to that which Messrs. Martin and Palgrave have just completed. Their method is similar to that which Newmarch employed to determine the circulation at the epoch 1843-4 (History of Prices, vol. vi.). They reason: As the percentage which the pre-Victorian sovereigns formed of the total circulation (previous to the recall of that coin) is to 100, so is the amount of pre-Victorian sovereigns to the total amount of sovereigns in circulation (previous to the recall); and similarly for the halfsovereigns. By means of circulars issued to bankers, Messrs. Martin and Palgrave ascertained that the percentage of preVictorian sovereigns was about 4 per cent.; and the number recalled was 2,335,000 nearly. Whence the total of sovereigns

previous to the recall is found to be about 58,375,000. Performing a similar computation for the half-sovereigns, deducting the coin recalled, and making an addition of £11,000,000 on account of the gold coin in the Bank of England which does not conform to the general average, Messrs. Martin and Palgrave (in their latest version, Economist, January 23) give £80,000,000 as the amount of the gold circulation.

Of the two data on which the inference mainly rests-the comparative and the absolute amount of the pre-Victorian coin -the former is corroborated, in the case of the sovereigns, by the close proximity between the observations for England and Wales, Scotland, and Ireland; 4.12, 4.1, 4.7 being the respective percentages formed by the pre-Victorian coin.1 This consilience is not presented by the half-sovereigns, for which the respective percentages are 84, 50, and 1.06. But it may be observed that the numbers on which the Irish and Scotch averages rest are very small. The second datum, the absolute quantity of the pre-Victorian coin recalled, is too little by the number of coins not given up-retained, it may be, as curiosities. Against this deficit Mr. Martin-in his letter to The Times of July 21, 1861, describing the method of calculation-puts the fact that some of the recalled pre-Victorian sovereigns "undoubtedly came from abroad." The total officially known to have come from abroad is £162,751.

Both the data have been subjected to severe criticism in recent numbers of the Economist (January 2, 16, 23, 30). The majority of the objections which have been made suggest that the result obtained errs in defect. This contention, if it is substantiated, will confer on the computation the important character of a lower limit to the amount of coinage in circulation; thus rendering the Martin-Palgrave method complementary to that of Jevons, which-in its simplest form at least, when unmixed with precarious calculations based on the export and import of coin 2-affords a higher limit. The two methods, if performed jointly, would give two limits between which the quantity of the coinage at the epoch to which the returns relate must lie.

II. Next may be noticed the brilliant attempt to estimate the rupee circulation which has been made by Mr. F. C. Harrison in the ECONOMIC JOURNAL.3 His method is that of Jevons as to its

1 This impression is confirmed by a more detailed inspection of the returns. The English sovereigns which were examined fall into four large classes, for which the percentages (of pre-Victorian coin) are respectively 4.2, 3.8, 3.5, 4·6. 2 See Jevons, Currency and Finance, pp. 266-7.

1891 and 1892.

essence, but with a specific difference; the foundation is the same, but Mr. Harrison's construction rests, so to speak, on a great number of props, and they support each other archwise. Jevons, seeking to determine the amount of the (sovereign) circulation in 1867, reasoned: As the percentage (ascertained by the inspection of samples) which the coinage of 1863-4 forms of the total circulation is to 100, so is the amount of the coinage of 1863-4 presumed to be in circulation to the total circulation. Mr. Harrison, seeking to determine the amount of the (rupee) circulation in 1890, utilises similarly, not only the amount of the coinage presumed to be in circulation, but also the corresponding data for preceding years, allowance being made for the greater diminution of the coinage of earlier years. How is the comparative degree of diminution ascertained? By observing the gradually diminishing proportion which the coinage of any year, say 1874, forms in the circulation of successive years, 1877,1 1878, 1879–1890. These proportions are respectively: 2.13, 1.8, 1.6, 1·55, 1·45, 1.4, 1.3, 1.15, 1.2, ·95, 9, 95, 9, 9 per cent. of the total circulation in 1890. They measure the decrease of the coinage of 1874, upon the hypothesis that the total circulation is stationary during the period 1877-90; which Mr. Harrison assumes as approximately true (op. cit. p. 722). How is this assumption justified? By the consistency of the various results obtained on this hypothesis, a consistency which cannot be ascribed to accident. To show this, let us suppose that the decrease indicated by the row of figures above cited is due, not to the diminution of the amount of the 1874 coinage in the circulation, but to the increase of the total circulation with respect to which the percentages are taken. Upon this supposition the whole coinage of 1874 has passed into the circulation of 1890. But that coinage amounted to 4.352 crores 2 of rupees (as shown in Mr. Harrison's Table A); and it forms 9 per cent. of the 1890 circulation (ibid.). Therefore (by Jevons's method) the circulation of 1890 4.352009 or 483 crores, a result which is violently inconsistent, not only with all Mr. Harrison's estimates, but also with common sense, since the whole amount of the coinage issued ab initio is only about 300 crores (Table F).

It may be suspected, however, that the downward slope of

1 Assuming with Mr. Harrison that the circulation of 1874 was three years in passing into circulation; and, after him also, at first leaving out of account the loss suffered by that coinage during those three years (op. cit. p. 733, and below, p. 411).

2 It may be well to remind the reader that a crore = 100 lakhs = 10,000,000 rupees. Thus 4.352 crores = 43,520,000 rupees.