Mr. Pierson, "it would cost Holland, exclusive of her colonies, about 52 million florins; France about 950 million francs." But if the 15 is to be adopted, then there is likely to be caused an immense appreciation of money throughout the East-the last result that a consistent Bimetallist can approve.

Mr. Pierson's own plan for keeping the relative value of gold and silver constant is a modification of the general idea that the central banks in Europe-in the United States the Treasuryshould be required to purchase the metals at a fixed price. But perhaps it is unnecessary to go into particulars, as Mr. Pierson admits that there are three fatal objections to the practical adoption of his plan.

While pointing out the difficulties of remedial action, Mr. Pierson does not, like the monometallists, deny the existence of monetary disease. It is a sad conclusion that things are in a bad way. It is a poor consolation that they might have been For instance, the success of the monetary experiment in the Dutch colonies is much more perfect than could with reason have been expected.

worse.

These conclusions are corroborated by a communication which Mr. Pierson has made to the January number of the Dutch monthly De Economist. In this number (p. 64) Mr. Pierson discusses the statistics of prices which Mr. Heinz, the chief of the Hamburg Statistical Bureau, has prepared in continuation of the work of Soetbeer [compare the statistics referred to in the ECONOMIC JOURNAL, Vol. IV. p. 201]. Soetbeer's series of index-numbers was interrupted by tariff-regulations which disturbed the prices of several commodities in such wise as to render them after 1891 no longer commensurate with the prices of the same articles for earlier years. Accordingly Mr. Heinz has to confine himself to articles which have been imported by sea into Hamburg. Operating with 137 articles of this class, Mr. Pierson constructs a series of index-numbers which is continuous from the year 1850 to the date of the most recent returns. Unfortunately for the comparison with Soetbeer's figures, Mr. Pierson has been compelled, by the imperfection of his materials, to take as the base of the new index-numbers the year 1850, instead of the period 1847-50 which Soet beer had taken. But in spite of this discrepancy, and the more serious difference in the mode of construction, the parallelism between the two sets of index-numbers is wonderfully close, as the annexed figures show :

1 1850 for Heinz's, 1847-50 for Soetbeer's index-number.

It will be observed that the new series like the old one, when the "inflation period" of 1871-75 is left out of account, shows no signs of appreciation due to monetary disturbances of that period thus confirming Mr. Pierson's view that the appreciation which we now experience is due to causes connected with goods rather than gold.

The congruity between the two Hamburg index-numbers is such that they mutually support each other. On the one hand additional strength is imparted to the conclusion which Mr. Pierson had obtained on independent grounds (above p. 352), that Soetbeer's index-numbers are more trustworthy than Mr. Sauerbeck's. On the other hand the index-numbers constructed by Messrs. Heinz and Pierson, having agreed closely with Soetbeer's for forty years from 1850 to 1891, may be presumed to be almost as trustworthy as Soetbeer's index-numbers would have been for the years after 1891.

If we compare the period 1886-93 with the period 1861-70 taken for base-as recommended by Mr. Pierson in his study on the scarcity of gold (above p. 352)—we shall find from Mr. Heinz's materials, the index-number 88.8; from Mr. Sauerbeck's, 69.5. According to Mr. Pierson the former measure of appreciation is the more trustworthy; but the latter is more convenient for the advocates of Bimetallism.

(L)

A DEFENCE OF INDEX-NUMBERS

[IN this article, which appeared in the ECONOMIC JOURNAL, March 1896, I ventured to differ from Mr. Pierson's "Further Considerations on Index-Numbers" (published in the same number of the Journal), on the ground that they ignore the character of Probabilities essential to the computation of index-numbers.]

.

.

It is justly observed by Adam Smith that the anxiety about public opinion is much greater among the candidates for excellence in some arts than it is in others. "The beauty of poetry is a matter of such nicety that a young beginner can scarce ever be certain that he has attained it. . . . Racine was so disgusted by the indifferent success of his Phèdre, that though in the vigour of his life and at the height of his abilities, he resolved to write no more for the stage. Mathematicians, on the contrary, who may have the most perfect assurance both of the truth and of the importance of their discoveries, are frequently very indifferent about the reception which they may meet with from the public.” 1 In the scale of susceptibility which is thus indicated, a high place must be assigned to the more refined parts of economic science. Even those investigations which at first sight appear to be wholly statistical-such as the calculation of index-numbers-may rest upon speculative assumptions, concerning which the consensus of authority is naturally desired. Accordingly, when the distinguished Dutch economist concludes in the immediately preceding paper that "all attempts to calculate and represent average movements of prices, either by index-numbers or otherwise, ought to be abandoned," those who have been making such attempts will anxiously reconsider the basis of their computation, and tremble for its safety. But the discouragement which such a condemnation coming from such an authority is calculated to produce may be mitigated by observing that the index-number which is the object of Mr. Pierson's crushing criticisms is one of a very peculiar character, differing in some essential attributes 1 Theory of Moral Sentiments, Part III. ch. ii.

from the operation as ordinarily conceived and practised. Racine
would not have been dejected by the indifferent success of his
tragedy if the play, so badly received, had been a version of his
masterpiece from which the characters of Phèdre and Hippolyte
had been left out. Two equally serious omissions are presupposed
by Mr. Pierson's animadversions.

There is, first, the character of probability. It is generally
implied that the problem now before us, in its data, method,
and result, is germane to the Calculus of Probabilities. The
nature of the problem is happily indicated by Professor Nicholson
when he compares the set of moving prices to a fleet of yachts
which under the influence of a common cause-it may be rising
wind or tide-are variously accelerated according to "the build
of the various yachts or seamanship of the crews." The type
of such problems is the investigation of what Mill calls a residual
phenomenon,1 illustrated by the discovery of the diurnal variation
in the height of the barometer by comparing the averages of a
great number of observations at different times of day. It is
postulated in such reasoning that the error or deviation of one
observation is independent of that which has been incurred by
another observation; 2 just as, when a die is thrown a number
of times, it may be assumed that the number of pips turned up
at each throw is unaffected by the preceding throws. It is true
that in concrete nature such ideal independence can hardly be
expected. Thus in barometrical observations it is possibly
not correct to treat the observation for each day as an independent
sample. Probably the weather sometimes follows suit for two
or three days together; but the deviation of the observations
is doubtless sufficiently random to justify Laplace's application
of the Calculus of Probabilities. So the grouping of human
statures is perhaps not perfectly sporadic; 3 but it is sufficiently
so to allow a Galton to infer with great probability that the con-
ditions of a particular class-e.g., boys in public schools, or men
in the Royal Society-as compared with less favoured classes.
are particularly favourable to growth. It is not necessary to

1 Mill, Logic, Book III. ch. xvii. Cp. Laplace, Probabilities, Book II. ch. v.
2 On this postulate see the present writer's "New Methods of Measuring
Variations in General Prices," Journal of the Royal Statistical Society, 1888,
p. 367, note. Cp. Laplace, loc. cit.: "Il faut avoir soin de varier les circonstances
de chaque observation."

As appears from the fact that in the group constituted by the measurements
of a nation there will be sub-classes with different averages.

Methods of Statistics," Journal of the Royal Statistical Society, Jubilee

discuss here whether the average would be of any scientific use if this condition of sporadic dispersion were not fulfilled-if all the observations were massed at two points, or collected into two sharply demarcated classes-e.g., dwarfs and giants.1 It is sufficient to observe that as a matter of fact the condition of sporadicity is very generally fulfilled both in physics and social phenomena: wherever there is at work a set of miscellaneous agencies, a mass of fleeting causes" in Mill's phrase.2

It is by ignoring this character of sporadic dispersion that Mr. Pierson's criticisms acquire their plausibility. He begins : "Let us suppose ten commodities, all equally important. Five of them are doubled in price, and five of them fall to exactly onehalf." But surely this is a very odd supposition, in view of the sporadic dispersion which very generally prevails in this world. It would have been more appropriate to suppose a number of figures representing variations of price (in one epoch as compared with another), not separately disposed in two heaps, but scattered about. Mr. Pierson's supposition would be appropriate if, for instance, Mr. Sauerbeck's percentages for the comparative prices of different commodities were massed at two points. But this is not so, as appears by considering his figures and diagrams representing annual or quarterly variations of price.3 A common trend comes out in the average, but the particular movements are independent.

The recognition of this sporadic character is fatal to Mr. Pierson's principal objection, which is in effect, though perhaps not apparently, that if the particular observations be weighted differently the average will be seriously different. This objection recurs in different forms. In his first paragraph Mr. Pierson supposes ten observations: five commodities of which the price has been doubled, five of which it has been halved; in the second as compared with the first period the data may accordingly be regarded as consisting of ten ratios, or percentages, five of them each = 200 (100); five of them each 50 (100). The simple arithmetical average of these may be written

=

1 See the reference given in the preceding note. See also p. 279 in the Memo. randum on Methods of Ascertaining and Measuring Changes in the Value of the Monetary Standard, by the present writer, published in the Report of the British Association for 1887. This Memorandum and the two supplementary ones, published in the Reports of the British Association for 1888 and 1889, should be referred to as containing justifications of statements made summarily in the present paper.

a Mill, Logic, loc. cit.

A similar scrutiny of Laspeyres' statistics of relative prices is attempted in the Memorandum of 1887 [H, above, p. 245.]