342 Illustrations of the Variate Difference Correlation Method 
Now such relations will undoubtedly be very approximately true, if the X's 
are random variates uncorrelated to each other, and provided on is small compared 
with n. These conditions seem amply satisfied when we proceed to fourth or sixth 
differences in barometric pressures, taken, say, over ten or twelve years ; the 
addition of four or five daily pressures will hardly affect sensibly either the mean 
or the standard deviation. But such extensive data, while not only involving a great 
deal of labour in the difference work* ai'e not those which, perhaps, most frequently 
demand the attention of the statistician, whether he be economist, sociologist 
or a student of scientific agriculture. In such cases it not infrequently happens 
that the available data only provide a range of 20 to, perhaps, at most 50 years ; 
and we need to discover whether there is a true relationship between our variates, 
apart from a continuous change in both due to the time factor. At present 
accurate statistics of annual trade or revenue, or satisfactory annual demographic 
data hardly extend at most beyond a period of 50 years. Very often — under 
even approximately like methods of record — we shall hardly have more than 
twenty years' trustworthy returns. Not only has the method of record been 
changed, but the conditions of transit and trade may have been immensely 
modified and in a manner which we could not suppose to be even approximately 
represented by a continuous function of the time. 
The object of the present paper is to illustrate the theory of the variate 
difference correlation method in its present stage of development on a short series 
of economic data, in order to test what approximation there is in such short series 
to stability, and further how nearly Dr Anderson's values for the successive 
standard deviations apply to such cases. We have selected as our data ten 
economic indices of Italian prosperity for the years 1885 to 1912, together with a 
" Synthetic Index," formed by taking the arithmetic mean of the ten economic 
indices referred to. These eleven indices are given by Professor Georgio Mortara 
in an interesting memoir : " Sintomi statistici delle condizione economiche 
d' Italia " which was published in the Giornale degli Economisti e Rivista di 
Statistica, for February, 1914, and form Tabella I, of that memoir, which we 
here reproduce in part as Table I. The indices in each case are obtained by 
dividing the returns for any year by the means of the returns for the years 
1901 — 05, inclusive, and multiplying as usual by 100. 
The indices are for returns of (i) Gross Receipts of Railways, (ii) Shipping, 
loaded and unloaded at the ports, (iii) Effective Revenue of the State, (iv) Inter- 
national Commerce, Value of Imports and Exports, (v) Number of postal 
Letters and private Telegrams, (vi) Amount of Stamp Duties, (vii) Savings Banks' 
Returns, (viii) Impoitation of Coal, (ix) Gross returns of consumption of Tobacco, 
(x) Returns of Coffee imported. Professor Mortara has drawn attention to the 
very high correlations of these individual indices with each other, and of each of 
them with the " Synthetic Index." The latter correlation is, however, to a certain 
* A discussion of tlie correlations of the higher differences in barometric pressures will we hope be 
shortly issued. 
