344: Illustrations of the Variate Diference Correlation Method 
which would have a substantial, but spurious value amounting to '316 if 
cr^ =0", = ... a J . If there be hiah correlation between the individual indices, of 
course, the correlation of each individual index with the arithmetic mean or 
synthetic index will also be high. Thus our Table IV shows it to range from "952 
in the case of Coffee to '998 in the case of Railways. Possibly a third of this 
correlation may in some cases have a spurious origin. But the individual indices 
are very highly correlated together ; only two such correlations are below "9, and 
the lower of these two is as high as "885. We are accordingly left with fifty 
correlations ranging from •885 to "997 between the individual indices, and if we 
accept these as true measures, then it is clear that any one of these ten indices 
might be used as a reasonable index of Italian prosperity; it would for practical 
purposes be idle to calculate them all or to table their arithmetic mean. 
But the high correlations found lay themselves open from our present stand- 
point to some suspicion of being solely due to the fact that during the 28 years 
under consideration Italy has progressively increased in population and ac- 
cordingly the consumption of innumerable goods and the means of interchange 
have all grown together with the time. In other words the correlations we give 
under the heading of " quantities " in each separate section of Table IV are very 
high solely because the individual indices are variates increasing one and all as 
continuous functions of the time. 
The material therefore seems especially suited to the application of the 
variate difference correlation method. For example, the correlation between the 
indices for tobacco and savings is "984 ; are we to interpret this to signify, that, 
if there are large savings this means that much will be spent on tobacco ? Or, 
is this high correlation simply in whole or part spurious, merely indicating that 
both savings and consumption of tobacco increased markedly with the time ? 
Actually the correlation of first differences drops from "984 to '766, that of 
second differences is negative if insensible, while from there onwards it steadily 
increases negatively, till with the sixth differences we reach — •431, which seems 
to indicate that, when time has been eliminated, expenditure on tobacco in any 
year means less money saved. Again the coffee and tobacco indices appear very 
highly correlated, "955, but by the third difference correlation we have reached 
about a third of the relationship, •319, which is scarcely altered in the sixth 
difference correlation, "326 ; we may assume therefore that there is probably a 
moderate " organic " relationship between the expenditures on coffee and tobacco, 
but the association is nothing like as close as would be suggested by the corre- 
lation of the raw indices. 
The work has been done in the following manner : The successive differences 
of the indices up to the sixth were found. The means and standard deviations 
of these differences were calculated, and the correlations were then worked out 
in the product-moment manner. This involved the very laborious work of 
determining 385 coefficients, and then to these coefficients were added the 
