VIL.—AVERAGES. 127 
form of average of the Y’s for any given year will afford an 
indication of the general level of prices for that year, provided the 
commodities chosen are sufficiently numerous and representative. 
The question is, what form of average to choose. If the geometric 
mean be chosen, and &,,, G,, denote the geometric means of the 
Y’s for the years 7 and 2 respectively, we have 
Go_(T Tn I's i 
Gy \Y') YI ; 
C5 2: SN 
ARR Se A LL 
- (Fy 7 BE) 
From the first form of this equation we see that the ratio of the 
geometric mean index-number in year 2 to that in year 7 is 
identical with the geometric mean of the ratios for the index- 
numbers of the several commodities. A similar property does 
not hold for any other form of average : the ratio of the arithmetic 
mean index-numbers is not the same as the arithmetic mean of 
the ratios, nor is the ratio of the medians the median of the 
ratios. From the second and third forms of the equation it 
appears further that the ratio of the geometric mean index- 
number in year 2 to that in year 7 is independent of the prices in 
the year first chosen as base (i.e. year 0), and is identical with the 
geometric mean of the index-numbers for year 2, on year 7 as 
base. Again, a similar property does not hold for any other form 
of average. If arithmetic means of the index-numbers be taken, 
for example, the ratio of the mean in year 2 to the mean in year 
1 will vary with the year taken as base, and will differ more or 
less from the arithmetic mean ratio of the prices in year 2 to the 
prices of the same commodities in year 7 ; the same statement is 
true if medians be used. The results given by the use of the 
geometric mean possess, therefore, a certain consistency that is 
not exhibited if other forms of average are employed. It was 
used in a classical paper by Jevons (ref. 4), though not on quite 
the same grounds, but has never been at all generally employed. 
26. The general use of the geometric mean has been suggested 
on another ground, namely, that the magnitudes of deviations 
appear, as a rule, to be dependent in some degree on the magni- 
tude of the average; thus the length of a mouse varies less than 
the stature of a man, and the height of a shrub less than that of 
a tree. Hence, it is argued, variations in such cases should be 
measured rather by their ratio to, than their difference from, the 
average ; and if this is done, the geometric mean is the natural 
average to use. If deviations be measured in this way, a