i= THEORY OF STATISTICS. 
i.e. the geometric mean of the numbers given by the two censuses. 
This result must, however, be used with discretion. The rate of 
increase of population is not necessarily, or even usually, constant 
over any considerable period of time: if it were so, a curve 
representing the growth of population as in fig. 24 would be 
continuously convex to the base, whether the population were 
increasing or decreasing. In the diagram it will be seen that 
the curves are frequently concave towards the base, and similar 
results will often be found for districts in which the population is 
not increasing very rapidly, and from which there is much 
emigration. Further, the assumption is not self-consistent in any 
case in which the rate of increase is not uniform over the entire 
area—and almost any area can be analysed into parts which are not 
similar in this respect. For if in one part of the area considered 
the initial population is P, and the common ratio R, and in the 
remainder of the area the initial population is p, and the common 
ratio r, the population in year = is given by 
Pt p,= Po B+ por”. 
This does not represent a constant rate of increase unless B=. 
If then, for example, a constant percentage rate of increase be 
assumed for England and Wales as a whole, it cannot be assumed 
for the Counties: if it be assumed for the Counties, it cannot be 
assumed for the country as a whole. The student is referred to 
refs. 14, 15 for a discussion of methods that may be used for the 
consistent estimation of populations under such circumstances. 
- 25. The property of the geometric mean illustrated by equation 
(13) renders it, in some respects, a peculiarly convenient form of 
average in dealing with ratios, 7.e. “index-numbers,” as they are 
termed, of prices. Let 
Kon wo i val 
EX i te 
Xn ) 
denote the prices of &/ commodities in the years 0,1, 2 . . . 
Further, let ¥,,=X,/X,, and so on, so that 
Y' 10 Yi Y 4 dele TT, 
Yop Yop Yop + + + + Yn 
represent the ratios of the prices of the several commodities in years 
1, 2, . . . to their prices in year 0. These ratios, in practice 
multiplied by 100, are termed index-numbers of the prices of the 
several commodities, on the year 0 as base. Evidently some 
+26