annually? The circumstances are not quite the same as in the former case, but the analogy is sufficiently close for our purpose. The answer is decidedly, No. If 100 articles of any kind are sold for £100, we say that the average price is £1. By this we mean that the total amount is the same whether the entire lot are sold for £100, or whether we split the lot up into individuals and sell each of these for £1. The average price here is a convenient fictitious substitute, which can be applied for each individual without altering the aggregate total. If therefore the question be, Will a supposed increase of 1 p. c.
in each of the 100 years be equivalent to a total increase to double the original amount?
we are proposing a closely analogous question. And the answer, as just remarked, must be in the negative. An annual increase of 1 p. c.
continued for 100 years will more than double the total; it will multiply it by about 2.7. The true annual increment required is measured by 100√2; that is, the population may be said to have increased ‘on the average’ 0.7 p. c.
annually.
We are thus directed to the second kind of average discussed in the ordinary text-books of algebra, viz.
the geometrical. When only two quantities are concerned, with a single intermediate value between them, the geometrical mean constituting this last is best described as the mean proportional between the two former. Thus, since 3 : √15 :: √15 : 5, √15 is the geometrical mean between 3 and 5. When a number of geometrical means have to be interposed between two quantities, they are to be so chosen that every term in the entire succession shall bear the same constant ratio to its predecessor. Thus, in the example in the last paragraph, 99 intermediate steps were to be interposed between 1 and 2, with the condition that the 100 ratios thus produced were to be all equal.
It would seem therefore that wherever accurate quantitative results are concerned, the selection of the appropriate kind of average must depend upon the answer to the question, What particular intermediate value may be safely substituted for the actual variety of values, so far as the precise object in view is concerned? This is an aspect of the subject which will have to be more fully considered in the next chapter. But it may safely be laid down that for purposes of general comparison, where accurate numerical relations are not required, almost any kind of intermediate value will answer our purpose, provided we adhere to the same throughout. Thus, if we want to compare the statures of the inhabitants of different counties or districts in England, or of Englishmen generally with those of Frenchmen, or to ascertain whether the stature of some particular class or district is increasing or diminishing, it really does not seem to matter what sort of average we select provided, of course, that we adhere to the same throughout our investigations. A very large amount of the work performed by averages is of this merely comparative or non-quantitative description; or, at any rate, nothing more than this is really required. This being so, we should naturally resort to the arithmetical average; partly because, having been long in the field, it is universally understood and appealed to, and partly because it happens to be remarkably simple and easy to calculate.
§ 4. The arithmetical mean is for most ordinary purposes the simplest and best. Indeed, when we are dealing with a small number of somewhat artificially selected magnitudes, it is the only mean which any one would think of employing. We should not, for instance, apply any other method to the results of a few dozen measurements of lengths or estimates of prices.
When, however, we come to consider the results of a very large number of measurements of the kind which can be grouped together into some sort of ‘probability curve’ we begin to find that there is more than one alternative before us. Begin by recurring to the familiar curve represented on [p. 29]; or, better still, to the initial form of it represented in the next chapter ([p. 476]). We see that there are three different ways in which we may describe the vertex of the curve. We may call it the position of the maximum ordinate; or that of the centre of the curve; or (as will be seen hereafter) the point to which the arithmetical average of all the different values of the variable magnitude directs us. These three are all distinct ways of describing a position; but when we are dealing with a symmetrical curve at all resembling the binomial or exponential form they all three coincide in giving the same result: as they obviously do in the case in question.