1878; pp. 1–76. This contains a very interesting discussion, especially for the statistician, of a number of different kinds of mean. His account of the median is remarkably full and valuable. But little mathematical knowledge is demanded. (3) A paper by Mr F. Y. Edgeworth in the Camb.
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for 1885, entitled Observations and Statistics. This demands some mathematical knowledge. Instead of dealing, as such investigations generally do, with only one Law of Error and with only one kind of mean, it covers a wide field of investigation.
§ 1. We have had such frequent occasion to refer to averages, and to the kind of uniformity which they are apt to display in contrast with individual objects or events, that it will now be convenient to discuss somewhat more minutely what are the different kinds of available average, and what exactly are the functions they perform.
The first vague notion of an average, as we now understand it, seems to me to involve little more than that of a something intermediate to a number of objects. The objects must of course resemble each other in certain respects, otherwise we should not think of classing them together; and they must also differ in certain respects, otherwise we should not distinguish between them. What the average does for us, under this primitive form, is to enable us conveniently to retain the group together as a whole. That is, it furnishes a sort of representative value of the quantitative aspect of the things in question, which will serve for certain purposes to take the place of any single member of the group.
It would seem then that the first dawn of the conception which science reduces to accuracy under the designation of an average or mean, and then proceeds to subdivide into various distinct species of means, presents itself as performing some of the functions of a general name. For what is the main use of a general name? It is to reduce a plurality of objects to unity; to group a number of things together by reference to some qualities which they possess in common. The ordinary general name rests upon a considerable variety of attributes, mostly of a qualitative character, whereas the average, in so far as it serves the same sort of purpose, rests rather upon a single quantitative attribute. It directs attention to a certain kind and degree of magnitude. When the grazier says of his sheep that ‘one with another they will fetch about 50 shillings,’ or the farmer buys a lot of poles which ‘run to about 10 feet,’ it is true that they are not strictly using the equivalent of either a general or a collective name. But they are coming very near to such use, in picking out a sort of type or specimen of the magnitude to which attention is to be directed, and in classing the whole group by its resemblance to this type. The grazier is thinking of his sheep: not in a merely general sense, as sheep, and therefore under that name or conception, but as sheep of a certain approximate money value. Some will be more, some less, but they are all near enough to the assigned value to be conveniently classed together as if by a name. Many of our rough quantitative designations seem to be of this kind, as when we speak of ‘eight-day clocks’ or ‘twelve-stone men,’ &c.; unless of course we intend (as we sometimes do in these cases) to assign a maximum or minimum value. It is not indeed easy to see how else we could readily convey a merely general notion of the quantitative aspect of things, except by selecting a type as above, or by assigning certain limits within which the things are supposed to lie.
§ 2. So far there is not necessarily any idea introduced of comparison,—of comparison, that is, of one group with another,—by aid of such an average. As soon as we begin to think of this we have to be more precise in saying what we mean by an average. We can easily see that the number of possible kinds of average, in the sense of intermediate values, is very great; is, in fact, indefinitely great. Out of the general conception of an intermediate value, obtained by some treatment of the original magnitudes, we can elicit as many subdivisions as we please, by various modes of treatment. There are however only three or four which for our purposes need be taken into account.
(1) In the first place there is the arithmetical average or mean. The rule for obtaining this is very simple: add all the magnitudes together, and divide the sum by their number. This is the only kind of average with which the unscientific mind is thoroughly familiar. But we must not let this simplicity and familiarity blind us to the fact that there are definite reasons for the employment of this average, and that it is therefore appropriate only in definite circumstances. The reason why it affords a safe and accurate intermediate value for the actual divergent values, is that for many of the ordinary purposes of life, such as purchase and sale, we come to exactly the same result, whether we take account of those existent divergences, or suppose all the objects equated to their average. What the grazier must be understood to mean, if he wishes to be accurate, by saying that the average price of his sheep is 50 shillings, is, that so far as that flock is concerned (and so far as he is concerned), it comes to exactly the same thing, whether they are each sold at different prices, or are all sold at the ‘average’ price. Accordingly, when he compares his sales of one year with those of another; when he says that last year the sheep averaged 48 shillings against the 50 of this year; the employment of this representative or average value is a great simplification, and is perfectly accurate for the purpose in question.
§ 3. (2) Now consider this case. A certain population is found to have doubled itself in 100 years: can we talk of an ‘average’ increase here of 1 per cent.