* There is much need of some good account, accessible to the ordinary English reader, of the nature and properties of the principal kinds of Mean. The common text-books of Algebra suggest that there are only three such, viz.[ TN: space] the arithmetical, the geometrical and the harmonical:—thus including two with which the statistician has little or nothing to do, and excluding two or more with which he should have a great deal to do. The best three references I can give the reader are the following. (1) The article Moyenne in the Dictionnaire des Sciences Médicales, by Dr Bertillon. This is written somewhat from the Quetelet point of view. (2) A paper by Fechner in the *Abhandlungen d. Math.[ TN: space] phys.[ TN: space] Classe d. Kön.[ TN: space] Sächs.[ TN: space] Gesellschaft d. Wiss.*[ TN: space] 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.[ TN: space] Phil.[ TN: space] Trans.[ TN: space] 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