If we inquire what is the mean in a mathematical point of view, the true answer is that there are several or many kinds of means. The old arithmeticians recognised ten kinds, which are stated by Boethius, and an eleventh was added by Jordanus.3
The arithmetic mean is the one by far the most commonly denoted by the term, and that which we may understand it to signify in the absence of any qualification. It is the sum of a series of quantities divided by their number, and may be represented by the formula 1/2(a + b). But there is also the geometric mean, which is the square root of the product, √a × b, or that quantity the logarithm of which is the arithmetic mean of the logarithms of the quantities. There is also the harmonic mean, which is the reciprocal of the arithmetic mean of the reciprocals of the quantities. Thus if a and b be the quantities, as before, their reciprocals are 1/a and 1/b, the mean of which is 1/2 (1/a + 1/b), and the reciprocal again is 2ab/a + b, which is the harmonic mean. Other kinds of means might no doubt be invented for particular purposes, and we might apply the term, as De Morgan pointed out,4 to any quantity a function of which is equal to a function of two or more other quantities, and is such that the interchange of these latter quantities among themselves will make no alteration in the value of the function. Symbolically, if Φ (y, y, y ....) = Φ (x1, x2, x3 ....), then y is a kind of mean of the quantities, x1, x2, &c.
The geometric mean is necessarily adopted in certain cases. When we estimate the work done against a force which varies inversely as the square of the distance from a fixed point, the mean force is the geometric mean between the forces at the beginning and end of the path. When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150 : 200 is not the same as 100 : 150. The mean ratio is as unity to √1·00 × 2·00 or 1 to 1·41. The difference between the three kinds of means in such a case5 is very considerable; while the rise of price estimated by the Arithmetic mean would be 50 per cent. it would be only 41 and 33 per cent. respectively according to the Geometric and Harmonic means.
In all calculations concerning the average rate of progress of a community, or any of its operations, the geometric mean should be employed. For if a quantity increases 100 per cent. in 100 years, it would not on the average increase 10 per cent. in each ten years, as the 10 per cent. would at the end of each decade be calculated upon larger and larger quantities, and give at the end of 100 years much more than 100 per cent., in fact as much as 159 per cent. The true mean rate in each decade would be 10√2 or about 1·07, that is, the increase would be about 7 per cent. in each ten years. But when the quantities differ very little, the arithmetic and geometric means are approximately the same. Thus the arithmetic mean of 1·000 and 1·001 is 1·0005, and the geometric mean is about 1·0004998, the difference being of an order inappreciable in almost all scientific and practical matters. Even in the comparison of standard weights by Gauss’ method of reversal, the arithmetic mean may usually be substituted for the geometric mean which is the true result.
Regarding the mean in the absence of express qualification to the contrary as the common arithmetic mean, we must still distinguish between its two uses where it gives with more or less accuracy and probability a really existing quantity, and where it acts as a mere representative of other quantities. If I make many experiments to determine the atomic weight of an element, there is a certain number which I wish to approximate to, and the mean of my separate results will, in the absence of any reasons to the contrary, be the most probable approximate result. When we determine the mean density of the earth, it is not because any