in the 17% of the several quantities. quiring fully into the grounds of this sential to observe that this one arithis really applied in at least three different 1-hit purposes, and upon different principles, t take great care not to confuse one applicathe process with another. A mente restilt, then, ve any one of the following significations.

It may give a merely representative number, pressing the general magnitude of a series of quantities, and serving as a convenient mole of comparing them with other series of quantities. Such a number is properly called The fictions mean or The average rest.

(2) It may give a result approximately free from disturbing quantities, which are known to affect some results in one direction, and other results equally in the opposite direction. We may say that in this case we get a Precise mean result.

(3) It may give a result more or less free from unknown and uncertain errors; this we may call the Probable

mean result.

Of these three uses of the mean the first is entirely different in nature from the two last, since it does not yield an approximation to any natural quantity, but furnishes us with an arithmetic result comparing the aggregate of certain quantities with their number. The third use of the mean rests entirely upon the theory of probability, and will be more fully considered in a later part of this chapter. The second use is closely connected, or even identical with, the Method of Reversal already described (p. 410), but it will be convenient to enter somewhat fully on all the three employments of the same arithmetical process.

E e

The significations of the terms Mean
and Average.

Much confusion exists in the popular, or even the scientific employment of the terms mean and average, and they are commonly taken as synonymous. It is desirable to ascertain carefully what significations we ought to attach to them. The English word mean is exactly equivalent to medium, being derived perhaps, through the French moyen, from the latin medius, which again is undoubtedly kindred with the Greek μeros. Etymologists believe, too, that this Greek word is connected with the preposition uera, the German mitte, and the true English mid or middle; so that after all the mean is a technical term identical in its root with the more popular equivalent middle.

If we inquire what is a mean in a mathematical point of view, the true answer is that there are several or many kinds of means. The old arithmeticians recognised at least ten kinds, which are stated by Boethius, and even an eleventh was added by Jordanus d.

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 any series of quantities divided by their number, and may be represented by the formula (a + b). But there is also the geometric mean, which is the square root of the product, axb, 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

d De Morgan, Supplement to the Penny Cyclopædia,' art. Old Appellations of Numbers.

quantities, as before, their reciprocals are and, the

I

a

mean of which is (+), and the reciprocal again is

a

2ab Other kinds of means might no doubt be invented

a+b

for particular purposes, and we might apply the term, as De Morgan pointed out, 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 o (y, y, y .

X3

= (x,, X2, X, ....), then y is a kind of mean of the quantities x,, x,, &c.

29

The geometric mean is necessarily adopted in certain cases. Thus 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 f. 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 of the one body (see p. 410).

In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. Thus if a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of 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 100 x 200 or 1 to 141. The difference between the three kinds of mean in such a case, as I have elsewhere shown, is very considerable, being as follows

e 'Penny Cyclopædia,' art. Mean.

f Thomson and Tait, Treatise on Natural Philosophy,' vol. i. p. 366. gJournal of the Statistical Society,' June 865, vol. xxviii. p. 296.

many

Wqpening the meat in the absence of express polkafon to the contrary as the common arithmetic mean we moet #1 distinguish between its two uses where it defines 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 reperiments to determine the specific gravity of a homogenoms pices of gold there is a certain definite ratio which I wish to approximate to, and the mean of my qurate results will, in the absence of any reasons to the contrary, be the most probable approximate result. When wo deforming on the other hand the mean density of the

earth, it is exceedingly unlikely that there is any part of the earth exactly of that density, and, as the crust is only about half the mean density, there must be other parts of greater density. I may also determine the mean specific gravity of a body composed of iron and gold, so that there will certainly be no portion possessing the mean density.

The very different signification of the word 'mean' in these two uses has been fully explained by M. Quetelet 1, and the importance of the distinction has moreover been pointed out by Sir John Herschel in reviewing his work i. It is much to be desired that scientific men would mark the difference by using the word mean only in the former sense when it denotes approximation to a definite existing quantity; and average, when the mean is only a fictitious quantity, used for the convenience of thought and expression. The etymology of this word 'average' is somewhat obscure; but according to De Morgank it comes from averia, ‘havings or possessions,' especially applied to farm stock. By the accidents of language averagium came to mean the labour of farm horses to which the lord was entitled, and it probably acquired in this manner the notion of distributing a whole into parts, a sense in which it was very early applied to maritime averages or contributions of the other owners of cargo to those whose goods have been thrown overboard or used for the safety of the vessel.

h Letters on the Theory of Probabilities,' transl. by Downes, Part ii, i Herschel's Essays,' &c. pp. 404, 405.

k 'On the Theory of Errors of Observations,' 'Cambridge Philosophical Transactions,' vol. x. Part ii. 416.