CHAPTER XVI.. The Method of Means.

THE METHOD OF MEANS.

All results of the measurement of continuous quantity can be only approximately true. Were this assertion doubted, it could readily be proved by direct experience. If any person, using an instrument of the greatest precision, makes and registers successive observations in an unbiassed manner, it will almost invariably be found that the results differ from each other. When we operate with sufficient care we cannot perform so simple an experiment as weighing an object in a good balance without getting discrepant numbers. Only the rough and careless experimenter will think that his observations agree, but in reality he will be found to overlook the differences. The most elaborate researches, such as those undertaken in connection with standard weights and measures, always render it apparent that complete coincidence is out of the question, and that the more accurate our modes of observation are rendered, the more numerous are the sources of minute error which become apparent. We may look upon the existence of error in all measurements as the normal state of things. It is absolutely impossible to eliminate separately the multitude of small disturbing influences, except by balancing them off against each other. Even in drawing a mean it is to be expected that we shall come near the truth rather than exactly to it. In the measurement of continuous quantity, absolute coincidence, if it seems to occur, must be only apparent, and is no indication of precision. It is one of the most embarrassing things we can meet when experimental results agree too closely. Such coincidences should raise our suspicion that the apparatus in use is in some way restricted in its operation, so as not really to give the true result at all, or that the actual results have not been faithfully recorded by the assistant in charge of the apparatus.

If then we cannot get twice over exactly the same result, the question arises, How can we ever attain the truth or select the result which may be supposed to approach most nearly to it? The quantity of a certain phenomenon is expressed in several numbers which differ from each other; no more than one of them at the most can be true, and it is more probable that they are all false. It may be suggested, perhaps, that the observer should select the one observation which he judged to be the best made, and there will often doubtless be a feeling that one or more results were satisfactory, and the others less trustworthy. This seems to have been the course adopted by the early astronomers. Flamsteed, when he had made several observations of a star, probably chose in an arbitrary manner that which seemed to him nearest to the truth.

Baily’s *Account of Flamsteed*, p. 376.

When Horrocks selected for his estimate of the sun’s semi-diameter a mean between the results of Kepler and Tycho, he professed not to do it from any regard to the idle adage, “Medio tutissimus ibis,” but because he thought it from his own observations to be correct. But this method will not apply at all when the observer has made a number of measurements which are equally good in his opinion, and it is quite apparent that in using an instrument or apparatus of considerable complication the observer will not necessarily be able to judge whether slight causes have affected its operation or not.

*The Transit of Venus across the Sun*, by Horrocks, London, 1859, p. 146.

In this question, as indeed throughout inductive logic, we deal only with probabilities. There is no infallible mode of arriving at the absolute truth, which lies beyond the reach of human intellect, and can only be the distant object of our long-continued and painful approximations. Nevertheless there is a mode pointed out alike by common sense and the highest mathematical reasoning, which is more likely than any other, as a general rule, to bring us near the truth. The ἄριστον μέτρον, or the *aurea mediocritas*, was highly esteemed in the ancient philosophy of Greece and Rome; but it is not probable that any of the ancients should have been able clearly to analyse and express the reasons why they advocated the *mean* as the safest course. But in the last two centuries this apparently simple question of the mean has been found to afford a field for the exercise of the utmost mathematical skill. Roger Cotes, the editor of the *Principia*, appears to have had some insight into the value of the mean; but profound mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and others, have hardly exhausted the subject.

*Several uses of the Mean Result.*

The elimination of errors of unknown sources, is almost always accomplished by the simple arithmetical process of taking the *mean*, or, as it is often called, the *average* of several discrepant numbers. To take an average is to add the several quantities together, and divide by the number of quantities thus added, which gives a quotient lying among, or in the *middle* of, the several quantities. Before however inquiring fully into the grounds of this procedure, it is essential to observe that this one arithmetical process is really applied in at least three different cases, for different purposes, and upon different principles, and we must take great care not to confuse one application of the process with another. A *mean result*, then, may have any one of the following significations.

(1) It may give a merely representative number, expressing the general magnitude of a series of quantities, and serving as a convenient mode of comparing them with other series of quantities. Such a number is properly called *The fictitious mean* or *The average result*.

(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, but it will be desirable to enter somewhat fully into all the three employments of the same arithmetical process.

*The Mean and the 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 necessary to ascertain carefully what significations we ought to attach to them. The English word *mean* is equivalent to *medium*, being derived, perhaps through the French *moyen*, from the Latin *medius*, which again is undoubtedly kindred with the Greek μεσος. Etymologists believe, too, that this Greek word is connected with the preposition μετα, 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 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.

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

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 (2*ab*)/(*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, 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(*x*{1}, *x*{2}, *x*{3} ....), then *y* is a kind of mean of the quantities, *x*{1}, *x*_{2}, &c.

*Penny Cyclopædia*, art. *Mean*.

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 case 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.

Jevons, *Journal of the Statistical Society*, June 1865, vol. xxviii, p. 296.

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 part of the earth is of that exact density; there may be no part exactly corresponding to the mean density, and as the crust of the earth has only about half the mean density, the internal matter of the globe must of course be above the mean. Even the density of a homogeneous substance like carbon or gold must be regarded as a mean between the real density of its atoms, and the zero density of the intervening vacuous space.

The very different signification of the word “mean” in these two uses was fully explained by Quetelet, and the importance of the distinction was pointed out by Sir John Herschel in reviewing his work. 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 convenience of thought and expression. The etymology of this word “average” is somewhat obscure; but according to De Morgan 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 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.

*Letters on the Theory of Probabilities*, transl. by Downes,