as the ‘Probable Error’;—a technical and decidedly misleading term. It is briefly defined as that error which we are as likely to exceed as to fall short of: otherwise phrased, if we were to arrange all the errors in the order of their magnitude, it corresponds to that one of them which just bisects the row. It is therefore the ‘median’ error: or, if we arrange all the magnitudes in successive order, and divide them into four equally numerous classes,—what Mr Galton calls ‘quartiles,’—the first and third of the consequent divisions will mark the limits of the ‘probable error’ on each side, whilst the middle one will mark the ‘median.’ This median, as was remarked, coincides, in symmetrical curves, with the arithmetical mean.
It is best to stand by accepted nomenclature, but the reader must understand that such an error is not in any strict sense ‘probable.’ It is indeed highly improbable that in any particular instance we should happen to get just this error: in fact, if we chose to be precise and to regard it as one exact magnitude out of an infinite number, it would be infinitely unlikely that we should hit upon it. Nor can it be said to be probable that we shall be within this limit of the truth, for, by definition, we are just as likely to exceed as to fall short. As already remarked (see note on [p. 441]), the ‘maximum ordinate’ would have the best right to be regarded as indicating the really most probable value.
§ 11. (5) The error of mean square. As previously suggested, the plan which would naturally be adopted by any one who had no concern with the higher mathematics of the subject, would be to take the ‘mean error’ for the purpose of the indication in view. But a very different kind of average is generally adopted in practice to serve as a test of the amount of divergence or dispersion. Suppose that we have the magnitudes x1, x2, … xn; their ordinary average is 1/n(x1 + x2 + … + xn), and their ‘errors’ are the differences between this and x1, x2, … xn. Call these errors e1, e2, … en, then the arithmetical mean of these errors (irrespective of sign) is 1/n(e1 + e2 + … + en). The Error of Mean Square,[3] on the other hand, is the square root of 1/n(e12 + e22 + … + en2).
The reasons for employing this latter kind of average in preference to any of the others will be indicated in the following chapter. At present we are concerned only with the general logical nature of an average, and it is therefore sufficient to point out that any such intermediate value will answer the purpose of giving a rough and summary indication of the degree of closeness of approximation which our various measures display to each other and to their common average. If we were to speak respectively of the ‘first’ and the ‘second average,’ we might say that the former of these assigns a rough single substitute for the plurality of original values, whilst the latter gives a similar rough estimate of the degree of their departure from the former.
§ 12. So far we have only been considering the general nature of an average, and the principal kinds of average practically in use. We must now enquire more particularly what are the principal purposes for which averages are employed.
In this respect the first thing we have to do is to raise doubts in the reader's mind on a subject on which he perhaps has not hitherto felt the slightest doubt. Every one is more or less familiar with the practice of appealing to an average in order to secure accuracy. But distinctly what we begin by doing is to sacrifice accuracy; for in place of the plurality of actual results we get a single result which very possibly does not agree with any one of them. If I find the temperature in different parts of a room to be different, but say that the average temperature is 61°, there may perhaps be but few parts of the room where this exact temperature is realized. And if I say that the average stature of a certain small group of men is 68 inches, it is probable that no one of them will present precisely this height.
The principal way in which accuracy can be thus secured is when what we are really aiming at is not the magnitudes before us but something else of which they are an indication. If they are themselves ‘inaccurate,’—we shall see presently that this needs some explanation,—then the single average, which in itself agrees perhaps with none of them, may be much more nearly what we are actually in want of. We shall find it convenient to subdivide this view of the subject into two parts; by considering first those cases in which quantitative considerations enter but slightly, and in which no determination of the particular Law of Error involved is demanded, and secondly those in which such determination cannot be avoided. The latter are only noticed in passing here, as a separate chapter is reserved for their fuller consideration.
§ 13. The process, as a practical one, is familiar enough to almost everybody who has to work with measures of any kind. Suppose, for instance, that I am measuring any object with a brass rod which, as we know, expands and contracts according to the temperature. The results will vary slightly, being sometimes a little too great and sometimes a little too small. All these variations are physical facts, and if what we were concerned with was the properties of brass they would be the one important fact for us. But when we are concerned with the length of the object measured, these facts become superfluous and misleading. What we want to do is to escape their influence, and this we are enabled to effect by taking their (arithmetical) average, provided only they are as often in excess as in defect.[4] For this purpose all that is necessary is that equal excesses and defects should be equally prevalent. It is not necessary to know what is the law of variation, or even to be assured that it is of one particular kind. Provided only that it is in the language of the diagram on [p. 29], symmetrical, then the arithmetical average of a suitable and suitably varied number of measurements will be free from this source of disturbance. And what holds good of this cause of variation will hold good of all others which obey the same general conditions. In fact the equal prevalence of equal and opposite errors seems to be the sole and sufficient justification of the familiar process of taking the average in order to secure accuracy.
§ 14. We must now make the distinction to which attention requires so often to be drawn in these subjects between the cases in which there respectively is, and is not, some objective magnitude aimed at: a distinction which the common use of the same word “errors” is so apt to obscure. When we talked, in the case of the brass rod, of excesses and defects being equal, we meant exactly what we said, viz.
that for every case in which the ‘true’ length (i.e.