As soon, however, as we come to consider the case of asymmetrical, or lop-sided curves, the indications given by these three methods will be as a rule quite distinct; and therefore the two former of these deserve brief notice as representing different kinds of means from the arithmetical or ordinary one. We shall see that there is something about each of them which recommends it to common sense as being in some way natural and appropriate.
§ 5. (3) The first of these selects from amongst the various different magnitudes that particular one which is most frequently represented. It has not acquired any technical designation,[1] except in so far as it is referred to, by its graphical representation, as the “maximum ordinate” method. But I suspect that some appeal to such a mean or standard is really far from uncommon, and that if we could draw out into clearness the conceptions latent in the judgments of the comparatively uncultivated, we should find that there were various classes of cases in which this mean was naturally employed. Suppose, for instance, that there was a fishery in which the fish varied very much in size but in which the commonest size was somewhat near the largest or the smallest. If the men were in the habit of selling their fish by weight, it is probable that they would before long begin to acquire some kind of notion of what is meant by the arithmetical mean or average, and would perceive that this was the most appropriate test. But if the fish were sorted into sizes, and sold by numbers in each of these sizes, I suspect that this appeal to a maximum ordinate would begin to take the place of the other. That is, the most numerous class would come to be selected as a sort of type by which to compare the same fishery at one time and another, or one fishery with others. There is also, as we shall see in the next chapter, some scientific ground for the preference of this kind of mean in peculiar cases; viz.
where the quantities with which we deal are true ‘errors,’ in the estimate of some magnitude, and where also it is of much more importance to be exactly right, or very nearly right, than to have merely a low average of error.
§ 6. (4) The remaining kind of mean is that which is now coming to be called the “median.” It is one with which the writings of Mr Galton have done so much to familiarize statisticians, and is best described as follows. Conceive all the objects in question to be marshalled in the order of their magnitude; or, what comes to the same thing, conceive them sorted into a number of equally numerous classes; then the middle one of the row, or the middle one in the middle class, will be the median. I do not think that this kind of mean is at all generally recognized at present, but if Mr Galton's scheme of natural measurement by what he calls “per-centiles” should come to be generally adopted, such a test would become an important one. There are some conspicuous advantages about this kind of mean. For one thing, in most statistical enquiries, it is far the simplest to calculate; and, what is more, the process of determining it serves also to assign another important element to be presently noticed, viz.
the ‘probable error.’ Then again, as Fechner notes, whereas in the arithmetical mean a few exceptional and extreme values will often cause perplexity by their comparative preponderance, in the case of the median (where their number only and not their extreme magnitude is taken into account) the importance of such disturbance is diminished.
§ 7. A simple illustration will serve to indicate how these three kinds of mean coalesce into one when we are dealing with symmetrical Laws of Error, but become quite distinct as soon as we come to consider those which are unsymmetrical.
Suppose that, in measuring a magnitude along OBDC, where the extreme limits are OB and OC, the law of error is represented by the triangle BAC: the length OD will be at once the arithmetical mean, the median, and the most frequent length: its frequency being represented by the maximum ordinate AD. But now suppose, on the other hand, that the extreme lengths are OD and OC, and that the triangle ADC represents the law of error. The most frequent length will be the same as before, OD, marked by the maximum ordinate AD. But the mean value will now be OX, where DX = 1/3DC; and the median will be OY, where DY = (1 − 1/√2)DC.
Another example, taken from natural phenomena, may be found in the heights of the barometer as taken at the same hour on successive days. So far as 4857 of these may be regarded as furnishing a sufficiently stable basis of experience, it certainly seems that the resulting curve of frequency is asymmetrical. The mean height here was found to be 29.98: the median was 30.01: the most frequent height was 30.05. The close approximation amongst these is an indication that the asymmetry is slight.[2]
§ 8. It must be clearly understood that the average, of whatever kind it may be, from the mere fact of its being a single substitute for an actual plurality of observed values, must let slip a considerable amount of information. In fact it is only introduced for economy. It may entail no loss when used for some one assigned purpose, as in our example about the sheep; but for purposes in general it cannot possibly take the place of the original diversity, by yielding all the information which they contained. If all this is to be retained we must resort to some other method. Practically we generally do one of two things: either (1) we put all the figures down in statistical tables, or (2) we appeal to a diagram. This last plan is convenient when the data are very numerous, or when we wish to display or to discover the nature of the law of facility under which they range.