If a

Another thing we must be careful about with averages of this sort is not to mistake the average for the ideal. child's growth is not up to the average, the physician has a right to suspect, though perhaps not to conclude, that something is wrong; but then the average by which the physician is guided is an average of children in good health, and then again it is only when the child's weekly growth falls below the average-not above it—that the physician is anxious. Thus he regards the average growth as a kind of minimum-not as the maximum to be striven for. So also with matters of conduct, the fact that 'everybody' does a certain kind of thing is no reason in the world for believing that that is an ideal kind of thing to do. In the case of a race perfectly adjusted to its environment and incapable of further improvement it might be; but, as things stand, the commonplace of to-day is the ideal of yesterday, and the ideal of to-day is the commonplace, not of to-day, but of to-morrow. Another reason for striving for something better than the average in the case of conduct is this: The average is made up of good, bad, and indifferent; and if the best people in a community should suddenly cease to keep as far above the average as the bad are below it, the average would necessarily fall, and would keep on falling until the community went to pieces or until some one arose again who was willing to be better than the average of his fellows. The third purpose for which we find a mean is convenience in representation-to have "a merely representative numThird use ber, expressing the general magnitude of a series of quantities, and serving as a convenient mode of comparing them with other series of quantities", as in group comparisons. "Such a number is properly called the fictitious mean or the average result.” *

of average.

The average weight of the players in a football team may

*W. S. Jevons, "Principles of Science" (1887), p. 359. The dis tinction which Jevons here makes between the use of the words Mean and Average is not always observed, and I have ignored it in the text.

not come anywhere near the weight of any one of them, and it is not a kind of type towards which football-players tend; for there is reason why the quarter-backs should usually be lighter and the centres heavier. There is therefore no one thing in the world which the mean employed in this way represents or attempts to represent, and yet it has a real use when we consider the group as a whole in its relations to something beyond: in this case in relation to some other football team and the chance of beating it. So when we give the mean temperature of Winnipeg we do not mean to say that that is the commonest temperature there, nor yet that that is a kind of type which the temperature of each day naturally tends to approach; for we know that most days are either hotter or colder and that it is natural for days to be much hotter in summer and much colder in winter. But with reference to places and relations that lie beyond, a statement of the average temperature may be full of meaning. If the mean temperature of Winnipeg is lower than that of San Francisco, this means that for some reason or other it receives less heat from the sun in the course of a year, or radiates more away, or perhaps both; and if there is any process of growth or manufacture which depends upon the total amount of heat (regardless of variations from day to day) which Nature gives in the course of a whole year, a knowledge of the mean temperature of each place would tell which of the two would be the more favorable in this respect. To quote again from Jevons:

Although the average when employed in its proper sense of a fictitious mean represents no really existing quantity, it is yet of the highest scientific importance, as enabling us to conceive in a single result a multitude of details. It enables us to make a hypothetical simplification of a problem, and avoid complexity without committing error. The weight of a body is the sum of the weights of infinitely small particles, each acting at a different place, so that a mechanical problem resolves itself, strictly speaking, into an infinite

number of distinct problems. We owe to Archimedes the first introduction of the beautiful idea that one point may be discovered in a gravitating body such that the weight of all the particles may be regarded as concentrated in that point, and yet the behavior of the whole body will be exactly represented by the behavior of this heavy point. This Centre of Gravity may be within the body, as in the case of a sphere, or it may be in empty space, as in the case of a ring. Any two bodies, whether connected or separate, may be conceived as having a centre of gravity, that of the sun and earth lying within the sun and only 267 miles from its centre.

While averages of this sort can represent the individuals in a group for certain purposes, it is only as members of the group. The average weight of the men in any athletic team is nothing more than the figure obtained by dividing the total weight by the number of players. The minute any one of them leaves the team that average ceases absolutely to represent him in any way whatever, and ceases at the same instant to represent the others either, whether individually or collectively. An average of this sort need not be in any sense either a representative of a single individual, or of a type towards which the individuals tend, or of an ideal. It is the mere product of an arithmetical process, useful for the estimation of certain outward relations of the things averaged.

The term Expectation of Life as used in insurance is likely to lead to the confusing of the two ideas which we are here trying to distinguish. To the insurance company it means merely the average time that insurable people of a given age and sex continue to live. To the layman it is likely to mean the time that he, as an individual, will probably continue to live-a very different thing, which should be calculated in an entirely different way.

*Jevons, op. cit., pp. 363-4.

Often it is well to add to an average some indication of the accuracy with which the average represents the quantities Ten is the arithmetical Measures of

11. It is also the arith

error.

whose average it is. mean between 9 and metical mean between 5 and 15. But in the first case the average comes much nearer to each of the .separate quantities than in the second. In the first case the difference between the average and each of the quantities averaged is only 1; in the second it is 5. When the average represents a large number of quantities, the simplest measure of the difference between it and each one of the quantities averaged is the average variation of the separate quantities from that average. The arithmetical average of the variations is found by finding the difference between the average and each one of the separate quantities (regardless of whether that quantity be larger than the average or smaller), adding all these differences together, and dividing by the total number of quantities. Thus the average of 5, 6, 7, 11, 13, 8, 6, 20, 10, 14 is 10; the separate variations from the average are respectively 5, 4, 3, 1, 3, 2, 4, 10, 0, 4; the sum of these separate variations is 36; and since there are ten quantities, the average variation is 3.

6.

When we are dealing with a number of separate quantities a knowledge of this average variation enables us to tell to what extent the average may be regarded as representative of each of them, and as thus serving the second purpose of an average, and to what extent, on the contrary, it must be regarded as a purely fictitious quantity serving the third purpose only. Of course the smaller the average variation the more accurately the average represents the separate quantities averaged.

When we are dealing with different measurements of the same quantity the average variation of the separate measurements from the average gives a measure of their accuracy. To be sure it does not tell anything about 'constant' or 'systematic' errors which affect all the measurements in the

same way; but it does tell how much importance must be attached to 'accidental' errors, or those which result from a large number of different causes and are as likely to affect a measurement in one direction as in the other. The larger the average variation, the more important are these 'accidental' errors and the less can we rely upon an average derived from a small number of measurements.

This average variation is easily found, and it is a good enough measure of error for some purposes; but mathematicians do not use it. What they do use is either the Median Error-generally known as the 'Probable Error'— or the Mean Square Error. The former is commonly used in English-speaking countries, the latter in Germany.

The Median Error, or so-called Probable Error, is the variation from the mean that half the separate measurements fall short of and the other half exceed. If we suppose all the separate measurements to be arranged in order of magnitude, the central quantity is the median, and the Median Error is the difference between that central quantity and the quantity half-way between it and the end of the line in either direction. The Probable Error' is thus the amount of error that any one of the quantities is as likely to fall short of as to exceed. It is not the amount of error or variation from the mean that will probably be made. If we denote the difference between each individual measurement and the mean (i.e., the 'errors' or 'residuals') by v1, v1⁄2, v, etc., and the total number of measurements by n, the formula for finding the Probable Error (r) of a single observation is this: