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 whose average it is. Ten is the arithmetical Measures of mean between 9 and 11. It is also the arith

error.

But in the first case the

metical mean between 5 and 15. 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, 2, 3, etc., and the total number of measurements by n, the formula for finding the Probable Error (r) of a single observation is this:

The formula for the Mean Square Error of a single measurement, e, or of the mean, e, is the same as that for r or 。, except that the factor .6745 is omitted.

The calculated error of a measurement is usually written after it, thus: 1287 ±3. Unfortunately, however, this is ambiguous, for sometimes the error indicated in this way is the error of any one measurement out of a series, and sometimes it is the error of the mean. By giving the latter where one expects the former an observer sometimes makes his results appear more accurate than they really are.

The student who wishes a fuller treatment of this subject is referred to Venn's "Logic of Chance", to Jevons' 'Principles of Science", or to some one of the many mathematical treatises on the theory of probability, such as Merriman's or Comstock's.

CHAPTER XXXII.

PROBABILITY.

In the last chapter we found that where several measurements or estimates do not agree it is often of practical advantage to assume that the truth lies somewhere between

Why needed.

them, and therefore to find out the average. We found, however, that sometimes the nature of the case is such that an average is out of the question. If two people both claim the same piece of land, it would hardly do to say that each of them owns half of it; if one toss of a penny gives heads and another gives tails, there is no practical purpose which will be served by assuming that the natural position of the penny is neither with heads up nor with tails up, but on its edge; if we do not know whether a certain act will please a person or annoy him, it will hardly do to assume that it will do neither one nor the other. In cases of this sort the mean is almost sure to be wrong, and so far wrong as to serve no practical use; it is therefore excluded and we have to choose between the extremes. theory of probability discusses this choice.

The

To the ordinary human mind and to all of the brutes this choice between extremes is often more natural than the search for a mean. If we do not know whether to fight a certain enemy as hard as we can or to run away from him as hard as we can, it is usually better to do one or the other-no matter which-than to follow the middle course and sit still and wait to be devoured; and, fashioned as we are for the