his own health during the period of growth, his food during that period, the amount of outdoor life which he had, the amount and regularity of his sleep, etc. Let us represent these different conditions by different letters from A to J; let us suppose for the sake of simplicity that each of these conditions is either distinctly favorable or distinctly un

T

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favorable; that each condition is as likely to be favorable as to be unfavorable; and that one favorable or unfavorable condition counts for as much as another. Condition A will be favorable in one case out of two; and in cases when A is favorable B will be favorable in one out of two. That is to say, A and B will both be favorable in only one case out of four. Similarly they will both be unfavorable in only one case out of four. But in one case out of four A will be favorable and B unfavorable, and in one case out of four A will be unfavorable and B favorable. Hence with only two variables there is one case out of four when both conditions are good; one when both are bad; and two when one is good and the other is bad. Working the problem out in this way we find that all ten conditions are favorable (or unfavorable) in only one case out of 210, i.e., in 1 out of 1024; but that cases in which some of the conditions are favorable and others unfavorable occur much oftener, and that the more evenly the favorable and unfavorable conditions are divided

To students acquainted with the binomial theorem these figures and the process by which they are reached will be familiar. The mathematical law on which they are based represents the facts in most cases well enough to show how much more frequently things of medium size or quality are produced than those at either of the extremes. Thus the man of medium size or attainments is usually the 'average man' in two or even three senses of the words. He is the man who stands in about the middle of the line; he is the commonest kind of man; and very likely the number that expresses his size or attainments is nearly the arithmetical mean of the numbers which express the size or attainments of all those in the group.

Before we have a right to expect the arithmetical mean or the median of a group to be the commonest size or kind found in that group we must make sure that we are really dealing with one group of homogeneous things and not with several. If we measure a group of men half of whom are Americans and the other half African pygmies, the average height will be too small for the Americans and too large for the pygmies; and it might well be that not a single man in the whole complex group came anywhere near it. In the same way, if we should find the average size of all the articles

in a given room, from tables and lounges to pins and collarbuttons, there is no reason to think that things the size of the average—even if such things existed-would be any more common than things of any other size that any one might happen to think of. The things in the room are not homogeneous; and neither are the Americans and pygmies. To be homogeneous, in the sense in which the word is used here, things must be of the same general kind,—i.e., produced by essentially similar groups of causes,—and the differences between them must be the compounded' result of a large number of relatively independent conditions of approximately equal value. The articles in the room were not of the same kind at all; and in the case of the Americans and the pygmies the one condition of ancestry, different for the two groups, overshadowed all the rest.

Even when the members of a group are perfectly homogeneous it does not always follow that those of medium size or attainments are the most numerous. In a previous paragraph I said that in a regiment of soldiers arranged in line according to their height there might be a giant at one end. of the line and a dwarf at the other. But if the soldiers are regular infantrymen recruited in time of peace, the dwarf would not be there, simply because the government refuses to accept recruits under a certain height. The well-marked curve at one end of the line which touches the men's heads is thus cut off, and in a regiment recruited in this way the commonest type of man would therefore be a little to the small side of the median and a little smaller than the arithmetical mean. So in any class at school or college, the students who are laziest and most stupid have been cut off by previous examinations; and consequently at the lower end of the class we do not find one person of extraordinary ineffectiveness, but rather a fairly large group who have barely succeeded in fulfilling the minimum requirements. Here also, therefore, the largest group is towards the lower end and somewhat below the arithmetical average.

Another thing we must be careful about with averages of this sort is not to mistake the average for the ideal. If a 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 distinction 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