that difference of one pound? If we take account of a difference in weight of one hundredth of a pound, should we not take account of a difference in age sufficient to produce that difference of one hundredth of a pound? much is this?

How

According to the tables the average weight of all the boys examined who are eight at their nearest birthday is 52.39 pounds; the average weight of all the boys who are nine at their nearest birthday is 57.41 pounds; and the average weight of all the boys who are ten at their nearest birthday is 62.38 pounds. This means that the boys gain about five pounds a year, or about a tenth of a pound a week, and the hundredth part of a pound in less than a day.*

This law of average growth means that we cannot draw any conclusion from an average difference in weight of one tenth of a pound between two groups of children of the same age', unless we have good reason to believe that the age really is the same' not merely to a year but to a week. A difference of a week in age would account for a difference of a tenth of a pound in weight. In the same way a difference of ten weeks in age would account for a difference of a pound in weight, and a difference of twenty weeks for the difference of two pounds which the tables show between the boys of nine in any two successive grades.

And now the question comes: Have we a right to believe that there is no such difference as this in the ages?

In the tables before us the children are grouped according to their age in years at their nearest birthday. No account is taken of months or days. In each group, then, there will be some children who are almost a year older than some others in the same group. But since it is fair to assume that there are about as many children a little under a given age

* We assume here for the sake of simplicity that the growth is uniform throughout the year. If we took account of the fact that it is not it would complicate the argument, but it would not affect the principle on which it is based.

as a little over it, the average age of all the children called nine would really be almost exactly nine; and so with each of the other ages. In this way we have a right to assume that the difference in average weight between all the boys called nine and all the boys called ten corresponds to a very definite difference in age of almost precisely one year. Thus this inference based upon the tables is perfectly correct, and we have a right to say that it really is a difference of one year in age which makes the difference of about five pounds in weight.

This, however, is very different from saying that the boys called ten in any one grade are on the average a year older than the boys called nine in the same grade, or that the boys called nine in one grade are on the average precisely as old as the boys called nine in another.

In fact the presumption is all the other way. A boy exactly eight years and six months of age is quite as likely to be in a grade with the boys of eight at their nearest birthday as with the boys of nine; and a boy of eight years and seven months is almost as likely to be. On the other hand a boy of nine years and six months is quite as likely to be with the boys of ten as with the boys of nine, and a boy of nine years and five months is almost as likely to be. In this way one boy of nine might easily be two grades ahead of another, not because he is any better developed for his age, but merely because he is ten or eleven months older. And thus, in general, there is every reason to believe that the boys of a given age in a higher grade are considerably older on the average than those of the same age' in a lower grade; and the difference in age might well account for a large part of the difference in weight. It could not account for a difference of five pounds, of course; for the difference in age must always be less than a year. Hence it could not account for all the difference which is found between boys of nine in the first grade and those in the fourth; but it might account for enough of it to make the conclusion that bright children are

larger than dull ones extremely doubtful. Precisely how much the actual difference in age really will account for we cannot tell until the age of each child is taken accurately enough to show precisely what this difference is.

There is a difference between this example of conclusions too precise for the data upon which they are based and the one given before it. When we compared two individuals

But

and concluded that the horse B was faster than the horse A because it trotted a few feet farther in the same time', our conclusion may have been wrong, but it may also have been right, since the times really may have been the same. in this other case when we compared several groups of individuals and said that the members of one were so much heavier for their age than the members of another, our conclusion was certainly wrong, since there is every reason for believing that the average ages of the members of the different groups were not at all the same. When the investigator compared the average weight of all the boys of one age with that of all the boys of another he had a perfect right to take the age of each individual very much more roughly than he would have done if he had been comparing two individual boys, because he had good reason to believe that the inaccuracies would balance each other. In this the method of group comparisons has a great advantage. But the very fact that the inaccurate measurements were good enough for one set of comparisons made him take for granted that they were good enough for another. Thus the peculiar advantage possessed by this method of group comparisons may conceal a great danger.

CHAPTER XXXI.

MEANS, OR AVERAGES.

In the last chapter we had occasion to show how an inference could be based upon a comparison of averages. Averages are used so much in various kinds of reasoning that a few definite statements should be made about them.

General

"The first vague notion of an average, as we now understand it, seems to me to involve little more than that of a something intermediate to a number of objects. The objects must of course resemble each other in certain conception. respects. Otherwise we should not think of classifying them together; and they must also differ in certain respects, otherwise we should not distinguish between them. What the average does for us, under this primitive form, is to enable us conveniently to retain the group together as a whole. That is, it furnishes a sort of representative value of the quantitative aspect of the things in question, which will serve for certain purposes to take the place of any single member of the group.'* In this respect an average is somewhat, though not precisely, like a general name. "The ordinary general name rests upon [i.e., is used to mark] a considerable variety of attributes, mostly of a qualitative character, whereas the average, in so far as it serves the same sort of purpose, rests rather upon a single quantitative attribute. It directs attention to a certain kind and degree of magnitude.

*John Venn, "The Logic of Chance", 1888, pp. 436 ff.

"We can easily see that the number of possible kinds of average, in the sense of intermediate values, is very great; is, in fact, indefinitely great. Out of the general conception of an intermediate value, obtained by some treatment of the original magnitudes, we can elicit as many subdivisions as we please, by various modes of treatment. There are, however, only three or four which for our purposes need to be taken into account.

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mean.

In the first place there is the arithmetical average or The rule for obtaining this is very simple: add all the magnitudes together, and divide the sum by their number. This is the only kind of average with which the unscientific mind is thoroughly familiar. But we must not let this simplicity and familiarity blind us to the fact that there are definite reasons for the employment of this average, and that it is therefore appropriate only in definite circumstances.''

kinds.

The Arithmetical Mean of a series of quantities is that quantity which can be substituted for each one of them when they are to be added together, and produce the same sum. Six is the arithmetical mean of 4, 5, 7, 8, Various because the sum of these four numbers and the sum of four sixes is the same. Hence "for many of the ordinary purposes of life, such as purchase and sale, we come to exactly the same result, whether we take account of "* the exact size of each separate quantity and the differences between them, or suppose each one of them to be equal to the average. If we are paying for melons by the pound it makes no difference in the price whether the dealer says that we bought one which weighed 4 pounds, one which weighed 5, one which weighed 7, and one which weighed 8, or whether he says we bought four that weighed about 6 pounds apiece.t The next kind of mean, or average, to be considered is the Geometrical. It is that quantity which can be substituted

* John Venn, loc. cit.

The arithmetical mean is the simplest case of the mean which is obtained by the method of least squares.