all the children at the end of the year whether they have accomplished very much during the year or not, it is evident that a child's grade in the schools of such a city would depend far less upon his mental development than upon the age at which his school career happened to begin. Thus there might be a double or triple reason why large children should be farther along in school than small ones, apart altogether from any difference in their own mental development.

When such objections as these are offered to the conclusions which any one draws from a set of statistics, the way to answer them is to find out by supplementary inquiries whether the causes suggested really are at work in the case in question; and if they are, to estimate the amount of effect which they are likely to produce, and thus see how much of the total effect is left for the causes originally assigned. Until conclusions based upon the method of group comparison have been subjected to much critical examination of this sort, we must not attach to them anything like absolute confidence. Another danger which confronts this method of group comparisons and indeed all methods that depend upon

Misplaced accuracy.

precise measurements-is that we shall infer the presence of some cause from numerical computations that are far too precise for the data from which they are derived. By this I mean too precise for the least accurate of the data. When mathematicians take two sets of measurements which are to enter into the same problem, and when they can only get a certain proportion of accuracy in one, they realize that the inaccuracy of these data will affect the problem as a whole in the same proportion, and so they make no effort to get a greater degree of accuracy in any of the other data. For example, suppose we know that one side, B, of a triangle is twelve times as long as the base, A, and that we measure A for the sake of finding the length of B. If A is really 101 inches long but we make it 100 inches, that will mean that B is really 101 feet long, though we cal

culate that it is just 100 feet. In this way, an error of one inch in the length of A corresponds to an error of twelve inches in the length of B. If we measured both A and B for the sake of comparing them and did not try to measure A more accurately than in even inches, it would be a waste of time to measure В more accurately than in even feet. Moreover, it would be a positive blunder to say that A measured exactly' 100 inches; that B measured 'exactly' 100 feet and I inch; and that B was therefore a little more than twelve times as long as A. If we measure B to an inch, we must measure A to a twelfth of an inch before we institute any such precise comparisons between them, or draw any conclusion from the existence of such slight discrepancies. When we multiply a measurement we multiply the error that we made in taking it; when we divide a measurement we divide the error. So in general we can say that any figure which has to be multiplied before it is added to, or subtracted from, or compared in any way with, another should be reached by more careful measurements than that other; while a figure which has to be divided before it is compared with another may be reached by less careful measurements than that other.

To take another example of this law of proportion in the accuracy with which we should take various measurements. A certain horse trots a mile in about two minutes and five seconds. The stop-watches by which he is timed will register fifths of a second, but nothing less. If we wish to find his speed as accurately as possible, how accurately should we measure the course over which he trots ? A horse trotting at the rate given goes more than eight feet in a fifth of a second, and since the watches will not register any time less than a fifth of a second, they are absolutely incapable of measuring the time that it takes the horse to go eight feet or less. It would therefore be a waste of time to measure the course for such a horse to a fraction of an inch.*

* Absurd, I mean, if we are measuring the track merely for the sake

Indeed, the very precision of such measurement might be misleading. Suppose, for example, that horse A trots over the mile track in New York in two minutes and five and onefifth seconds, and that horse B trots over the mile track in Toronto in the same time. Suppose also that accurate measurements show the Toronto track to be really six feet longer than the New York track. How natural it would be to say that since B went six feet farther than A in the same time, he must have gone faster! But this conclusion is absolutely unwarrantable; for when we say that the two distances were covered in the same time' we mean that in each case the time was at least as much as 2.5 and less than 2.5%. In other words, we mean that the difference between the two was less than a fifth of a second. But with a possible difference in the time of almost a fifth of a second it may be that A really trotted faster than B after all.*

In our example of group comparisons I think we find a blunder of this same sort. The other objections which we made to the conclusion based upon the weights of school children in different grades were largely hypothetical. This objection is real. The investigator's object is to find out what difference there is in the weights of pupils of the same age who are in different grades. In the tables which he gives for comparison the average weights are all calculated to the hundredth part of a pound. How accurate should he have been in finding the average ages? If we take account of a difference in weight of one pound, should we not take account of a difference in age that is sufficient to produce

The

of timing that particular horse with that particular kind of watch. accuracy is justified by the fact that some time we may have a better watch or wish to time a slower animal.

*It is assumed for the sake of simplicity in the argument that the stop-watch really will measure with accuracy to the fifth of a second. When we remember that the starting and stopping of the watch depend upon human action in the midst of exciting surroundings, it is evident enough that there is still less accuracy in the measurement of the horse's time.

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 ciled 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