up an average to include exceptional cases. But the only safe way to overcome the unfairness is to base the average on so many cases that it will make practically no difference in that average whether a few exceptions are included or not. If any such large number of cases is not available, then the statistical method is not reliable. We appeal to statistics because we wish impartial witnesses instead of mere individual opinions, and it is obviously absurd to run the slightest chance of selecting some of the witnesses because they will testify in our favor and excluding others because they will testify against us.

Accidental selection.

However successful we may be in avoiding the influence of our own personal equation or that of others in the selection and measurement of cases, we may still choose our cases according to some plan that does involve an unfair selection though we do not realize it. The method of group comparisons is based on the assumption that the force under investigation is the only one which does not act about as much upon the members of one of the groups compared as upon the members of the other. But if this assumption is not correct, if some other force is present which really does act more upon the members of one group than upon the members of the other, and if we overlook its presence, it is clear enough that we shall attribute too much or too little to the force we are investigating; we supposed we had excluded all other constant causes, and we had

not.

Suppose, for example, whether it be true or not, that the further a child gets in school the better able he is either to help his parents at home or to earn money for them outside. It is evident that the poorest and most shiftless parents will be tempted to take their children out of school as soon as their labor is worth a very few cents a day; and unless the law is enforced very rigorously they will find means of doing Others less poor or less improvident will allow their children to go a little farther. This process would continue

SO.

from grade to grade, until in the high school nearly all the children represented homes that are fairly comfortable and well-ordered. It is evident that causes of this sort would produce a process of partial selection on some basis other than that of the children's individual mental development. Many of the children in the lowest grades would be paupers; those in the highest would not. In such a case it might well be that the difference in weight between children of the same age in lower and in higher grades is due, at least in part, to the fact that the latter come on the whole from more prosperous families, and are therefore better fed and cared for.

If it should turn out upon investigation that there is nothing in this suggestion, we might offer another somewhat different. Suppose that amongst the most ignorant classes in a city there is for some reason or other a rather hostile feeling towards the public schools, so that the most ignorant parents keep their children out of them as long as they possibly can; while parents of the better classes, on the other hand, send them just as soon as they can. This would of itself tend to put children from the better homes in higher grades than those of the same age from the worse homes. Moreover, in the better homes children are helped with their lessons and encouraged to keep up with their class, or ahead of it, while in the worse they are not. In this way children who are better cared for at home, and therefore probably larger, would tend, apart altogether from any mental superiority of their own, to be farther along in school than those who are less well cared for at home. But there is still another, more direct, consideration in the case we have supposed. The smaller the child the longer can a parent prejudiced against the schools pass him off as too young to be sent; and the larger the child the easier it is for the intelligent parent who wants him to go to school to persuade the authorities that he is really mature enough to begin, even if he is not quite up to the legal age.

If it is customary in the city in question to promote nearly

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 develop ment.

When such objections as these are offered to the conclu sions 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. Unti 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

of timing that particular horse with that particular kind of watch. The 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.