CHAPTER XXX.

GROUP COMPARISONS, OR THE METHOD OF STATISTICS.

and uses.

We

In the examples already given, the Method of Concomitant Variations was applied in the following way. We noticed the state or action of some individual thing, then introduced a change into the surroundings and noticed what change. followed in the state or action of the same individual. compared the time which it takes for a pendulum Principle to make a complete oscillation (or a hundred continuous oscillations) when it is swung at the level of the sea and when it is swung at various distances above it, or we compared the distance that a spring is stretched or twisted by some heavy object attached to it at the level of the sea and at various distances above it. In all the experiments necessary for such comparisons we swung the same pendulum or weighed against our spring the same lump of lead. This, however, was not absolutely necessary. were quite certain that two pendulums or two lumps of lead were exactly alike in every essential respect, we could use one at the level of the sea and the other at the mountain-top, and still draw our conclusion from the difference in the results. But whether the individuals compared were identical or not, in either case we compared the states or actions of single individuals. Moreover it will be remembered that to draw any conclusion from the results of such a comparison we had to be sure that the change in elevation was the only change made that might affect the downward pull in

If we

question. If that downward pull had been subject to variation from a hundred other causes known and unknown, whose influence we could not possibly estimate, then the experiment could have told us nothing about the effect upon that pull of the changed distance from the centre of the earth.

In cases of this latter sort inferences that cannot be based upon what happens in the case of a single individual can be based upon the total or average change produced in a very large number. We know, for example, that the growth of a child as measured by its weight depends not only upon its age, but also upon a vast number of other influences: its own natural vitality, the size of its parents, the nature and amount of its food, the amount of sunlight and fresh air to which it is exposed, its freedom from disease, its exercise, its happiness, and so on; and we cannot possibly tell how much influence any one of these has exerted upon the growth of an individual child. Consequently we can tell very little about the growth due to age by weighing a child when it is six and again when it is seven; much less by comparing the weight of one child of six with that of another child of seven. Yet if we compare the average weight of several thousand children of six with that of several thousand children of seven selected in the same way from the same neighborhood, then we shall have a right to infer that the difference in the average weights is due, to a fraction of one per cent, to the difference in age. If one child of seven has attained its growth under better conditions than some child of six, there is undoubtedly some child of six that has attained its growth under better conditions than some child of seven; so that on the whole the favorable and the unfavorable influences in the two groups are balanced evenly enough to be disregarded. Thus by comparing large enough groups we can often eliminate the effects of all the causes but those under consideration and get a very accurate measure of the effect exerted by the latter.

This Method of Group Comparison is used very commonly at the present time, especially in physiology, psychology, and economics, to estimate the effect of various influences that cannot possibly be isolated from a great many others, and whose results cannot be estimated with accuracy in any other way. To take a practical example.

Is there any connection between a child's size, as distinguished from its age, and its mental development? We should be able to answer this question if we can find out whether large children are better developed mentally than smaller children of the same age, or, what amounts to the same thing, whether children of better mental development are larger on the whole than children of the same age of poorer mental development. To find this out an investigator takes through the teachers of a large city the ages and weights of some thirty thousand pupils in the public schools. He then finds the average weight of all the pupils of the same sex and age throughout the schools, as well as the average weight of all the pupils of the same sex and age in each grade'. Comparing these averages together he finds that in practically every case children in higher grades weigh more than children of the same age and sex in lower grades. Accepting the progress which a child makes in school as a fair enough test of its mental development, and the grade in which it is found as a fair enough test of its progress, he concludes from this that children with greater mental development are on the whole larger than children of the same age with less mental development. From such a conclusion, rightly established, we should have a right to infer that a child's mental development depends not merely upon its age, but upon the development of its body, or, conversely, that the development of the body depends not merely upon age, but upon the development of the mind, or else that the development of the mind and the develop

*

* See Transactions of the Academy of Science of St. Louis, Vol. VI, No. 7.

ment of the body both depend largely upon the same conditions; in short, that there is a close causal relation between them.

Here is an example of the measurements:

Twenty-two boys out of the total of 2188 who were measured are not accounted for in this table. These twenty

two boys must have been in the Kindergarten and the grades higher than the fourth. Their measurements are not averaged because the investigator thought, very properly, that an average could not be depended upon unless it were based upon at least twenty individual measurements.

The table shows a difference in weight of about two pounds between boys of nine in one grade and those in the grade above it; so that between the boys of nine in the first grade and those in the fourth there is the very decided difference in the average of over six pounds. The very considerable size of these differences, the almost unbroken regularity with which they appear from one grade to the next, not only with boys of nine, but with both boys and girls of all ages between six and sixteen or seventeen, and the large number of total measurements, to say nothing of certain other relations brought out in the original article,—all these prove that the different average weights in different grades cannot be the result of mere chance-that there must be some real cause at work which tends to make the boys or girls of a given age in a higher class heavier than those in a lower class. In this way, by comparing the average measurements of several large groups we can often prove the existence of causal relations which we could never prove by merely comparing a few individuals,

Number of data.

The errors to which we are liable in such investigations as this are very serious. In the first place we must make sure that enough measurements are made to eliminate the effects of purely individual idiosyncrasies. One boy with twenty pounds of extra fat would make a difference of two pounds in the average when he is one of only ten; but he makes a difference of only one fiftieth of a pound when he is one of a thousand. In general, the larger the groups from which we get our averages, the less chance there is that the distribution of peculiar individuals will be uneven enough to make difference worth considering in the results. In the table it will be noticed that the difference between nine-year-old boys in the third grade and those in the fourth is slightly greater than that between any other two successive grades. This extra difference might well disappear if instead of forty-four boys in the fourth grade to examine there had been a thousand. The accuracy with which we should read our averages always depends upon the number of measurements from which the average is computed.

A second blunder to be avoided in statistical investigations where the data are supplied by different observers arises from

Personal equation, etc.

what is called the 'Personal Equation' of the observers. Where there is any doubt about a quantity some people constantly tend to overestimate it, others to underestimate it. If one object or set of objects happens to be measured by a person who habitually overestimates, and another by one who habitually underestimates, it is evident that the difference between the two will appear a little larger or a little smaller than it really is. Personal equation of this general sort appears in many different forms. If each of two people has to press a button the moment he sees a certain sight or hears a certain sound, the chances are that they will not both press it at exactly the same time. One will nearly always act a fraction of a second later than the other. The more prac