have been made and after all the means for avoiding and minimizing error have been employed, there yet remains a margin of uncertainty. In such cases it is possible to obtain a close approximation to the true measurement by taking a number of measurements and striking an average. After constant errors have been eliminated any given measurement is as likely to be too great as it is to be too small; hence, in a large number of measurements there will probably be as many of those which exceed the true magnitude as there are of those which fall short of it. If the number of measurements is small this is more doubtful, but if a great many measurements have been made, we can rely upon the average with safety. The average of all these measurements is the closest approximation which we can get. Different kinds of average are used according to circumstances. The closeness with which the several measurements are grouped about the average will be indicated here, as in all cases of the use of average, by the size of the error. If the error is small, the measurement is reliable, if large, more doubtful. The Comparison of Quantities which Cannot be Measured. In the study of many phenomena the prob lem of quantitative comparison is made very difficult by our inability to find an exact quantitative equivalent for the phenomena. "Many mental phenomena elude altogether direct measurement in terms of amount. How many thefts equal in wickedness a murder? If the piety of John Wesley is 100, how much is the piety of St. Augustine? How much more ability as a dramatist had Shakespeare than Middleton? What per cent. must COMPARISON OF QUANTITIES 211 be added to the political ability of the Jewish race to make it equal to the Irish race? Nevertheless, such phenomena can be measured and subjected to quantitative treatment." 13 The method to be employed in such cases, as Professor Thorndike goes on to show, is to arrange the individuals (or other unmeasurable data) according to their rank. We may not be able to say how much more eminent A is than B, but if we can say that A is in the first rank, whereas B is in the tenth, we have a true basis of comparison. We cannot measure directly the intelligence of students in a class, but we may be able to say that one is in the first group, whereas another is in the fourth. Thus, with any number, it would be possible to give each his proper place in the group. This method can be applied to any trait whatever. The great difficulty is in making sure that the ranking is correct. Single observations and individual judgments are subject to the same errors here as in all other cases of observation. EXERCISES. 1. What sort of average should be employed in determining the standard size of an article to be manufactured in large quantities-say window shades? 2. What sort of average should be employed in getting a number to represent the value of articles in a large and varied invoice of merchandise? 3. If a college had 400 students in 1880 and 1000 students in 1905, how many did it have in the year which falls 13 Thorndike, An Introduction to the Theory of Mental and Social Measurements, p. 18. This book is an exposition of the methods of measuring individuals, groups, variability of performances, etc., including an exposition of the necessary modes of presenting the facts, making calculations, and so on. half way between, provided that the rate of increase was constant? 4. What averages might be employed and which would be preferable in comparing the stature of soldiers in the French army with those in the American army? In comparing the standing of successive classes in college? In comparing the salaries of members of the faculty in two universities? In comparing the rate of growth of a large university and a small college? 5. How would you indicate the degree of closeness with which a series of quantities approached their average? 6. What is the difference between "Average Error "Probable Error?' and CHAPTER III PROBABILITY THE Conclusions at which we arrive by the assistance of statistical methods and the employment of averages often fall far short of the certainty attaching to scientific laws. The conditions required for establishing a scientific law are not fully present, and consequently many of such conclusions, if not all of them, lack complete verification. It does not follow, however, that these are valueless. As a matter of fact, most of the generalizations which we use in everyday life are incompletely verified; they are extremely valuable as instruments of knowledge and practice; indeed, in the absence of scientific laws, they are indispensable. So long as their provisional character is remembered, there is no serious danger in using them. A generalization of this character is said to be probable or to possess some degree of probability. Probability belongs also to particular propositions. What do we mean by probability, by saying that a statement is probably true, that an event will probably happen? As we use the term ordinarily, it means that we believe we have a right to accept a statement or expect an event, without feeling perfectly certain of it. This attitude, when it has any justification, is based upon the belief that the grounds for accepting the statement are stronger than those for rejecting it. It may be that we know of no positive reasons against it, but do not regard the reasons in its favor as conclusive; or it may be that there are positive reasons against it, but that those in its favor are stronger or more numerous. These reasons or grounds may be of various kinds. There may be many things pointing toward the occurrence of such an event; as, for example, in the statement that life will probably at some time cease upon the earth. Or conditions at the present time may be similar to those in which the event has happened before; the outcome of an examination of instances according to the principles of the method of Agreement gives a result which is usually only probable. In all these cases it is impossible not to feel that a great deal of vagueness attaches to our statement that anything is probable. We are not able to say how probable it is. There is such a thing, however, as mathematical or quantitative probability. It is based upon the comparative number of times an event or connection of events has occurred. If a given circumstance A has been observed 1000 times, and if, in 700 cases of its occurrence, a phenomenon B has also been present, we have definite grounds for inferring that A will probably be accompanied by B again. Every time A and B have occurred together in the past is an argument in favor of their occurring together in the future, and every time A has occurred without B is an argument against this connection; if the cases of the latter sort are many in comparison with those of the former, we say that the connection in the future is improbable. In the case just mentioned we should express the degree of probability by the fraction 10. Now in dealing with the matter in this quantitative way, the term "probability " has a meaning which is somewhat different from that in which we |