from the start; but when P opens low, its operation is unmistakable (III and I).

The above conclusions may need scrutiny and revision to accommodate new evidence when it appears; but the facts we have are so clear and accordant in purport and trend, and confirm so well the demands of theory at every stage, that these formulations, so far as they go, may—as we venture to believe-be expected to hold without substantial modification. The problem in any case has not turned out a 'wandering fire'; for as soon as we engage the question with care, instead of denouncing it as unworthy of investigation, it yields a convincing and reasonable answer.

Insofar as practice means high proficiency, it connotes high normality as well. This generalization we construe to mean that as the organism works up to the ultimate levels of performance, as the determinants of its behavior come to be stabilized and refined, as the whole process culminating in a differential judgment is better controlled, its responses come to distribute with approximate normality. (7) would then by logical implication seem to be the limiting curve which our performance tends to describe in at least one extensive and important field of controlled activity (differential sensitivity); even as, with increasingly refined conditions, V = 1/2 gť2 is the limiting formula for the course of a falling body: a formula never in practice completely attained but always approached. A careful and competent Obs. achieves a high degree of fit in the early stages of practice; and if for any reason it begins low, goodness of fit is found to improve insofar as proficiency betters at all.

Whether the inductions above proposed hold for all cases of practice-change is a wider problem, of course, than our evidence alone can solve; but they may well cover not 'psychophysics' alone but wide regions of quantitative psychology as well, wherever practice takes a rôle. In any event we here have optimal conditions for examining the whole matter, for the reason that we have known units of stimulation with which to deal. In so many psychological problems (as,

educational and alertness scales) a (y)-curve is postulated to begin with and the difficulty of a given task or question (its 'stimulus-value') then related to that assumption. Here, on the contrary, no assumptions of any kind are used; the stimulus values can be measured and controlled with any desired degree of precision, and a set of judgment-frequencies thereby derived under optimal conditions of control. That these empirical frequencies constitute, with growing exactitude as practice proceeds, a normal curve, is we think a matter of the first consequence for quantitative theory in psychology; it implies that the performance of an organism, when freed of casual and irregular determinants, tends to approach a standard type.

XII. ON THE T-METHOD: A PROCEDURE FOR FINDING THE TREND OF ANY SERIES OF NUMBERS

In the article (XI) just preceding we have had occasion to use a simple device for discovering the trend of an irregular series of numbers. What we may call the T-method (from its dependence upon Tchebycheff's criterion) seems to enjoy certain advantages over the procedures commonly used. As opposed to the correlation coefficient, the T-method, being extremely general, may be used with all sorts of distributions, no matter how widely they depart from the gaussian form, whereas r is reliable only when the correlated series are closely normal; it may be used with series too brief to warrant the product-moment procedure; and its criterion of significance does not presuppose normality as does the p.e. of r. The least-square procedure, furthermore, becomes laborious with more than a few observation equations, while its reliability is cumbrous and difficult to determine; whereas the T-method is simple and short, and its validity can be found in a moment.

Given a set of data like the 28 figures for P and for h, Obs. II, lighter.1 We may ask two questions: (a) Does P (goodness of fit) tend to increase as trials proceed? (b) Do P and h tend to rise and fall together? In reply to (a), we count the number of increases from every figure to all the 1 Article XI, just preceding.

following; clearly if certain effects are due to practice or to rank in a serial order, any one value will be conditioned by all which precede and will in turn affect all which follow. We find now that the first P (.953) is exceeded by two of the remaining 27 (.963 and .995), the second P by one of the later 26, and so on; in a total of 378 possibilities (27 + 26 + +1), we thus have 200 increments, a ratio of 200/378 or .529, as compared with an expected ratio of .500 in a random series of numbers.

· ..

The question now appears: What is the variability of this ratio (.529)?

Its s.d. may be found by resorting to the usual urn-schema, a white ball denoting an increment (p) and black a decrement (q); p + q = 1. So long as p and q remain constant, we have what is called a Bernoulli series, in which case the s.d. of p equals √pq/n; here we have instead a change in the likelihood of an increase (a white draw) at every comparison; the situation alters from trial to trial so that no two probabilities can be assumed the same. In the present case, then, we have a Poisson series, where the proportion of white and black varies from draw to draw. In this type of situation the s.d. is bound to be smaller than in the Bernoullian and can be determined only if the value of p is given for each successive draw; the Poisson σ is therefore indeterminable with an experimental situation like the present, where we know nothing about the likelihood of an increase (white ball) from trial to trial, but only the empirical proportion derived from the whole set of 378 comparisons or 'draws.' Our only recourse, then, is to use the Bernoullian sigma; in so doing we at least know that any conclusions based upon it will be conservative and a fortiori true of the Poisson situation. Solving .529 X .471/378, we have ± .0257 as s.d. of .529.

Does the empirical ratio (.529), then, depart significantly from the 'chance' ratio of .500? In answer, we cannot resort to the customary criteria of significance, which are all based upon a normal frequency-surface. There are but two abscissas, two possibilities of increase in our distribution, zero and unity; the base runs from o to I, the zero-ordinate being

178, the unit-ordinate 200, the mean .529; the curve is far indeed from gaussian, being a tall, narrow trapezoid. To show how widely the two forms of distribution differ: normally over 99.7 per cent. of the whole area is included within ± 3 s.d. of the mean; here only about 16 per cent. Otherwise put, in a normal surface all the values composing the mean lie within 3 to 4 s.d. of it; here every value is either o or I and thus falls 18 to 20 o from the mean .529. In Tchebycheff's criterion, fortunately, we have a thoroughly general device for measuring significance quite apart from the form of distribution; by this theorem the probability that a given proportion (.529) will deviate positively or negatively from the expected (a priori) value (here .500; in general the mean ratio of increases with a large number of random series of 378 each) by more than t times its s.d. is less than 1/2. In our case I/t2 = .02572/.0292 = .78; the chance that with 378 comparisons between numbers in random order the proportion of increases will deviate from the expected value of .500 by more than ± .029 is less than .728, or that the proportion will fall within the range .500 .029 is greater than .22. If, for purposes of comparison, we now correlate P with the rankorder (the first P, .953, with 1, the second P with 2, and so on for the 28 cases), we get r equal to .093 with s.d. ±.187; the probability that 28 numbers chosen at random would correlate with their serial order more than ± .093 is about .62 (as compared with the more general T-criterion: less than .78). It should be noted that the criterion (1/2) applies only when t is not less than unity.

In like manner, we find (b) whether P varies with h by counting how often a rise (or fall) in P is attended by a rise (or fall )in h, or per contra how often one rises when the other falls. In all, P varied with h 242 times in 378, a ratio of .640 with s.d..0247; whence we have 1/ť2 equal to .02472/.1402 = .031. Upon correlating P with h for the 28 cases, we get .475 with s.d.±.146. Hence by the T-criterion the likelihood is less than .031 that two series of 28 numbers written at random would rise and fall together in more than the stated proportion (.500 ± .140) of cases; whereas by the r-method,

the corresponding probability that two random series of 28 numbers would yield an r as high as .475 is about .0013. It will be noted that the chances favoring validity are higher by the r-method in both cases. The T-procedure is conservative in two ways: (a) we are forced to use a s.d. (Bernoullian) which is invariably larger than the true (Poisson) value; (b) Tchebycheff's theorem states the probability always in terms of greater (or less) than a given fraction, the precise degree of excess or defect being of course unknown. Apart from the general conservatism of T, we noted above (Table VIII) how nearly the two methods agree in the conclusions they justify and the reliability they afford.

With a limited number of cases it is well to take the amount of change (+ or -) from each value to all that follow it instead of merely counting the number of changes up and down; in a long series the mean positive may be expected to equal the mean negative change, but in a short series this cannot be assumed. Thus, with Obs. III, Lighter, h rises 1.455 in 25 increases and falls .391 in II decreases: a proportion of 1.455/1.846 or .788 in magnitude and 25/36 or .694 in number -an appreciable discrepancy. Of the two the former is clearly better.

In like manner, we may determine whether P correlates with h by finding how much P changes when moving with h and how much when moving against h. So in III, Lighter: when his increasing, P registers a total increment of 11.59 and a decrement of 1.35; when h is falling, P rises 1.46 and drops 2.95. P thus varies with h in the sum of 11.59 +2.95 and versus h by 1.35 +1.46; whence the proportion of increase to the whole extent of change is 14.54/17.35 or .832; the reliability of which is found in the usual way.

To sum up: in the T-method, here demonstrated by means of concrete examples, we have a simple and convenient device for analyzing any body of material. Its cardinal advantage over the common modes of treatment lies in its generality, its universal applicability; it can be used with all manner of distributions, though they depart widely from the gaussian form; the question of 'normality' does not arise. The