On the Relation of the Methods OF Just Perceptible Differences and Constant Stimuli By SAMUEL W. FERNBERGER A thesis presented to the Faculty of the Graduate School of the University of Pennsylvania in partial ful- filment of the requirements for the degree of Doctor of Philosophy. 1912 On the Relation of the Methods OF Just Perceptible Differences and Constant Stimuli By SAMUEL W. FERNBERGER A thesis presented to the Faculty of the Graduate School of the University of Pennsylvania in partial fulfilment of the requirements for the degree of Doctor of Philosophy. 1912 »2> Gift The Uoi-o^rsHy 3 JUN 13 TABLE OF CONTENTS PAGE. I. Introduction I & 5 1 . Statement of the problem 1 2. Differences between the two methods 1 A. Formal 2 B. Experimental 2 3. Historical background of the problem 3 II. Arrangement of the experiments 7-18 1 . The subjects 7 2. Form and adjustment of stimuli 7 3. Experimental arrangement 10 A. Elimination of space errors 11 B. Controlling of the time error 12 C. Order of presentation of stimuli 12 D. Adjustment of the variable stimulus 13 E. The judgments 15 III. The method of constant stimuli 19-48 1. Form of the results 19 2. Constancy of conditions throughout the experiment 19 A. Coefficient of divergence 19 a. Method of calculation 22 b. Results 23 c. Discussion 24 B. The psychometric functions 28 a. Theory and method of calculation 29 b. Results 38 c. Discussion 41 3. Method of constant stimuli as a measure of sensa- tivity 46 A. Interval of uncertainty 46 B. Point of subjective equality 47 C. Time error 47 iv INDEX IV. The method of just perceptible differences 49-81 1 . Various forms of the method 49 2. Analysis of the four fundamental differences .... 49 3. Calculation of the probabilities of these differences from the results of the method of constant stimuli 53 A. Theory and form of the calculations 55 4. The observed and calculated results of these differ- ences J2> 5. Form of the calculations of the observed values. . 73 A. The probable error as a measure of accuracy . . 75 6. Comparison of the results by the method of con- stant stimuli with the observed and calcu- lated results by the method of just perceptible differences 78 I. INTRODUCTION Psychophysics has evolved a number of procedures by means of which the sensitivity of a subject can be determined. These procedures differ widely as to the experimental arrangement which they require and as to the calculation to which the results are subjected, but they have the common purpose of measuring the sensitivity of the subject. We may say, in general, that a psychophysical method is a prescription for the collecting of data and their evaluation in such a way that the result enables us to compare the sensitivity of different subjects, or of the same subject under different conditions or at different times. All these methods agree in this one point, that by them we undertake to give measures of sensitivity. The quantities which are used as the measures of sensitivity are widely different and, perhaps, not always directly comparable. One is confronted with the situation that the comparison of the sensitivity of different subjects by a given method yields perfectly satisfactory results, but that these results do not always agree with those obtained by other methods. One is then led to ask, what the relation of these different meas- ures of sensitivity may be. This problem is frequently stated in the form of the question of the relation of the method of con- stant stimuli and the method of just perceptible differences. It has been frequently pointed out that the methods of constant stimuli and just perceptible differences show variations of an experimental or of a mathematical nature in consequence of which their results are not comparable. Numerous investigators have noted this fact and have made different classifications of the methods on the basis of differences in their experimental pro- cedure or in the treatment of the results. In all of these classifi- cations the method of just perceptible differences is always the representative of one group and the method of constant stimuli is given as an example of another group [cf. Titchener, Experi- mental Psych., II, II, pp. 315-318]. At present these two methods have been developed to a more or less standard form, evidenced 2 SAMUEL W. FERNBERGER by the fact that different authors describe them almost uniformly. The method of just perceptible differences is described by Wundt [Phys. Psych., 5th Ed., vol. I, pp. 470] ; G. E. Miiller [Die Gesichtspunkte nnd die Tatsachen der Psychophysischen Metho- dik, pp. 179] and Titchener [Experimental Psychology, vol. II, part I, pp. 55-69; part II, pp. 99-143]. A description of the method of constant stimuli is to be found in G. E. Miiller [Ge- sichtspunkte, pp. 35] and in Titchener [Exp. Psych., II, part I, pp. 92-118; part II, pp. 248-318]. The solution of the problem of the relation of the psychophysi- cal methods certainly would have been much simpler, had it been possible to subject the same experimental data to the different calculations. The results of experiments made according to these descriptions cannot be treated as material for both methods. It was therefore necessary to divide the experiments into two groups, in one of which the data was taken by the method of just perceptible differences and in the other by the method of con- stant stimuli; and this brought with it the difficulty of deciding whether a given difference between the results was due to chance variations; to changes in the attitude of the subject, or to differ- ences in the methods themselves. The differences in the arrange- ment of the experiments by the two methods, indeed, are so great that one may suspect the existence of differences in the attitude of the subject. It has been argued that the influence of expectation, found in the method of just perceptible differences, where the subject has a knowledge of the stimuli, is a serious handicap. This influence is obviously absent in the method of constant stimuli since the experiments are arranged in such a way that the subject is given no clue whatsoever, as to the objective relation of the stimuli. In any study of the relation of these two methods, an effort must be made to eliminate all such influences and to have the data in such form that any variations between the results will be due to differences in the methods themselves and not to any of the influ- ences due to the experimental procedure. One means of elimi- nating the influence of expectation is to mingle the experiments INTRODUCTION 3 by the two methods in such a way that the subject has no informa- tion as to the stimuli compared. The conditions in the two groups of experiments will be very nearly identical, if the results are taken simultaneously, but these conditions may undergo certain changes in the course of the experimentation. This belief cannot be disposed of offhand since the collection of the data neces- sarily requires a considerable time. One of the most important of these conditions is the psychophysical make-up of the subject. It is very likely that the psychophysical make-up itself does not remain constant since at least one factor, namely practice, changes. The difficulty, therefore, is twofold and requires an investigation of the changes due to variations that have an experimental basis, and also an investigation of the purely formal character of the methods. These two sides of the problem must not be confused. It may well be that the two methods are formally identical but in actual experimentation do not give the same results, since they are performed under different conditions. The suspicion that the conditions are not the same for the methods of just per- ceptible differences and constant stimuli is very strong, since the entire experimental arrangement is such as to produce different attitudes on the part of the subject. Thus it cannot be expected that the two methods will give the same results unless the experi- ment is arranged in such a way as to make the conditions directly comparable. Many experimenters have noted the fact that if results were taken by one of these methods and compared with the results of the other, that they do not agree. We may mention Meinong [Ueber die Bedeutiing dest Weberschen Gesetses. Zeits. f. Psych, und Phys. der Sinnesorgane, XI, 1896, pp. 244] ; Ebbinghaus [Psych., I, pp. 504] ; Merkel [Phil. Studien, IV, p. 543] and Boas [Pflilger's Arch., XXVIII, 1882, pp. 562]. The investigation of the formal character of the methods is the problem which Urban set for himself, and he devised for this purpose the notion of the probability of a judgment, which logi- cally led to the notion of the psychometric functions. [The Application of Statistical Methods to the Problems of Psycho- Physics, 1908, pp. 106.] Forming a judgment on the comparison of two stimuli is a chance event and there exists a certain proba- 4 SAMUEL W. FERNBERGER bility with which a certain judgment will occur under certain conditions. If our experimental data are extended enough, the observed relative frequency of that judgment may be regarded as an empirical determination of its unknown probability. One of the conditions of a judgment is obviously the intensity of the stimuli. The psychometric functions give the probabilities of the different judgments as functions of the intensities of the comparison stimulus. Urban showed that the method of just perceptible differences could be analysed by means of these notions and that its final results could be stated in terms of the probabili- ties of the different judgments and the intensities of the stimuli. Empirical determinations of the probabilities of the extreme judg- ments, i. e. of the judgment greater and smaller, are the basis for the calculation by the method of constant stimuli, and it is therefore possible to test the formal character of the methods under discussion on one and the same set of results. Experience showed that both methods of treatment gave essentially the same results. -We are, therefore, confronted with the fact that the methods of constant stimuli and just perceptible differences give the same results if the calculations are made on the same material; but different results if the materials are different. From this we con- clude that the methods are formally identical but that the condi- tions, under which the experimental data must be gained, are materially different; that is, that the methods favor different attitudes of the subject. One must, therefore, devise an experi- mental procedure by which the two methods can be performed under as nearly identical conditions as possible, in order to study the agreement of the results from the two methods. The collection of a large amount of material enables one to study incidentally, an entirely different problem ; that of whether the conditions remain approximately the same throughout the experiment, or if they undergo certain changes. If the physical conditions of an experiment are kept as constant as possible, so that no variations in the conditions can be attributed to them, we must attribute any varying conditions to a change in the psycho- physical make-up of the subject. One factor, at least, making INTRODUCTION 5 such a change extremely probable is the one due to the practice acquired by the subject during the performance of the experiments. Extended material, therefore, enables one to study the influence of progressive practice. If the experiments are made on two subjects, one of whom has a high degree of training in this kind of experimentation, while the other has none, one has the oppor- tunity to study the influence of practice in a very advanced stage and to compare it with the practice in the initial stage. These factors have led to the selection of the experimental arrangement for this study. It enables us to investigate the prob- lems of the formal and experimental relations of the methods of just perceptible differences and constant stimuli, and inciden- tally, the effect of progressive practice. My thanks are due to Prof. F. M. Urban for suggesting this problem to me and for acting as a subject. I also have to thank Prof. E. B. Twitmyer for revision of manuscript. II. ARRANGEMENT OF EXPERIMENTS This paper is based on the results of experiments in lifted weights on two subjects. The experimentation began January 3, 1912, and was completed February 20, 1912. During this time records were taken almost every day and many times during the morning and afternoon. Subject I, Dr. F. M. Urban, was highly trained in the technique of lifted weights and was the same as the one designated as subject II in a former study by Urban [Statisti- cal Methods]. Subject II, the writer, had some experience in psychological experimentation but, at the beginning of this experi- ment, had no training in judging small differences of lifted weights. He was given one day's practice, in order to become acquainted with the experimental procedure and with the sensa- tions produced by weights that differ but slightly in intensity. From the second day his judgments were recorded. The weights used in this experiment were hollow brass cylin- ders, closed at one end. They were approximately 2.5 inches in diameter and 1 inch high and the wall was 0.0625 inches thick; and they were brought to any desired weight by filling them with shot and parafine. Although these weights had the same outward appearance to the subject, each weight could be recognized by the experimenter by means of small numbers stamped on them with a steel die. The set consisted of 15 weights; of which 7 were standard weights of 100 grams each. The comparison weights for the set used with the method of constant stimuli were 84, 88, 92, 96, 104 and 108 grams. There were also two weights of 84 grams and 97.44 grams which served for the preparation of the variable stimulus in the method of just perceptible differences. These cylinders had been made slightly lighter than the weight desired and a very delicate adjustment could be obtained by inserting shot and parafine until they were heavier than the proper intensity. The parafine was then carefully scraped out until the desired weight was obtained. An effort was made to use as little parafine as possible since it is susceptible to greater variation from + OWOi^ioooOOCOOO 0000"">0"~'"">OOOOio Ou-)00"^"^000000000 + T i i T T T + mO O o ir )O u ^ iri O l ^O O O O O + + ►J < O u^O O LOinino m in in o in O O + OLOioioOOOOLnOLOOOu^io + + + I •a « o o o is "-> m n ■* m ic r^ o\ ^ « n t*)Tj-u-)0 IT) , l-H 1-1 |1 hi M 8 SAMUEL W. FERNBERGER atmospheric changes than the shot. It was not deemed necessary to determine the weights within a smaller value that 5 mgr. The weights were tested daily for the first week and then once a week throughout the experiment. Whenever a weight was found to vary more than 10 mgr., it was readjusted, but this was necessary only nine times during the experiment. During the first three weeks only one adjustment was necessary and no single weight had to be adjusted more than once. Table I contains the weights of the cylinders and also the amount of variation discovered in them, these variations being given in -f- or — mgr. from the correct weight. The first column contains the numbers stamped on the cylinders ; the next column gives the correct weight of each, and the succeeding columns contain the variations of the cylinders found on the date at the head of the column. A star indicates that a cylinder was found to vary more than 10 mgr. and that it was readjusted. In the choice of the materials which compose the weight, there are three essential points to be considered. In the first place, the weight must show very little variation from atmospheric changes ; secondly, there must be no distinguishing marks on them by means of which the subject can tell one from another. And thirdly, they must be the same in temperature and must arouse the same tactile sensations. Various types of weights have been suggested in the past. The weights used by Fullerton and Cattell were wooden boxes weighted to the proper intensity with shot and raw cotton. They were 6 cms. in diameter and 3 cms. high [Fullerton and Cattell, The Perception of Small Differences, 1892, pp. 118]. Jastrow [Amer. Journal of Psych., V, p. 245] used weights of the same type. Galton [Inquiries into Human Faculty, 1883, p. 373] used weights that were made by placing shot in a cart- ridge shell. Urban [Stat. Meth., p. 1] used the same brass cylin- ders that were employed in the present study. Brown [The Judgment of Difference, Univ. of California Pub., V, I, No. 1, 1910] used cylindrical tin boxes 2.5 cms. high and 4.5 cms. in diameter, which were weighted with shot and parafine. In the choice of weights for our experiment, the wooden boxes must be at once thrown out of consideration, since they do not ARRAXGEMEXT OF EXPERIMENTS 9 fulfil the first of our requirements. The variations in these weights are quite considerable as was shown by Urban [Stat. Meth., p. 173]. In this study Urban gives a table showing the- variations in his set of weights, which were similar to those used in this experiment and a set of Cattell weights that were adjusted but not used. The cartridge weights are open to the same criti- cism, although probably not to as high a degree as the wooden weights. An effort should be made to have the weights consist of as anhygroscopic materials as possible. With the weights used in this experiment, the brass cylinders themselves and the shot are practically anhygroscopic. The parafine was included to keep the shot stationary' and to simplify the adjustment and readjust- ment of the weights. The weights were as nearly alike as it was possible to turn them out, making it impossible for the subject to distinguish between them. The table that follows contains the height and diameter of each weisrht to 0.001 of an inch, and it can be seen that these Height in Diameter in Weight inches inches 1 0.996 2.466 2 0.998 2468 3 0.908 2.470 4 0.995 2.470 5 0.997 2.468 6 0.997 2.464 7 0.098 2.464 9 0.990 2.468 12 0.997 2.468 13 0.999 2.469 14 0.098 2.467 15 0.997 2.468 19 0.996 2.467 52 0.997 2.466 Blank 0.996 2.468 differences can be disregarded. Numbers were stamped on them so that they could be identified by the experimenter. The sur- faces were polished and lacquered, rendering them similar to the touch. The weights were kept under exactly the same conditions and furthermore, they were handled the same number of times, so that there was no perceptible difference in temperature. Thus the cylinders used in this experiment seem to conform to all the requirements of a set of weights and furthermore, have the ad- vantage of being easily adjustable. jo SAMUEL W. FERNBERGER The weights were placed at regular intervals around the circum- ference of a circular table with a revolving top, which was 75.5 cms. in diameter and was raised 68.0 cms. from the floor. The top was covered with a layer of prepared cork, which deadened the sound of the weights when replaced on the table. The posi- tion of each weight was indicated by a small number on the table. The standard and comparison weights were placed alternately — the standard weights at the odd, and the comparison weights at the even numbers on the table. The subject was seated in a comfortable position with his right arm supported by a table in such a way that the hand, from the wrist down, hung over the edge. An effort was made to have the edge of the supporting table strike approximately the same posi- tion of the forearm of the subject. The turn top table was then brought into such a position that, with merely a downward move- ment of the wrist, the hand would grasp one of the weights. The cylinders were lifted with the right hand ; most of the weight being sustained by the thumb, second and third fingers, and the first and little fingers resting on the edge. The movements of the hand were regulated by the beats of a metronome, which was adjusted to 92 beats per minute, while every fourth beat was accentuated by the automatic stroke of a bell. These hand move- ments were regulated in the following manner. At the start of each trial, the hand of the subject was raised at the wrist, with the forearm remaining on the table. At the stroke of the bell, the hand was dropped and the weight, which had been brought directly underneath by turning the table, was grasped. At the second stroke of the metronome the weight was lifted, and at the next stroke, it was replaced on the table. Finally at the fourth stroke, the empty hand was lifted, returning to its original position. Between the third stroke of the metronome and the bell following, the experimenter turned the table so that the next weight to be lifted, was directly under the hand of the subject, and everything was ready for the next lifting. In a very short time these wrist movements became quite automatic. The weights were lifted from 2 to 4 cms. and an effort was made to have the height of lifting constant for each subject. Due to the control ARRANGEMENT OF EXPERIMENTS n of the metronome, each weight was in the air approximately the same length of time. A screen was placed between the subject and the table so that it was impossible for the subject to see the weights; his hand passing through a slit in this screen. Furthermore both subjects voluntarily closed their eyes while making judgments, as they believed that they could make their judgments with more accuracy in that way. Previous investigators have not made a very clear analysis of the space error. They have sought to avoid it or eliminate its influence rather than explain it. The obvious method of elimina- ting the space error and the one that has been most frequently used, is to perform the experiment twice, in both of the spatial relations; that is, with the standard weight to the left and right of the comparison weight. Then the error of the one spatial relation counterbalances the error of the other, which is in the opposite direction. In the present study the space error was avoided in a simpler manner; since by means of the revolving top of the table, all side movements of the hand were eliminated. The table was turned so that the weight to be lifted was brought directly under the hand of the subject. Thus the only movements necessary were in one direction merely — directly downward. Care was taken that the subject did not reach to one side or the other in grasping a weight, since in this case a space error would have occurred. An effort has been made before to avoid the space error by experimental technique rather than eliminate it by repeat- ing the experiment in the two spatial relation. [L. Steffens, Zeitschrift f. Psych, und Physiol, der Sinnesorgane, XXIII, 1900, pp. 279, and J. F robes, Zeit. f. Psych, und Physiol, der Sinnesorgane, XXXVI, 1904, pp. 234]. In these studies the weights were placed on a board which was pushed along under the hand of the subject, so that the weights in turn were directly underneath. This procedure, although it eliminated the space error, had the disadvantage that when a series had been taken, the board had to be replaced in its original position before a second series could be begun. With our experimental arrangement, how- ever, any number of series could be run off in succession. Besides 12 SAMUEL W. FERNBERGER this, the table, which revolves very easily, can be moved with greater regularity and accuracy than could ever hoped to be ob- tained with a sliding board. The time error was present in our experiments and no effort was made to either avoid or eliminate it. The standard stimulus was always lifted first and the comparison stimulus second. The metronome controlled all hand movements and kept them regular so that the time error was constant throughout the experi- ment. As the investigation of the sensitivity of the subjects was not of primary interest, and as a constant time error should not effect the relationship of the results obtained by the two methods under discussion, no attempt was made to avoid this error. The comparison weights were placed about the table in a care- fully arranged order. This order was changed four times dur- ing the experiment, partly to eliminate the influence of the particular arrangements, partly to counteract the influence of the knowledge about the arrangement used, which the subjects might have acquired. Table II gives these orders. The first column Table No. 1/3/12 1/10/12 1/22/ 1 2 2/18/12 i and 2 96 84 84 88 3 and 4 104 104 104 C 5 and 6 108 C 104 7 and 8 84 C 88 96 9 and 10 02 92 92 84 11 and 12 C 108 108 92 13 and 14 88 88 96 108 Table II gives the table numbers in pairs : 1 being the first standard stimulus, 2 the first comparison stimulus, 3 the second standard stimulus and so forth. The other columns give the comparison weights at the even numbers of that pair. The C indicates the comparison weight of the method of just perceptible differences series. Each column is under the date at which the order was adopted. In every order, no matter how carefully planned, there ARRANGEMENT OF EXPERIMENTS 13 are certain landmarks by means of which the subject can tell in what part of the series he is. Such a landmark is seen in the first series where the two heaviest weights (104 and 108 gms.) come together. Another occurs in the second order where the two lightest comparison weights [84 and 88 gms.] came together. Both subjects acted as experimenters and so necessarily became acquainted with the order in which the weights were presented. Furthermore, the experiments were conducted daily and so the subjects became acquainted with the orders more rapidly than if there had been a longer interval between experimentation. One complete revolution of the table involved the passing of seven judgments : six on invariable comparison weights for the method of constant stimuli, and one on a variable comparison weight for the method of just perceptible differences. This seventh comparison weight, indicated by C in table II, was adjusted for the method of just perceptible differences in the following manner. At each revolution of the table, a judg- ment was passed between it and a standard weight in the same manner as the other pairs. After a judgment had been taken on each of the seven pairs, steel bearings of a given number were placed in cylinder C and another complete revolution of the table was made. Then the same number of bearings were placed in cylinder C. This was continued until C weighed over 108 gms., the weight of the heaviest stimulus used in the method of constant stimuli. The bearings which weighed 0.42 gms., were of a sur- prising uniformity in weight. All the bearings were weighed and they did not show any variations within 5 mgr., which was the limit of exactitude in our weighing. We at first intended to use shot for the purpose of weighting our variable stimulus, but found that the differences among them were quite consider- able. Cotton wool was placed in the bottom of the cylinder so that the total weight was 84.80 gms. The cotton wool was placed in the cylinder to keep the bearings from moving about and thus by the noise, indicating to the subject which cylinder he was lifting. The bearings made a noise only three or four times during the lifting, and each time the judgment was thrown out and another taken. i 4 SAMUEL W. FERNBERGER Ten series were taken by means of this method. In series II the variation was two bearings for each revolution of the table. For series III the variation was three bearings, and so on until series X, when a variation of ten bearings was used. In an effort to test the "carefully graded approach" of the central intensities of the comparison stimulus, which is considered by some psychologists to be the keynote of the method of just per- ceptible differences, another series was planned [Titchener, Ex- perimental Psychology, II, II, p. 103]. In this series the two extreme variations were seven bearings; the next two variations toward the central values were six bearings; then in order five, four, three, two, and the five central values varied only one bear- ing. This is designated as series I. It was not deemed profitable to perform a series with the variation of only one bearing, as this would have necessitated 54 revolutions of the table to complete each series. Ten determinations of each series were taken; of these five were ascending and five descending. In the ascending series the proper number of bearings were placed in the cylinder, after each revolution of the table; while in the descending series, the total number of bearings were placed in the cylinder at the start of the experiment and after each revolution of the table, the proper number were removed. It is obvious that these series varied in length ; the series X required only seven revolutions of the table while series II required 29 revolutions. As a matter of technique, two weights were used for this comparison stimulus. The first with the cotton wool weighed 84.80 gms. and the second with the cotton wool, 98.24 gms. In an ascending series, the 84.80 gm. cylinder was first judged empty, then successive judgments were taken until it weighed equal to or just heavier than 98.24 gms. Then the other cylinder was substituted and the lifting continued. The opposite procedure was used in a descending series. The use of only one cylinder would have necessitated the handling of twice as many bearings as were used. The series were not taken in any regular order but entirely at haphazard; the determining factor in the choosing of a series being the amount of time at our disposal. If we had sufficient ARRANGEMENT OF EXPERIMENTS 15 time, a long series of short steps was chosen ; while if our time was short, a short series of long steps was used. A different starting point was chosen for each successive revolution of the table. This was done so that, even though a subject knew an order fairly well, he would not be able to tell at what part of that order he had started. It was furthermore deemed advisable, on each revo- lution of the table, to allow the subject to make two judgments that were not recorded. This was done so that the movements of the wrist might become as automatic as possible and also to give the subject an opportunity to concentrate his attention. At the beginning of experimentation each day, the subjects made one complete revolution of the table grasping the weights as strongly as possible and lifting them high and vigorously. This was done for a double purpose : it assured the experimenter that the weights were in the correct position on the table in rela- tion to the hand of the subject. At the beginning of experimenta- tion, the weights give the impression of great lightness which is lost as the lifting process proceeds. This process can be hastened by the sort of lifting just described, and after such a warming up, both subjects made introspections that they had little trouble with the absolute impression. After each five to seven revolutions of the table, the subject was allowed to rest, until he was willing to resume experi- mentation. If a subject declared that he felt fatigued or unfit he was not asked to experiment. In the manner just described, results by the method of just perceptible differences were taken simultaneously, and therefore under as nearly indentical conditions as possible, with results by means of the method of constant stimuli. The former are the results obtained with the weight C and a standard weight, and the latter are the judgments on the six other pairs of weights. Immediately after each comparison weight had been replaced on the table, a judgment was given in terms of the comparison weight. These judgments were given verbally and by saying one word, three terms being used: heavier, equal and lighter. A heavier judgment signified that the comparison weight was subjectively heavier than the standard weight just preceding it. i6 SAMUEL W. FERNBERGER A lighter judgment signified that the comparison weight was subjectively lighter than the standard weight just preceding it. The equality judgment was more complex as it not only included cases of actual subjective equality between the standard and comparison weights, but also all those cases where it was impos- sible for the subject to give either a lighter or a heavier judgment, usually termed doubtful cases. The cases of absolute subjective equality were much more frequent with subject I than with subject II. The results were recorded by the experimenter on printed blanks of the following form : 96 h e e 104 h h h 108 h h h 84 1 1 1 92 1 e 1 c 1 1 6 1 12 88 1 1 1 The first column of these blanks gives the comparison weights, C being the variable weight for the method of just perceptible differences. The succeeding columns give the judgments for a complete revolution of the table, H signifying a heavier; E an equal, and L a lighter judgment. The small numbers (o, 6, 12, etc.) indicate the number of bearings in the comparison weight C during that revolution of the table. The above chart is a portion of the record of a series VI, as the weight C is regularly varied by six bearings. From Fechner down, there has been a great deal of discussion about the choice of judgments; Fechner himself, objecting to the use of the equality or doubtful judgments [Elemente der Psycho- physik, I, 72, 94. Revision der Hauptpunkte der Psychophysik, 67]. He suggested that the equal and doubtful judgments be divided in half and that one half be added to the right and the other half to the wrong cases. This procedure was followed by ARRANGEMENT OF EXPERIMENTS 17 Fullerton and Cattell [Perception of Small Differences, pp. 59]. Merkel suggested that the equal judgments be not only excluded from the calculations but also from the records [Philosophische Studien, IV, 131] and this method was followed out by Kraepelin in his experimental procedure [Philosophische Studien, VI, pp. 496]. Jastrow [Amer. Jour, of Psych., I, 282] and Higier [Phil. Stud., VII, 247] contended that there should be no equal or doubtful judgments and that when the subject does not know whether the judgment should be greater or less, he should guess. Sanford [Course, pp. 357] follows this same procedure. Brown [Judgment of Difference, 1910] is the latest adherent of the exclusion of the equality or doubtful judgments. He even asserts that if the subject is forced to give a judgment of either greater or less, that he can do so. Brown apparently fails to notice that this places his uncertain judgments in exactly the same category as those of Jastrow where the subject is forced to guess. Nor should the numerical results of these variations differ widely from those of Fechner, because the laws of chance would give approximately an equal division of this class of judgments be- tween the two other classes. The only difference is that Fechner makes this division frankly while the others hide it under the technicalities of experimental procedure. Urban [Application of Statistical Methods, pp. 5] uses a very complicated system of judgments. He uses the classes of greater, equal and lighter. The degree of confidence with which the judgment is given is designated by the subject by the numerals 1, 2 or 3. Besides these he uses two guess classes, "Guess- heavier and guess-lighter". Later in the paper [Stat. Meth., pp. 14] he states that this system is too complicated and that the guess judgments are not advisable as they afford a loophole for the subject who does not wish to commit himself. Among the men who favor the use of the equality judgment we find Ebbing- haus, Wundt and Muller. Titchener [Experimental Psychology, vol. II, part II, pp. 290] points out the original reasons why the equal judgments were first excluded. He shows that it was because three classes pre- sented difficulties in mathematical treatment that were at that i8 SAMUEL W. FERNBERGER time unsurmountable. But these difficulties have since been solved and a further exclusion of judgments of this class for that reason, he calls unscientific. Titchener further points out that we have the evidence of introspection of trustworthy- observers that equal and doubtful judgments do actually occur. "If we are dealing with mind as mind presents itself to us for examination, we cannot ignore these judgments." This seems to be an unanswerable criticism. Brown, in his paper, gives a case of subjective equality. His subject stated [Judgment of Differ- ence, pp. 30] "those two are exactly the same". This may have been only one case in 4000, as Brown states, but still it must be accounted for. The subjects in our experiment both reported actual subjective equality; subject I gave this introspection quite frequently. The criticism of Brown [pp. 28] that the classes should be mutually exclusive is just as applicable to three classes as to two. His criticism that [pp. 32], for the equality judg- ment, the subject must maintain a mental standard of equality is not very impressive. Undoubtedly the subject must maintain such a standard, but he must also maintain, in the same sense, a standard for heaviness and lightness. The classes of judgments chosen for this experiment fulfil all the requirements of the case. They are susceptible to mathe- matical treatment and they are not so complicated that the sub- ject has any difficulty in giving them. The subject is not put to the strain of forcing a guess judgment. Lastly and of pri- mary importance they fit the facts of actual experience. III. METHOD OF CONSTANT STIMULI The record sheets enable us to find the relative frequencies of the heavier, equal and lighter judgments for all of the six comparison weights. In order to study the effect of practice, these observed frequencies were divided into groups of ioo in the order in which they were taken. These results are given in Table III for subject I and in Table IV for subject II. The numbers in the first column give the groups of ioo judgments in the order in which they were taken. Three columns are given to each com- parison weight in which appear the lighter, equal and heavier frequencies of the judgments on that weight. The numbers in the same columns show, on the whole, a rather close agreement. but that they are by no means identical. For this reason it is not possible to say offhand whether a constant effect of practice has taken place or whether the conditions remain the same. This decision may be affected in two ways ; by the determination of the coefficient of divergence for the probabilities for the different judgments, and by the determination of the constants for the psychometric functions. The data in tables III and IV are results of repeated observa- tions of certain unknown probabilities and the question arises whether the conditions, which determine these probabilities, remain the same or undergo variations. The coefficient of diver- gence [Lexis and Dormoy] enables us to make this decision systematically. • If the coefficient gives a value close to unity we have a normal dispersion, and may know that the conditions have been approximately constant during the experiment. If the coefficient of divergence is considerably smaller than unity we obtain an under normal dispersion and we conclude that there is a law or rule which tends to produce always the same value of the relative frequencies. If the value comes out considerably greater than one, we speak of a more than normal or over normal disper- . sion which indicates that the conditions during the experiment have varied. We may illustrate these three kinds of dispersion by 00 o J3IAB3JJ d M ON d d d d On d in ON d o\ d 00 On d 6 CO On d g d rx On d d Itjnbg 6 rx o d no o d ^8 d o d 2 d d ^8 d 8 d "8 6 O d 8- d co O d cq d ja;qSiq 8 d O d CO o d CO O d CM o d CM o d I— i O d co O d 8 d 8 d co O d o d 8 d cq O d 2 n J3IAB3JJ co *x d NO tx d CO 00 d rx rx d IN 00 d m rx d rx d 00 d oo d On tx d in ix d ON rx 6 in rx d CO 00 d IBtibg CO d oi CO d ix oi d 01 co 6 Ov 01 d 00 CO d 00 CO d in CO d co d 01 d o CO d 01 d ient>3 o d tx 01 d oi 0) 6 \o d VO 01 d oo 6 O0 oo d VO CO d tx CO d IX CO d in CO d d 3- d CO CO d J^qgi'j Ov CO d d tx d 01 d d tx co d 00 CO d VO d in 01 d 00 01 d CO d 01 CO d o CO d VO d oi Ov J31ABSJJ tx M d d rx d o" O Ol d d in d in 6 01 d o 01 d 00 d "8 6 8 d 1? d jenbg VO 01 d -3- d Ov d co 01 d VO d 00 01 d VO Ol d Ov 01 d CO CO d Ov d o CO d CO Ol d co 01 d 00 01 d aa;qSiq d o VO d VO d CO VO d m d 00 in d in d VO in 6 in in d t-H VO d 01 in d Ov VO d 9S vo d 8 d §8 J3IAE3JJ o d "8 6 m O d O d in O o" 8 d in O d 6 3 d s d 8 d 8 d 01 o d 8 6 01 d IBnbg o N d ix 6 d 8 d ix O d d VO d rx d d d CO Ol d 8 d d j3;ijSiq o jx d ix d Ov d d 00 00 d m 00 d Ov rx d Ov d 01 00 d 00 d in tx d d 8 d 00 00 d CO J3IAB9JJ o> d d t-f o d 5 d 8 d 8 d 8 d h-4 o d 8 d 8 d 8 d 8 d 5 d O O d ienbg d d oi O d 8 d 3" d 8 d in O d 3 d d 01 o d 8 d d d 01 O d ja;qSrq IT) 00 d d ix Ov o" o d VO Ov d O q in O d in Cv d 3 d o\ d 8 rx Ov d VO o o" 00 Ov d 09 •a c 3 oi co -* in VO tx oc Ov o ol co ■*j- < 22 SAMUEL W. FERNBERGER considering the drawing from an urn of black and white balls of a given relative frequency. Several urns are prepared and n numbers of drawings are made from each urn, and each ball is replaced after its colour has been noted. Under such conditions if the number of drawings is great enough, we will obtain a coefficient of divergence approximating unity, or in other words, a normal dispersion. An over normal dispersion will be obtained when the urns, before the drawings are made, contain different relative frequencies of black and white balls. Again each ball will be replaced in the urn after its colour has been noted and the same number of drawings will be made from each urn. But the varying frequencies in the urns represent changed conditions so that our results will give a coefficient of divergence greater than unity. If we can by some kind of device eliminate certain cases we will obtain an undernormal dispersion. For example, if we decide that every time three white or black balls are drawn in succession we shall record the third ball as having the opposite colour, we would obtain a coefficient of divergence less than unity. From this we conclude that in an under normal dispersion the re- sults have been tampered with in some way. We must calculate the coefficient of divergence for the probabilities of each judgment for every intensity of the comparison stimulus that we used. The formula by which the coefficient of divergence is caluclated is o \ s * v * V - \(„— 1) p(\—p) in which s is the number of observations in each series; 2v 2 is the sum of the deviations of the relative frequencies from their average, n is the number of series and p the average of the rela- tive frequencies, which is the most probable determination of the probability of the judgment. The quantity (i < — p) will then be the probability that this judgment will not occur. Table V is a double table that gives the coefficients of divergence for both subjects I and II. In the first column are found the intensities of the comparison weights. The next three columns give the co- efficients of divergence for the heavier, lighter and equal judg- ments for subject I. In the vertical column, headed Average, is given the average of the coefficients of all three judgments for bo o CO in (N CO 8 C-l CO 00 Os 3" 0> CN -*■ in OS ■ i-i* < c/j u +^ Cu — ._ ID O CO ■* CO °s SO > P cy bo 00 co in 00 in Os Os CO M M O w w O w 3 t— » en a § ■5 E op (jo J" 53 1-4 s (H t-t l_l i-i o M |_* < Ol cu tl .« CU lO "2 0) in jf Tf > p «« E, CU 00 hrCO CO t^ Os CN r^ OS Os co CN 00 M q )— 1 o" d w i-5 d W w i-1h g o 1—1 ^ en 3 Cn — C a! cy CO CN 09 so ^ 1 8 <8 00 CN tM in CO 00 so M M M HH O O I- « 3 l— » « u C «3 H* • en cu "3 c£ 00 00 Os SO OS S •s bo CO E 1-1 m cu > en -! 04 ►H m o 5 8 co 00 CO CO 01 VO vg 0) CO sj> in ■<* tx 6 00 CM CM VO 3- CM 00 VO Ix co CO ^ in tx CO 0) CM CM 00 o in ix VO in O VO CM tx CM CO CO CO 00 o\ CM a- CO tx 00 00 VO o\ tx CM 00 VO ov 00 CM tx CM Ov tx CO VO h-t CO CO 1 ^ a Ix CO CM VO Ix VO CO oo" I I I — N — — n 00 11 Ov CO m CM CM o' o' o' o" CM CM CM o CO CO O VO 00 00 Ov CM n w in q | -" 6 6 6 Ph X CM O 00 CO Ov £ VO in « 00 Ov Tf CO 00 in •<*■ n in in in vr> tx tx CO 00 CM Ov 00 Oi 00 CM CM CO CM CM CM co o •*• Ov O Ov ix CM CO n c? CO co CM o vS 1 CO -t- o H "3- CM ui VO O -* vp CM co 01 o CM CO M M in VO M- 5 00 rx CM CM < 0, £ 8 « 00 CO CO o CO PL. X on O ■* Tt On •* Tf ■ in 0) Ov vO tN. in •■h CO O VO o tN. OO 00 CO X o\ ci CO vd t< in H cvi VO 00 ■* vd tN d jr; CO 1 CN 1 in I T CN CO CVI 1 CO 1 5 vo *3- oo ^ CN in tN. Ov 00 - I Tf Tf r— 1 in 00 o 2* ■ ^ in P4 in i CN CO tN, oo in. CN CN 00 a' o in tN, VO vS 00 in S 8 cvi O od tN, Cv 8 tN. vO vd SB in 8 CO i .S! §8 8 i IN. vd tN. ci VO CVI cvi 1 Ov in o rr in co 00 Ov IN. VO tN CO CO CO cs cn CVI CO cvi 0» cn N CI CN CN * 01 '■Q. oo £5 Ov O o IN M Ov Ov VO 57 q o\ ^t CO -+ CO tN. in ■* tN. JN. in di 00 CO tN, CO ■* ■4 oo vd 8 ci M 00 in. ^ tN, in CO o o 00 00 ■* CO CO IN cn CO CO CO CO CN CO CN N C\ CO CO o\ in tN, in £ 1-4 % CN Ov en CO vo 8 CN 00 co CN £n £1 Tf o CO oo tN, in CN CO CO M CO O O. ■4 ^ CO NO NN On 00 VD 00 VO NO Tf 00 00 Oh X on 00 N I cq q rj- in On i^ On 00 o 00 On O On 00 l-H 8 1 N M 1-1 M I I o 00 m \o NO o m in w ih ON VO w 65 00 NO H hh in n Ph d d 00 On 00 R o 0) o CO O o o o o X! W pq < Ph X X $ o iO <0 On jt On o< Of) t >-H rx M On CO M co 04 (g 00 Ph X ^ 00 & C3 *■ in CO ■* •>*■ ■* s- # Tt Tf in •4 ON On ON ON On ON On On On On On ON On W C .G bfl J l-H On tx NO ■* in NO ON tx rf- 8 tx Tf NO m NO NO tx w CO in tX tx Ix 00 On ^ O tx (M 8 tx "tf *-* CO ■* 0) CM tx J3 l-H O O *-< METHOD OF CONSTANT STIMULI 41 By a close study of these two tables [XI and XII] we can dis- cover many facts that were only hinted at in the results of the coefficient of divergence. We will consider the effect of the un- conscious learning of the order of presentation of the stimuli. The order of the series was changed after the first, fifth and thir- teenth groups of 100. If there was any effect of this knowledge of the series, we should expect to find that the h for the group immediately before the change would be larger than the h in the group following. As has been noted above, the h controls the steepness of the curves and with a knowledge of the order we may expect a greater accuracy in the judgments. An examination of the tables will show that for subject I, the h's for the groups immediately after these changes are slightly smaller for the most part, than those of the series just preceding, but this is not true for subject II. This would indicate that for the latter, the influ- ence of the knowledge of the order was of no consequence. For subject I this knowledge may have had some effect upon the re- sults, but as the differences are small this effect was not considerable. For the convenience of studying the effect of practice, it was deemed advisable to plot the four curves of the course of the h's shown in figures I and II. Upon the abcissa are laid off the series in order; while the ordinates represent the values of h. These curves are strikingly different in form when we compare those of the different subjects; but for the the same subject they are quite similar in character. Although all the curves show unsys- tematic variations, the curves for subject I start with values that are fairly high and the general trend is upward. This is particu- larly noticeable in the curve h 2 for subject I. The curves hj and h 2 for the other subject start with comparatively small values and, for the first four groups with h 1; and the first six groups for h 2 , they show a very rapid rise. Both curves then fall from this maximum, and from the seventh series on, show a very gradual upward tendency. For reasons stated above we have ruled out the influence of the knowledge of the orders as being negligible. Changes in sensi- tivity, due to the weather conditions, the tone of the subject and 1500 75 50 25 1400 75 50 25 1300 75 50 25 1200 75 50 25 1100 75 50 25 1000 Curves for the values of hi. and hz. for 5ubj ect I. ■ • ! ■ \ 1 \ \ 1 \ 1 \ 1 i 1 1 1 \ 1 \ I \ i ;1 • 1 \ 1 \ I \ 1 \ 1 < \ 1 ' » 1 / l 1 i / ' 1 / V / 1 \ / / 1 \ fl / > \ A » /' i i i /l « I 1 / 1 * 1' / » / 1 J- hi. i V^ A / /\ ' / / \ i / l ' / \ l ' /^ \ \ / \ . / A. / \ ' ' /\ / \ 1 ' / \/ \ V 1 \ 1 Figure L I 3 4 5 6 7 8 9 10 II 12 13 14- tO it 12 13 14 44 SAMUEL W. FERNBERGER like influences may account for the unsystematic variations or the chance irregularities in the curves. But we have still to ac- count for the systematic tendencies that give the curves for the two subjects their distinctive character. We have eliminated all of the factors that could influence our judgments except that of practice; in fact the curves for subject II might be taken for typi- cal practice curves. This agrees with the facts regarding the subjects themselves, as it will be remembered that subject II was absolutely untrained at the start of the experiment; while subject I was highly trained, due to his having acted as subject in previous experiments. The values of h start high for the latter, but, in spite of all his former practice, the curve for subject I still shows a gradual rise. The curves for subject II indicate, by their rapid rise in the first six series, that during this time the subject became practised in the method of lifting the weights. This means that the hand move- ments became so automatic that the space error was entirely elimi- nated. This may perhaps need some explanation. At the start of the experiment there was a tendency on the part of subject II to turn his hand slightly in the direction from which the weights approached, and thus a space error, even though a slight one, was committed. This, however, was subsequently eliminated when the hand movements of the subject became automatic, and there- fore accurate in regard to movement in space. The hand move- ments improved in regularity as to time and so the time error be- came more constant as the experiment proceeded. There was still another effect of the perfecting of the automatism, since when perfected it no longer requires any of the attention of the subject, and this then, can be entirely focused on the judgments. In this connection, it will be remembered, that the results of the calcula- tion of the coefficient of divergence argue for the same notion. The drop in the curves after the maximum had been reached cannot be accounted for unless the maximum itself is caused by a chance variation that accentuates it. If a similar curve be con- structed for subject I from the results of the former experiment in which he acted as subject, it will show a similar rapid rise in the early series. Several years have elapsed between that experi- METHOD OF CONSTANT STIMULI 45 merit and this one. If the sense organs had been trained due to the practice several years ago, to a higher state of efficiency; it is reasonable to believe that due to the disuse during the interval between the experiments, they would have degenerated somewhat to the condition in which they were before they had received any training at all. The rise in the curves for subject II are not very great and this would argue against any considerable education of the sense organs. It must further be remembered that the measure of the sensi- tivity of the sense organ is not to be found in the abruptness of the curve of the psychometric functions, h, but by the interval between the two thresholds. This is the basis of the practicability of the method of constant stimuli. An examination of Tables XI and XII will show that the thresholds remain fairly constant. These values show chance variations but it is impossible to dis- cover systematic tendencies in their course. They remain con- spicuously constant for subject I. Thus the training of the sense organ, which might directly effect the sensitivity seems to have little influence, if any, as the sensitivity remains very constant and at least shows no systematic variations. This is an important consideration, because if practice effected the measure of sensi- tivity, the method of constant stimuli would have to be abandoned as a practical means of arriving at that value. It will be noticed, however, that, although the values of the h and c vary considera- c bly, — the threshold, remains practically constant, h The characteristics of the curves for the two subjects agree with the values of the coefficient of divergence obtained for the same set of results. For subject I we obtain a coefficient of diver- gence that approximates unity more closely than that for subject II, and the curves of the former show not only smaller unsyste- matic variations, but smaller systematic ones as well. Our ex- periments were probably not extended enough to show long periodicity and the results were not regular enough to reveal short periodicity. 46 SAMUEL W. FERNBERGER It then seems that the important variations in the curves of the h and c values are : 1. Those of an unsystematic or chance character, that are due to the condition of the subject and similar chance influences. 2. Variations of a systematic nature that seem to have their origin principally as the result of practice which allows a concen- tration of the attention primarily upon the judgments, when the hand movements become so automatic that they no longer require attention. 3. This effect of practice is not evenly distributed throughout the experiment, but for an untrained subject is comparatively great in the first few hundred experiments. 4. This effect of practice, however, does not seem to effect the real purpose of the experimentation, which is to ascertain the sensitivity of the subject, as the ratios that define the measure of sensitivity remain practically constant and show no systematic variations. We will turn now from the study of the effect of practice, as shown in this set of results, to a consideration of the results them- selves. The real purpose of the psychophysical methods is to ascertain the sensitivity of the subject. The problem is how much larger or how much smaller a comparison weight must be from the standard weight until a difference can be perceived between them. This statement implies that there is a distance on each side of the standard weight, within the limits of which a compari- son stimulus, although physically lighter or heavier than the standard weight, will not be judged lighter or heavier with a probability of 0.50. This distance is called the interval of uncer- tainty and is defined as the difference between the two thresholds. The measure of sensitivity is one half this interval. For example, if we have a standard weight of 100 grams and our interval of uncertainty is 4 grams, the measure of sensitivity will be 2 grams. This means that either a weight of 102 or of 98 grams will be distinguished from a weight of 100 grams with a probability of 0.50. But these are not all the considerations that are necessary to tell METHOD OF CONSTANT STIMULI 47 the sensitivity of our subject. It will be remembered that our results are effected by constant errors. Two weights of ioo grams if lifted in either different spacial or temporal relations will not be subjectively equal. In our experiments the space error was eliminated by the turning top table, but the time error was present, although it was controlled by the regular beats of the metronome. We must now find the point of subjective equal- ity, which is that weight which, under the conditions of our experi- mental procedure, will be subjectively equal to our standard weight. This is found by applying the formula Ci +c 2 K + h When this point of subjective equality has been ascertained, we can find the influence of the constant errors almost directly, by finding the difference between the point of subjective equality and the standard stimulus. We will consider how this effects our example given above. Suppose we find that the influence of the time error is — 3 grams. In this case our point of subjec- tive equality is 97 grams. The measure of sensitivity remains the same, however. Thus a restatement of our results indicates that without the influence. of the time error, weights of 95 or 99 grams can be distinguished from one of 97 grams. Table XIII is a double table containing the values for our subjects of these quantities we have just discussed. For each subject will be found in order the values of the interval of uncer- tainty ; the point of subjective equality ; and the time error. These will be found opposite the number of each of the groups of 100 experiments into which our results were divided. At the bottom will be found the averages for each of the columns. It will be noticed that the interval of uncertainty remains very constant for both subjects although the values are by no means identical. The variations are obviously due to changes in the sensitivity of the subject. These are the changes that are caused by the variations in the psychophysical makeup of the individual. There does not seem to be any correlation between the character of 48 SAMUEL IV. FERNBERGER these changes for the two subjects. We see that the averages of the interval of uncertainty are almost identical for the two sub- jects. This is of course a matter of chance and merely indicates that the sensitivity of the two subjects happened to be almost the same. The time error was a negative one in the Fechnerian sense ; that is, the second weight lifted was relatively heavier. This means that the point of subjective equality is smaller than the standard weight. Thus, on the average, for subject I for our arrange- ment of the experiment, a standard weight of ioo grams would be subjectively equal to a comparison weight of 97.31 grams. It will be noticed that the point of subjective equality is practically midway between the two thresholds. This indicates that the amount that must be added in order to perceive a difference be- tween the two weights is approximately the same as the amount that must be subtracted in order to just perceive a difference. For our calculations the point of subjective equality remained com- paratively constant for each subject, but there is considerable difference [one gram on the average] for the two subjects. The time error for subject I seems to become less as the experiment Subject I Subject II Point of Point of Number Interval of Subjective Time Error Interval of Subjective Time Error of Groups Uncertainty Equality Uncertainty Equality I 5-49 96.83 — 3-17 4.90 95-30 — 4-70 II 5.92 96.97 — 3.03 4.80 97.09 — 2.91 III 4.98 96.76 — 3-24 3-23 96.80 — 320 IV 4.85 97.64 — 2.36 370 96.17 — 3.83 V 5.52 96.76 — 3.24 3.18 96.24 — 3-76 VI 5-42 96.87 — 3-13 4.28 95-72 — 4-28 VII 4-83 97-34 — 2.66 4.98 96.29 — 371 VIII 4.42 97.12 — 2.88 5-70 95.42 -4.58 IX 4.26 97.21 — 2.79 5-47 95-50 — 4-50 X 3.84 98.06 — 1-94 4.80 96.16 — 3-84 XI 5-52 9747 — 2.53 5-72 95-67 — 4-33 XII 5.08 9749 — 2.51 5-84 97.62 — 2.38 XIII 4.80 98.32 — 1.68 6.13 96.95 — 3-05 XIV 548 97.5o — 2.50 5.25 97-43 — 2.57 Average 503 97.31 — 2.69 4.86 96.31 — 3-69 Table XIII METHOD OF JUST PERCEPTIBLE DIFFERENCES 49 proceeds. The physical conditions remained the same throughout the experiment as this constancy is attested by the fact that there are no systematic variations in the course of the time error for subject II. This systematic change, found in the results of sub- ject I is not very great and thus may be due to the chance arrange- ment of the results. If it is significant, however, it must be due to some change in the psychophysical organism. It is not due to an effect of practice, since on the basis of the discussion of prac- tice given above, we should expect to find an even stronger tendency in subject II, where it is entirely absent. IV. METHOD OF JUST PERCEPTIBLE DIFFERENCES The classical form of procedure of the method of just per- ceptible differences is to start with two stimuli at subjective equality and increase the comparison stimulus C by equal steps only up to the point where a difference is first noticed ; this point we call the just perceptible positive difference. We then start again, to obtain the just imperceptible positive difference, with the comparison stimulus C subjectively greater than the standard stimulus S ; then C is decreased until no difference is perceived between C and S. Again we start with C and S subjectively equal and decrease C until it is first judged less and thus obtain the just perceptible negative difference. Finally, to obtain the just imperceptible negative difference, we start with C subjectively less than S and increase C to the point where it is first impossible to perceive a difference. Then the just perceptible and imperceptible positive differences are averaged and we obtain the threshold in the direction of increase. The threshold in the direction of de- crease is found by averaging the just perceptible and imperceptible negative differences. This is the form in which this method was described by Fechner [Elemente der Psychophysik, I, p. J2\, Wundt [Phys. Psych., I, 1902, p. 475], Muller [Gesichtspunkt und die Tatschen der psychophysischen Methodik, 1904, p. 179], and Titchener [Exp. Psych. II, I, p. 56]. 50 SAMUEL W. FERNBERGER Another form of the method of just perceptible differences is the one used in this experiment, which decreases the amount of experimentation considerably and has also other advantages. We start with C so much lighter than S that there is a very small probability of any but a lighter judgment, and then increase the comparison stimulus C by successive steps until there is a very small probability of any but a heavier judgment. The four differences, from which the thresholds are obtained, are thea picked out of the results and are defined in the following manner. The just perceptible negative difference is the greatest stimulus upon which a lesser judgment is passed. The just imperceptible negative difference is the smallest stimulus upon which a not- lighter — either equal or heavier — judgment is given. The small- est stimulus upon which a heavier judgment is given becomes the just perceptible positive difference. Finally, the greatest stimulus upon which a not-greater — equal or less — judgment is passed is the just imperceptible positive difference. Thus in one experimental operation we obtain all the values that in the other form of the method required four distinct series. After these differences are obtained, the thresholds are found in the same manner as in the other method. Besides the matter of expediency, there is another great advantage of this form of the method over the classical form. The original method requires that we start at the point of subjective equality; but this term is not defined and the method gives no indication as to what intensity of the comparison stimulus is to be used for it. In actual practice this method is handled in such a way that the comparison stimulus is used, as a starting point, on which an equality judgment is given. This definition, however, is very loose since a great number of comparison stimuli fulfill this requirement. The variation of the method now under consideration, avoids this dis- advantage by not making use of the notion of subjective equality at all. The fact that this variation gives satisfactory results indi- cates clearly that only the so-called upper and lower limits of the interval of uncertainty are of consequence for the determination of the sensitivity of the subject. From this it seems to follow that METHOD OF JUST PERCEPTIBLE DIFFERENCES 51 the discussion as to whether the arithmetic mean or the geometric mean ought to be taken as representative of the point of subjec- tive equality, cannot be decided on the ground of this method; since it does not give any definition of that value whatsoever. [Wundt, Phys. Psych., I, 5th Ed., p. 477.] In order to avoid the so-called errors of expectation, a descend- ing series of just the reverse experimental procedure was taken for every ascending series, of the form that has been described. This influence of expectation may be twofold; either, if the sub- ject knows that he is in an ascending series, he may give a heavier judgment too soon, or that knowledge may bias him against giving such a judgment. Similarly he may give a lighter judg- ment too soon or not soon enough, if the subject knows that he is in a descending series. There is, however, a great difference of opinion as to the extent of the influence of expectation. For example, Wundt [Phys. Psych., I, 1902, p. 479, 491] believes that this tendency may be overcome by practice; and Fullerton and Cattell [Small differences] think that this influence is so great that they strongly recommend that the subject should have no knowledge of the type of series that he is judging. In many cases it is not possible to keep this knowledge from the subject. For example, in the present study, the subject could tell from the sound made by the bearings being placed in the cylinder, whether he was judging an ascending or descending series. In the old form of the experimental procedure of this method, it was not possible to keep this knowledge from the subject because, after the first judgment, he could realize which type of series he was judg- ing. On the other hand, Titchener [Exp. Psych. II, II, p. 128] states that with good subjects this expectation has no influence whatsoever. Whether this is true or not, the method of double procedure, used in this study, allows for this error if present. Since it is not possible to avoid this error, one tried to eliminate it, or at least, to minimize its influence by using ascending and descending series alternately. The errors being in opposite direc- tion in the series may be expected to cancel one another. We will now analyze the nature of the four differences upon 52 SAMUEL W. FERNBERGER which this method is based. It was believed for a long time, that the method of just perceptible differences uses notions which are foreign to the error methods. Urban, however, has shown that the common source of both methods is to be found in the notion of the probability of a certain judgment. The just perceptible nega- tive difference may be defined as the largest stimulus on which a lighter judgment was given. This implies that none of the greater comparison stimuli were judged lighter, but that on this one a lighter judgment was given. Thus by definition of the other dif- ferences, the just imperceptible negative difference implies that all the smaller stimuli were judged lighter but this one was judged not-lighter. The just perceptible positive difference implies that all smaller stimuli were judged not-heavier while on this one a heavier judgment was passed. Finally the just imperceptible positive difference implies that, on all greater stimuli, a heavier judgment was passed but on this one a not-heavier. Now let q be the probability of a lighter judgment on a given intensity of the comparison stimulus. Then by definition, the probability of a not- lighter judgment will be 1 — q, as these probabilities are mutually exclusive. In this same way 1 i — p will be the probability of a not-heavier judgment, if the probability of a heavier judgment is p Now suppose we have a series of comparison stimuli r 1? r 2 , . . . r n arranged in the order of their intensity, in which r x is the heaviest, and we use this series for the determination of the just perceptible negative difference. Let us designate the probabilities of a lighter judgment on the first, second, . . . n com- parison stimuli by q 1} q 2 , . . . q n . There exists for each stimulus a certain probability that it will be obtained as a determination of the just perceptible negative difference, which we call O l9 Q 2 , • ■ • a . Q 1 = & ; £»-! = (i-?x)(i-?,) U-O*. . ; Q n = (l-ft)(l-fc) (1-^-xK • The probability that a stimulus will be obtained as a determina- METHOD OF JUST PERCEPTIBLE DIFFERENCES 53 tion of the just imperceptible negative difference can be similarly analyzed. For the comparison stimuli r ls r 2 . . . r n we have Q\ = ?n ■ ?n-i 9 2 (1— ?0 ; Ql = 9 n ■ 9 n -i • • 9% (i— 9,) ; Ql-i = 9 n (i—9 n - l ) ; Ql = (l-O . The probabilities that the just perceptible positive difference will fall on the stimuli r l5 r 2 , • • • r n are expressed A = (l-AXl-A-0 (1— A)A ; P % = kl—pn) a—pn-i) (I" A) A 5 p n -, = (i— a)/«-i ; P n =Pn ■ Finally the probabilities with which the different stimuli will be obtained as a determination of the just imperceptible positive difference for the stimuli r l5 r 2 , . . . r n are p\ = (i-A) ; ^i = Ad-A) ; p \-x=Px -A — • • /«_ (l— A-0 ; Pl=Pi -A A-x(l— A) • We now turn to a discussion of the results of our experiments. It will be remembered that one of the comparison stimuli was changed systematically so that the results on it could serve as a determination of the thresholds of the method of just perceptible differences, and at the same time, the judgments on the other comparison stimuli enabled us to apply the method of constant stimuli; the 'relation of these two methods being the principal problem of this paper. Obviously there is no difference in the form of the individual judgments of these two methods, since they consist in the passing of a judgment of whether the comparison stimulus is subjectively lighter, equal or heavier than the standard stimulus. Our results of the method of just perceptible differ- ences are of two forms; observed values that were taken simul- taneously with those by the method of constant stimuli, and calculated values. One way of obtaining these calculated values 54 SAMUEL IV. FERNBERGER I II III IV V J VI VII VIII 1 1 IX X Total i 84.80 10 10 10 10 10 10 10 10 10 100 85.22 10 85.64 10 10 86.06 10 10 20 86.48 10 10 10 86.00 10 30 87.32 10 10 10 20 87.74 10 10 10 30 88.16 10 10 20 88.58 10 | 10 30 89.00 10 10 10 40 89.84 10 10 10 90.26 10 10 20 90.68 10 10 20 91.10 10 10 10 30 91-52 10 10 10 50 92.36 10 10 10 10 10 10 40 93-20 10 10 20 93-62 10 20 94.04 10 10 10 10 10 50 94.88 10 10 10 20 95.30 10 10 95-72 10 10 30 96.14 10 10 10 10 40 96.56 10 10 10 96.98 10 10 10 10 60 97.40 10 10 10 97.82 10 10 10 40 98.24 10 10 20 98.66 10 10 10 99.08 10 10 10 30 99.50 10 10 10 10 50 99-92 10 10 20 100.76 10 10 10 101.18 10 10 10 10 50 101.60 10 10 10 50 102.44 10 10 10 10 20 103.28 10 10 10 30 103.70 10 10 104.12 10 10 104.54 10 10 10 10 50 104.96 10 10 10 10 105.38 10 10 30 105.80 10 10 106.22 10 10 20 106.64 10 10 107.06 10 I 10 10 40 107.48 10 10 10 10 10 107.90 10 10 40 108.32 10 10 108.74 10 110.00 10 10 10 30 1 11.26 10 10 20 112.52 10 10 METHOD OF JUST PERCEPTIBLE DIFFERENCES 55 is that employed by Urban [Stat. Meth.], where the experimental data were in such a form that it was possible to apply both forms of mathematical treatment to the same set of results. This is not possible in the present study as a glance at the table below will convince one that not enough judgments were taken on each intensity of the comparison stimulus to enable us to apply the treatment of the method of constant stimuli. In this table are found the number of judgments passed on the intensities of the comparison stimulus for each series, and the last column gives the totals for each intensity used. The other way of approaching this problem is to take the results of the method of constant stimuli and calculate from them the results we are likely to obtain from the method of just per- ceptible differences. This is possible because the determination of the quantities h and c determines the entire course of the psychometric functions. We, therefore, are able to obtain the probabilities of all three judgments on any comparison stimulus, if the constants h and c for the lighter and heavier judgments are given. These probabiltiies are indeed, all that we need for the calculation of the method of just perceptible differences, and we therefore can find the most probable value of any one of the four differences. The conditions under which our results for the methods of just perceptible differences and constant stimuli were taken, were very much the same, if not entirely alike, and we therefore, may expect to obtain calculated results which agree with the observed results within the limits of accuracy obtainable in this kind of experiments. The calculation of the probabilities with which the different intensities of the comparison stimulus will be obtained as determinations of the different perceptible and imperceptible positive and negative differences is easy but rather long and somewhat complicated. So it is necessary to plan the whole work systematically in a manner which will now be described. The first step consisted in the collection of the material for the determination of the constants of the psychometric functions. The records, as they were taken during experimentation, were entered in tables, similar in form to those described above as the 56 SAMUEL W. FERNBERGER records of first entry, but the records of each series of the method of just perceptible differences were kept separate. The relative frequencies of the judgments in this division of the results by series, are found in tables XIV and XV for subjects I and II re- spectively. These are similar in form to tables III and IV, with the exception that the numbers in the first column, instead of rep- resenting groups of ioo judgments, as in the former calculation, now indicate the series of the method of just perceptible differ- ences with which they were taken simultaneously. As has been pointed out above, the length of the different series of the method of just perceptible differences varied ; there being only seven judg- ments for series X ; while series II required the passing of twenty- nine judgments. Thus it is obvious that the number of judgments from which these relative frequencies were calculated vary in a similar ratio. From these relative frequencies, the constants for the method of constant stimuli, h and c, were calculated by the same method that has been described above. In order to show the steps of this calculation, tables XVI and XVII give, for the lighter and heavier judgments of subject I, the values of [P], [xP], [xxP], [yP], [xyP], [sP] and [xsP]. The corresponding values are found for subject II in tables XVIII and XIX. These four tables are similar in form to tables VII-X, which have been described above. The numbers in the first five columns give the coefficients for setting up the normal equations and the data of the columns [sP] and [xsP] show that all the necessary checks are fulfilled. Tables XX and XXI, which are similar in form to tables XI and XII, give the constants h and c for the lighter and heavier judgments for both subjects. The thresholds S x and S 2 are obtained in the same way as in the former calculation and are included in these tables. The series of the method of just perceptible differences were performed in haphazard order, so that the results that go to make up the values included in these tables, are from various parts of the experiment at different stages of practice, and we, therefore, must not look for the effect of practice in these constants. In fact the results that were taken at the end of the experiment in each series, where the practice was high, tend to cancel any J3IAE3]-[ l*n&3 jajqSi-j J3IAE3H jBnb 3 j3}q.oig J31AB3JJ * pnbg J3}q§i1 J3IAB3U jEnbg jaiqSi^ J3IAB3JJ pnbg J3}q3iq J3IAB3JJ lEnb 3 js;qSi'j in h O "3-00 MOOOC aN r^H ino Tfioooo OnOnOnOnOnCnOnOnOn nmo tow co oooog N© mNO N ^"00 m OOOO ooooooooo 04 ^J- O oo Q< nn " m o o o o o o O N N t o o o o OnnO inNin04OOIQON N tx ncO n on n « 5 oi n n nno n noo oo oo oo in m M - 04 NtOM WNO M h-thHt-HOJl-HMMtHh-tHH c<3« inKClOOOOO K no t-voo oo wkn nt^m oooooooooo inocoOONwi-' on "*oo o u^no n n hNNNmmNN NNfoOOO O co 00 Hinno KO O iono in M- o in in\o i-> O noo * ■* t|- n tr* t|- io to ci w mnmKO p con co oj i-it^i-. -to^cn whmOwi-imO roOmOOOOOO^t- »0 HKtiinOHinOH NNNNtltOflNNtO lOOOMnlOO OK? N movo noo m in co o NO NO NO NO NO invoke nno OOOOooNOOl 8 m ot 04 en w m in < o o o o o o o < in rj- o co o O o in in t(- O « O NO in^-N w m invo OKNflOlO mNQ ON N00 OvQ OiON txOO 00 00 00 ONCO 00 00 00 00 00 co oo -o^c\o\ I — I I — I l-*^!— 11— I J3IAB3JJ pnbg jsjqSiq J3IAB3fJ Ftibg js^qSil J3IAB3fJ[ pmbg j3;qSiq 04inoO04coQ00m< ■*m(D M TJ-00 OMM 0\ On On On Onoo On On On < NONOOONOOM ■>r in -q-vo "OVCMnH OOOOOOOOO novo g oooyjo « !^o\ HflONOfJHHBN oooooooooo Qitsv)Knfi o O O Tf On On "t'OI'NO O « cn. is.00 oq oo. oq oo. oq qoq co in in O ^Q O in in 04 " " M . M . " M . M . ". Q •"! 00 00 O CO 0000 O WiO't in-^-t^cocoincoN04 m q q q q q O O O q q no "^" in o oo tj- o cooo o 01 m tJ- CO00 ton H (OO co co co CO 04 CO co 04 04 oo N On O N 04 00 O 04 04-O "NO r^oo ^- O in m no oo ■^"04 04 04 OO CO CO Tf CO 04 kkuio moo Omo "t "? "00 00KiOi-KO h 04 TfrooooocooorOTf^j- aaiABafj pmbg j3;qSiq J3IAB3JJ jEnbg J3;q3ig J31AE3JJ pmbg jsjqSja ^g „4>.0404Oin0400 ^fON^ONifM km Tj- Wl-iOOll-ll-ll-lM On £-. m O m co O o O ■*)- f>,£°° JMOmSS" iono inNO ininNONONONO Nh OOmNOO"lM t?^O40O04«O4in04TJ- qqoooooobo O NO co O inini-i 04 NO O 00 OO0Q In. 4> mm04i-iOOOO SSO co in Q oinONi On in^5 m n 5 o no O c i>oq oo n NOO Onoo onc o co o k inNO O OOMnMMH q q O o o o o HinmoOKOMn^ g oo 04 04 «">no nnw h OOOOOOOOOO On04 in co m N oo 04 no 00NO inr>.04 "NOnonoOO 0\C^0}0\ONONChO\0NQ HMHk >'i MHH, y"y' HI — II — I-" • — ^ f — IHMrN HH l^K,HM > X in in in oo oo in o inoo oo CJ>3f04i-ico04000400Tj- jjq 04. 0404040O04040404O4 ^ Cm Cm P-, X Cm PL, X Ch C\ 0> i-h OWO m.VO VO O CO iown Q\ r-s i-^ ts. cwo Tt- 1-^ 0\ coOO ^t-"0 «uih h ooo ooo n r^oo r^vo oo !ON^N N N t( N fl M t+ O O M mOO fl KO<^ pj oj •3-oq q invq o "p "p tJ- in i-< -4 in in in co ts.00 0)ONOOO\00 r^ o> cocococo Oim« m oq q o© <>vq i>oo w oo rit^iOMKKHOO'nir) co t» tj- n n w III I I I I I Wh OOO ^VO O "1 1^ N h Of) tC^O OiM Oih VO VO riroxfH 00 vo t^O o^^ts covo O cvi vo N r^ c*vo oo Tf Tf OKN N r<)K tJ-00 O r^ m oi KO\Kn cooo w ix r^oo xj- ts. tN.og ooioo h nONOO&o ovoo o rx covo O r^vo N oi h c; VO VO invo O O co inoo w ■*« ■*« N m km OOI cocococococococooi CO CO Cm > X < OINNOIOINPINN i-i in Qv r^ ON covOvn M3 O 00 CV) O CN N CV) Tt-OOO i-i NN N l^VO K1T1M h 01 00 h^O OiN N"^ H ■*""■ i^w ■ O ooofi O\oq Pi O CiVO coOO' co rf rt- •<* h h roc^ CO co Cq oj in c» ^j- moo O vo cooo ui^TrNOMHOroto ►-• 0) co in in en iov)h o 6 6 6 6 6 6 6 6 6 6 m 00 N N NfO 1 * Tj-00 O Cl h\o -d- q Tf O Qv co covq m in ei o\vo K. ~^-co cta rj- -**• ■* -4oo cn N ts. invo "* in invo co ©vod vo KKO nk invo in O OICVICVICOCVINNCNCNCN fOM h h O fONOO COOO 00 0OOO0 MN COOO ON O vo r^oo O 00 i^vo vo m i-i oi cvi cvi co cvi cvi cs pj cn cvi *l=!^>>trssxx y^ » — ^ i — ; — ni I-H ^Iv.HH > > < P-, X to oo oivo oofiinoo co oi 19 m m a Tt O\oq q m vq oiK.oiuSts.M-HTfuSrv co oi Q\vo " -r oi invo oo t^ in cvoo ^U5>« «oo OOOOOOsOOlVOO r^vO oioooioioooooioocvioi p7 vO h\o KN m t^vO -too moo m>o OvinoO mopvo vo ^ 01 in r^ t)-vO Ov O m ■4 oo d 00 vd «' K. o\ d ■* O 01 m i-i 0) inoO m GvOO co oo oo oo co oo oi cooi oi p7 X VO oo vo ■* t^\o "* Q\ O vo m N ish ^t-vo 01 Os moo t-* o\ tN. ©.vo -+kq m Tf co m oi Tt- co rf moo h t}- m q cocococococococococo CO 'u m 57 X to I— 1 ON Tf COOO t^I^O CO O 00 o\ ts.vq rj-oq rs. ^-00 00 oi d oi d in t^. in invd oi vo in co Qnvo O i-i os 01 co OOOOvOr^OlOOoOcoO OICOOICOOICOOIOIOICO 57 inOinoivOolt^ovo^t in moo 00 Q co t+ oi vO On >-< OvinqoOvo •*■>*■ o\h'\o fivd li^d k ao hoo n» o\Tto OICOCOCOOICOOIOIOICO 57 X oiooiot^coOOv inoo 00 vo co t^ -3- o\oo in fx 01 01 rv.- H rq oioq oi vq vo oi invo' •* M- oi w ts.vd MH N f|tO HHOl-rfl-. l l M l l l l l | i— i 1^01 KNtOvOmNK vooicooicoooococoin 01 co tj- -3- in oi oi ^t-vo oi 6666666666 1 1 1 1 1 1 1 1 1 1 57 X ->tinvo oi tJ-oo " moo m KOoo q hoo q TtH **■ t^. 6 invd ■* oi in oi co 01 Q Q 01 00 in inoo On •>*• CnCO vO 01 00 co •*■ m ^vp oooovi-ir^oioooscoo 01CO01CO01CO0101 Olco 57 X 00 Nm inoo t^vo -* 01 oi CO IN.0O 00 ON « Tf 01 CO CO co oi co dv ^t dod t^. oi d Ovi-i O "00 0100 Cs-fi-i OICOCOCOOICOOIOIOICO 57 ■ — i 00 Q ■* COOO vOOim mN 00 On co On 01 vo co tJ- 1-1 in Ov 1-1 1-; 01 ov co Ov q in 1-1 oicococooicooicooico CO t/3 1 — 11 — I ^*° i^* ^ I — 11 — irS?S MM K^^i—ll— 1 6o Subject I SAMUEL IV. FERNBERGER Lighter Judgments Heavier Judgments Series h, c, s, h 2 c 2 s 2 I 0.11214 10.655 95.02 0.12064 12.076 100.10 II 0.1 1266 10.705 95-Q2 0.12071 12.050 99.82 III 0.1 1354 10.732 94.52 0.12315 12.287 99-77 IV 0.1 1883 11.266 94.81 0.1 1088 11. 121 100.30 V 0.12186 11.480 04.20 0.12373 12.286 09.30 VI 0.1 1265 10.644 9448 0.12340 12.332 99-93 VII 0.1 1650 10.958 94.06 0.12359 12.246 99.09 VIII 0.11833 11.211 94-74 0.12528 12.459 9945 IX 0.132 13 12.519 9475 0.12345 12.323 99.82 X 0.12429 11.670 93-89 0.14430 14.276 98.93 Table XX Subject II Lighter Judgments Heavier Judgments Series h. Ci s, h 2 c 2 s 2 I 0.1 1 824 11.027 93-26 0.1 1 783 11.659 98.94 II 0.11210 10.587 04.44 0.1 1 134 11.018 98.96 III 0.1 1295 10.636 94.16 0.1 1297 11.092 98.19 IV 0.1 1 189 10.491 9376 0.10936 10.752 98.32 V 0.1 1007 10.263 9324 0.1 1875 11.740 98.86 VI 0.10147 9.472 93-34 0.10204 10.038 98.37 VII 0.13067 12.273 93-93 O.I 1000 11.777 98.96 VIII 0.1 1490 10.831 94-27 0.1 1320 11.208 99.00 IX 0.12538 11.810 94.20 0.13827 13-597 98.34 X 0.12024 "■371 9457 0.10909 10.805 99-04 Table XXI variation of the results at the beginning, where practice was low. Thus these values of the threshold show remarkably little varia- tion. The numbers under the headings S x and S 2 give the thresholds in the direction of increase and decrease by the method of constant stimuli, for the group of experiments that were made simultaneously with the corresponding series of the method of just perceptible differences. After having obtained the constants h and c, we can ascertain the probability of a heavier or lighter judgment for any intensity of the comparison stimulus, by the formula p = y 2 ± y 2 *(y) . This calculation is made in two steps ; as first y must be calculated by the formula 7 = hx — c in which x is the value of the intensity of the stimulus for which we are calculating the probability. The value of *(y) is then looked in a table of the probability integral [Czuber. Wahr- METHOD OF JUST PERCEPTIBLE DIFFERENCES 61 scheinlichkcitsrechniing, 1908, pp. 388-391]. This value of *(y), which takes the sign of its argument, is substituted in the formula given above and we obtain the probability of that judgment for the weight x. These values must be found for every intensity of the comparison stimulus that was used in the corresponding series for the method of just perceptible differences. In actual practice this calculation is very much simplified. If we desire the probabilities of the lighter judgments q, for series II, we first find y 1 for x = 84.80, this being our lightest compari- son weight. We then find the value of y for x equal to the amount of difference of the steps of this series. This value is subtracted from y x and we obtain y 2 or that of the next heavier comparison stimulus. Then y is subtracted from y 2 and we obtain y 3 for the next heavier comparison weight, and so forth. As our y values are only carried out to four places, corrections have to be made when necessary. A check on this step of the cal- culations is effected by finding the y of the heaviest comparison weight by means of the formula, and ascertaining if it coincides with the value obtained by the series of subtractions. These y's are then successively introduced into the formula for finding the probabilities, given above, and the different probabilities of the lighter judgment — q ■ — are obtained for all of the comparison stimuli used. The probabilities that a not-lighter — equal or heavier — judgment will be passed are obtained by subtracting the probabilities of a lighter judgment, in each case, from unity. From this series of the q's and (I — q)'s we may obtain the just perceptible negative difference and the just imperceptible negative difference, by means of the formulae given above. This calculation is much simplified by the use of logarithms. These are looked up for every value of q and 1 — q. Then the probabil- ity that our lightest weight will be obtained as a determination of the just imperceptible negative difference is Q 1 = (1 — q 2 ) and is found directly. Adding log q ± to log (1 — q 2 ) gives log Q 2 which is the probability that our next heavier comparison stimulus will be obtained as a determination of this difference. Then log q x is added to log q 2 and to their sum is added log ( 1 — q 3 ) which gives Q 3 . To the sum of log q x and q 2 is added log q 3 and to this sum is added log ( 1 — q 4 ), which gives the 62 SAMUEL W. FERNBERGER probability that this difference will fall on the fourth comparison weight. This scheme of calculation is very simple and is learned rapidly ; its mechanism will be easily understood from the follow- ing example, where the formulae are found alongside part of one of our calculations. log Q x log q x log (i-qj log (q 2 ) log Q 2 = log (i-qj q 2 log (i-qi) log (i-q 2 ) log (i-qi)(i-q 2 ) log q 3 log Q 3 = log (i-qi)(i-q 2 ) q 3 log (i-qi)(i-q 2 ) log (i-q 3 ) log (i-qi)(i-q 2 )(i-q 3 ) log q 4 = -94448 — 3 = .99616 — 1 = .44716 1 — 2 = -44332 — 2 = .99616 — 1 = -98767 — 1 = .98383 - 1 = .81558 - 2 = -79941 — 2 = .98383 - 1 = 97063 — 1 = .95446 — 1 = .07700 — 1 log Q 4 = log (i-qi)(i-q 2 )(i-q 3 ) q 4 — -03146 — 1 In this way the calculation is reduced to a number of successive additions by carrying the sums of the terms that accumulate. These additions give us the values of Q and Q lt or the probabil- ities with which the just perceptible and imperceptible negative differences will fall on the different values of the comparison stimulus. A similar calculation with the h and c for the heavier METHOD OF JUST PERCEPTIBLE DIFFERENCES 63 judgments gives the values of P and P 1} or the probabilities that the different intensities of the comparison stimulus will be found as a determination of the just perceptible and imperceptible posi- tive differences. Both sets of calculations for the heavier and lighter judgments, had to be performed for each of the ten series for both subjects. A check on this part of the calculations is effected by the fact that, as these are probabilities of mutually exclusive events, all the values in one set must add up to unity. This had to prove within a limit of 0.0001, which was the exactitude of our calcula- tions. It was found for the longer series that this calculation did not have to be carried through for all of the values of x, as fre- quently the probabilities of the differences became so small as to no longer affect the last decimal place. In the short series, on the other hand, it frequently happened that there was a remainder, and this had to be calculated as it was necessary for a check of the correctness of our work. This remainder has some significance since it gives the probability that the series of comparison stimuli r t , r 2 , . . . r n will be gone through without giving a determination of the quantity sought fbjr. Thus the remainder for the just perceptible positive difference gives the probability that no stimulus of the series will be obtained as a determination of this quantity or — what is the same — that no heavier judgment will be given on any of the stimuli used. This remainder has the form of a product of probabilities and obviously becomes very small if any factors are small. Thus in our short series there was greater likelihood of having larger individual probabilities and we find a tendency for greater remainders to occur in the short series rather than in the long ones. This is the reason of the rule that the series of comparison stimuli should be extended so far that we will have a very high probability of obtaining only heavier judgments on the largest comparison stimulus, and only lighter judgments on the smallest stimulus used. It is, therefore, not desirable to work with series in which the remainders are of at all considerable size. None of the remainders of our series are large enough to invalidate our results in any way. Tables XXII-XXXI give the results of these calculations for Series I Comparison Stimulus q Q Q 1 P p P 1 84.80 0.947 O.OOOI 0.0526 0.005 0.0046 8774 0.876 0.0003 0.1 177 0.018 0.0174 90.26 0.774 0.0014 0.1871 0.047 0.0456 92.36 0.663 0.0034 0.2164 0.093 0.0869 94.04 0.561 0.0066 0.1870 0.151 0.1273 95-30 0.482 O.OIIO 0.1239 0.206 0.1482 O.OOOI 96.14 0.429 . 0.0172 0.0658 0.250 0.1422 0.0001 96.56 0.403 0.0270 0.0295 0.273 0.1168 0.0004 96.98 0.378 0.0406 0.0124 0.297 0.0925 0.0013 97.40 0.353 0.0586 0.0049 0.322 0.0705 0.0038 97.82 0.328 0.0809 0.0018 o.349 0.0516 0.0106 98.24 0.304 0.1082 0.0006 0.376 0.0362 0.0269 98.66 0.282 0.1392 0.0002 0.403 0.0242 0.0640 9950 0.234 0.1548 O.OOOI 0.459 0.0165 0.1261 100.76 0.181 0.1437 o.545 0.0106 0. 1948 102.44 0.1 19 0.1075 0.655 0.0058 0.2255 104.54 0.065 0.0630 0.776 0.0024 0.1889 107.06 0.028 0.0278 0.883 0.0006 O.I 120 110.00 0.009 0.0088 0-954 O.OOOI 0.0456 2 I. OOOI 1. 0000 I.OOOO I. OOOI R 0.0000 0.0000 0.0000 0.0000 Table XXII Series II Comparison Stimulus q Q Q 1 P p P 1 84.80 0.948 0.0518 0.005 0.0052 85.64 0.932 0.0641 0.008 0.0077 86.48 0.913 0.0767 0.013 0.013 1 87.32 0.890 0.0886 0.016 0.0160 88.16 0.863 0.0986 0.023 0.0222 89.00 0.831 0.1047 0.032 0.0301 89.84 0.796 O.OOOI 0.1054 0.044 0.0398 90.68 0.755 0.0002 0.1003 0.059 0.0513 91-52 0.712 0.0007 0.0893 0.078 0.0635 92.36 0.664 0.0020 0.0740 O.IOI 0.0761 93-20 0.614 0.0048 0.0565 0.129 0.0871 94-04 0.562 O.OIOI 0.0394 0.161 0.0949 94.88 0.509 0.0187 0.0248 0.199 0.0982 O.OOOI 95-72 0.456 0.0308 0.0140 0.242 0.0953 0.0004 96.56 0.403 0.0456 0.0070 0.288 0.0863 0.0014 97.40 0.352 0.0615 0.0031 0.339 0.0723 0.0039 98.24 0.304 0.0764 0.0012 0.394 0.0554 0.0090 99.08 0.259 0.0877 0.0004 0.449 0.0384 0.0182 99.92 0.218 0.0944 O.OOOI 0.506 0.0238 0.0322 100.76 0.180 0.0953 0.563 0.0131 0.0507 101.66 0.147 0.0914 0.618 0.0063 0.0716 102.44 0.119 0.0836 0.672 0.0026 0.0915 103.28 0.094 0.0732 0.722 0.0009 0.1076 104.12 0.074 0.0619 0.768 0.0003 0.1 167 10496 0.057 0.0505 0.809 O.OOOI 0.1 186 105.80 0.043 0.0399 0.846 0.1 132 106.64 0.032 0.0309 0.878 0.1026 107.48 0.024 0.0232 0.904 0.0886 108.32 0.017 0.0170 0.926 0.0736 2 0.9999 1. 0000 1.0000 0.9999 R 0.0000 0.0000 0.0000 0.0000 Table XXIII Series III Comparison Stimulus q Q Q 1 p p p 1 84.80 0.941 0.0592 0.005 0.0046 86.06 0.913 0.0819 0.008 0.0084 87.32 0.876 0.000 1 0.1062 0.015 0.0148 88.58 0.830 0.0007 0.1279 0.026 0.0249 89.84 0.774 0.0030 0.1412 0.042 0.0396 91.10 0.709 0.0095 0.1408 0.065 0.0594 92.36 0.636 0.0233 0.1248 0.099 0.0836 0.0001 93-62 0.557 0.0461 0.0965 0.142 0.1088 0.0006 94.88 0.477 0.0754 0.0636 0.197 0.1294 0.0029 96.14 0.397 0.1043 0.0349 0.264 0.1388 0.0 1 00 97.40 0.322 0.1245 0.0156 0.340 0.1318 0.0263 98.66 0.253 0.1312 0.0055 0.423 0.1083 0.0543 99.92 0.193 0.1239 0.0015 0.510 0.0753 0.0904 101.18 0.142 0.1067 0.0003 0.596 0.0431 0.1251 102.44 0.102 0.0848 0.0001 0.678 0.0198 0.1470 10370 0.070 0.0630 0.752 0.0071 0.1505 104.96 0.047 0.0441 0.816 0.0019 0.1367 106.22 0.030 0.0293 0.869 0.0004 O.I 122 107.48 0.019 0.0186 0.910 0.0001 O.0847 108.74 0.0 1 1 0.01 14 0.941 O.O592 2 0.9999 1.0000 1. 0001 1. 0000 R 0.0000 1 0.0000 0.0000 O.OOOO Table XXIV Series IV Comparison Stimulus q Q Q 1 ' P P P 1 84.80 0.954 0.0464 0.008 0.0076 86.48- 0.919 0.0004 0.0770 0.015 0.0151 88.16 0.868 0.0026 0.1157 0.028 0.0278 89.84 0.798 0.0I2I 0.1537 0.051 0.0480 91.52 0.710 ■ 0.0370 0.1762 0.084 0.0761 0.0003 93-20 0.606 O.0803 0.1696 0.133 0.1096 0.0019 94.88 0.495 O.I297 0.1320 0.198 0.1415 0.0091 96.56 0.384 O.1634 0.079*3 0279 0.1603 0.0293 98.24 0.282 O.1672 0.0357 0.374 0.1547 0.0680 99.92 0195 0.1434 0.0113 0.476 0.1235 0.1 194 101.60 0137 0.1 166 0.0024 0.581 0.0788 0.1646 103.28 0.077 0.0713 0.0003 0.680 0.0387 0.1851 104.96 0.044 0.0424 0.768 0.0140 0.1749 106.64 0.023 0.0231 0.840 0.0036 0.1433 108.32 O.OII 0.0106 0.896 0.0006 0.1041 2 1. 0001 0.9999 0.9999 1. 0000 R 0.0000 0.0000 0.0001 0.0000 Table XXV Series V Comparison Stimulus q Q Q l p p P 1 84.80 0.947 0.0006 0.0526 0.006 0.0056 86.90 0.896 0.0059 0.0986 0.015 0.0149 89.00 0.815 0.0288 0.1570 0.036 0.0351 0.0002 91.10 0.703 0.0837 0.2055 0.076 0.0716 0.0025 93-20 0.568 0.1566 0.2101 0.143 0.1250 0.0161 95-30 0.425 0.2040 0.1587 0.242 0.1813 0.0586 9740 0.291 0.1970 0.0833 0.370 0.2099 0.1315 99-50 0.181 0.1493 0.0280 0.514 0.1832 0.1978 101.60 O.IOI 0.0928 0.0056 0.656 0.1 138 0.2133 103.70 0.051 0.0494 0.0006 0.779 0.0465 0.1757 105.80 0.023 0.0227 0.852 O.OII2 0.1380 107.90 0.009 0.0092 0.934 O.O0I8 0.0662 2 1. 0000 1. 0000 0.9999 0.0999 R 0.0000 0.0000 O.OOOI 0.0000 Table XXVI Series VI Comparison Stimulus q Q Q 1 p p P 1 84.80 0.939 0.0024 0.0610 0.004 0.0040 87.32 0.873 0.0176 0.1 193 0.019 0.0189 O.OOOI 89.84 0.770 0.0677 0.1885 0.039 0.0381 0.0013 92.36 0.632 0.1509 0.2323 0.093 0.0873 0.013 1 94-88 0475 0.2161 0.2094 0.189 0.1610 0.0619 9740 0.321 0.2151 0.1287 0.329 0.2272 0.1556 99-92 0.193 0.1602 0.0490 0.499 0.2313 0.2328 102.44 0.103 0.0953 0.0105 0.669 0.1553 0.2299 104.96 0.048 0.0466 0.0012 0.810 0.0623 0.1629 107.48 0.019 0.0188 O.OOOI 0.906 0.0132 0.0890 110.00 0.007 0.0070 0.960 0.0013 0.0394 112.52 0.002 0.0020 0.986 O.OOOI 0.0140 2 0.9997 1. 0000 1.0000 1.0000 R 0.0002 0.0000 0.0000 0.0000 Ti> BLE XXVII Series VII Comparison Stimulus q Q Q 1 p p P 1 84.80 0.936 0.0090 O.0635 0.006 0.0062 87.74 0.851 0.0546 0.1395 0.024 0.0235 O.OOI2 90.68 0.71 1 0.1581 0.2301 0.070 0.0684 0.0158 93-62 0.529 0.2494 O.2672 0.168 0.1519 0.0837 96.56 0.340 0.2434 O.I977 0.328 0.2456 0.2067 99.50 0.185 0.1623 O.083I 0.529 0.2670 0.2734 102.44 0.084 0.0802 O.OI73 0.721 0.1713 0.2243 105.38 0.031 0.0307 O.OOI5 0.865 0.0572 0.1260 108.32 0.009 0.0094 O.OOOI 0.947 0.0085 0.0523 1 1 1.26 0.002 0.0023 0.983 0.0005 0.0166 2 0.9994 1. 0000 I.OOOI 1. 0000 R 0.0006 0.0000 0.0000 0.0000 Table XXVIII Series J 'II I Comparison Stimulus q Q Q 1 P P P 1 84.80 0.952 0.0116 0.0480 0.005 0.0048 O.OOOI 88.16 0.865 0.0775 0.1288 0.023 0.0227 0.0035 91-52 0.704 0.2137 0.2433 0.080 0.0779 0.0409 94.88 0.491 0.2927 0.2952 0.209 0.1872 0.1678 98.24 0.279 0.2308 0.2053 0.416 0.2943 0.2979 101.60 0.126 0.1 189 0.0695 0.648 0.2678 0.2768 104.96 0.043 0.0427 0.0096 0.835 0.1214 0.1551 108.32 0.012 0.0116 0.0004 0.942 0.0225 0.0580 2 0.9995 1. 0001 0.9986 I.OOOI R 0.0006 0.0000 0.0014 0.0000 Table XXIX Series IX Comparison Stimulus q Q Q l P P P 1 84.80 0.968 0.0187 0.0316 0.004 0.0044 0.0002 88.58 0.876 3.1358 0.1203 0.025 0.0249 0.0084 92.36 0.673 0.3192 0.2773 0.097 0.0940 0.0801 96.14 0.397 0.3120 0.3443 0.262 0.2295 0.2500 99.92 0.167 0.1573 0.1887 0.507 0.3280 0.3295 10370 0.047 0.0468 0.0360 0.752 0.2400 0.2203 107.48 0.009 0.0086 0.0018 0.909 0.0720 0.0886 1 1 1.26 O.OOI O.OOIO 0.977 0.0070 0.0229 2 0.9994 1. 0000 0.0998 1. 0000 R 0.0006 0.0000 O.OOOI 0.0000 Table XXX Series X Comparison Stimulus q Q Q 1 P P p 1 84.80 0.94s 0.0544 0.0550 0.002 0.0020 0.0006 89.00 0.805 0.2375 0.1842 0.021 0.0214 0.0287 93.20 0.548 0.3583 0.3435 0.121 0.1 183 0.2130 9740 0.269 0.2402 0.3050 o.377 0.3239 0.3999 101.60 0.088 0.0860 0.1023 0.707 0.3779 0.2662 105.80 0.018 0.0182 0.0097 0.919 0.1440 0.0796 110.00 0.002 0.0024 0.0002 0.988 0.0125 0.0120 2 0.0970 0.9999 1. 0000 1. 0000 R 0.0031 0.0000 O.OOOI 0.0000 Table XXXI subject I; one table being given to each of the ten series [I-X]. In the first columns are found the intensities of the comparison stimuli used in each particular series. The second columns give the probabilities of a lighter judgment on these comparison stimuli. The third and fourth columns give, under the head- ings Q and Q 1; the probabilities with which the different com- SAMUEL W. FERNBERGER parison stimuli appear as determinations of the just percep- tible and of the just imperceptible negative difference. The next column gives the probabilities with which heavier judgments may be expected on the stimuli. These last values serve as a basis for the calculation of the probabilities of the different stimuli for being observed as determinations of the just per- ceptible and of the just imperceptible positive difference, the values of which are found in the last columns of the table. The numbers at the bottom of each column — marked 2 — give the sums of all of the terms and the numbers R give the values of the remainder. These remainders and the sums of the proba- bilities when added up come to unity within the limit of o.oooi and thus prove the correctness of the computation. Tables XXXII-XLI are similarly constructed and give the correspond- ing values of subject II. It will be noticed that the size of the remainder is in no case so large as to invalidate the series. There is a tendency for the size of the remainder to increase as the series decreases in length. A glance at the table that follows, which contains all the remain- ders for both subjects, shows this tendency. In the first four Series I (G) Comparison Stimulus q Q Q 1 p p P 1 84.80 0.921 0.0007 0.0786 0.0 10 0.0100 8774 0.822 0.0036 0.1642 0.031 0.0305 90.26 0.692 0.0099 0.2331 0.074 0.0712 92.36 0.560 0.0183 0.2308 0.136 0.12 10 94.04 0.448 0.0265 0.1618 0.207 0.1587 O.OOOI 95-30 0.367 0.0342 0.0833 0.272 0.1653 O.OOOI 96.14 0.315 0.0429 0.0330 0.320 0.1418 0.0006 96.56 0.291 0.0558 0.0108 0.346 0.1042 0.0016 96.98 0.267 0.0699 0.0032 0.371 0.0733 0.0040 9740 0.244 0.0847 0.0009 0.398 0.0494 0.0097 97.82 0.223 0.0995 0.0002 0.425 0.0317 0.0218 98.24 0.202 0.1 132 0.0001 0.453 0.0195 0.0457 98.66 0.183 0.1254 0.481 0.01 13 0.0902 99-50 0.148 0.1 192 0.532 0.0065 0.1 501 100.76 0.105 0.0941 0.619 0.0035 0.1095 102.44 0.062 0.0596 0.720 0.0015 0.2036 104.54 0.030 0.0292 0.824 0.0005 0.1550 107.06 0.0 10 0.0105 0.912 O.OOOI 0.0853 110.00 0.003 0.0026 0.067 0.0328 2 0.9998 1.0000 1. 0000 1. 0001 R 0.0001 0.0000 0.0000 0.0000 Table XXXII Series II Comparison Stimulus q Q Q 1 p P P 1 84.80 0.938 0.0621 0.013 0.0130 85.64 0.919 0.0763 0.018 0.0178 86.48 0.897 0.0890 0.025 0.0240 87.32 0.871 0.0998 0.033 0.0316 88.16 0.841 0.1072 0.045 0.0408 89.00 0.806 O.OOOI 0.1098 0.058 0.0510 89.84 0.767 O.OOOI 0.1061 0.076 0.0621 90.68 0.725 0.0005 0.0963 0.096 0.0731 91-52 0.679 0.0015 0.0814 O.I2I 0.0829 92.36 0.629 0.0037 0.0638 O.I49 0.0901 93-20 0.578 0.0080 0.0457 O.182 0.0937 94.04 0.525 0.0153 0.0297 0.219 0.0921 O.OOOI 94-88 0.472 0.0261 0.0173 O.260 0.0854 0.0004 9572 0.420 0.0400 0.0090 O.305 0.0739 0.0012 96.56 0.368 0.0555 0.0041 0.353 0.0595 0.0031 97.40 0.319 0.0708 0.0016 0.403 0.0439 0.0071 08.24 0.273 0.0834 0.0006 0.455 0.0296 0.0141 99.08 0.231 0.0916 0.0002 0.508 0.0180 0.0252 99.92 0.193 0.0946 O.OOOI 0.559 0.0098 0.0404 100.76 0.158 0.0922 0.612 0.0047 0.0581 101.60 0.128 0.0856 0.661 0.0020 0.0766 102.44 0.102 0.0762 0.708 0.0007 0.0931 103.28 0.081 0.0653 0.752 , 0.0002 0.1 054 104.12 0.062 0.0539 0.796 O.OOOI 0.1089 104.96 0.048 0.0433 0.828 0.III2 105.80 0.036 0.0337 0859 0.1055 106.64 0.027 0.0256 0.887 0.0959 107.48 0.019 0.0191 0.910 0.0835 108.32 0.014 0.0139 0.930 0.0703 2 1. 0000 1. 0001 1. 0000 1. 0001 R 0.0000 0.0000 0.0000 0.0000 1 ABLE XXXI [I Series III Comparison Stimulus q Q Q 1 P p P 1 84.80 0.933 0.0673 0.016 0.0162 86.06 0.902 0.0910 0.026 0.0260 87.32 0863 0.0002 0.1155 0.041 0.0395 88.58 0.814 O.OOII 0.1351 0.062 0.0573 89.84 0.755 0.0042 0.1446 0.091 0.0785 91.10 0.688 0.0123 0.1394 0.129 0.1006 0.0002 92.36 0.614 0.0284 0.1 187 0.176 0.1200 0.0009 93.62 0.535 . 0.0533 0.0876 0.233 0.1308 0.0036 04.88 0.454 0.0829 0.0550 0.298 0.1286 0.0109 96.14 0.376 O.I 100 0.0286 0.372 0.1 125 0.0262 9740 0.303 0.1270 0.0120 0.450 0.0855 0.0509 08.66 0.236 0.1297 0.0040 0.530 ' 0.0554 0.0820 99.92 0.179 0.1 198 0.0010 0.609 0.0299 O.I 120 101.18 0.13 1 O.IOI2 0.0002 0.684 0.013 1 0.1326 102.44 0.093 0.0790 o.75i 0.0046 0.1387 103.70 0.064 0.0579 0.81 1 0.0012 0.1302 104.96 0.042 0.0402 0.860 0.0002 0.1117 106.22 0.027 0.0263 0.000 O.OOOI 0.0887 107.48 0.018 0.0165 0.931 0.0656 108.74 O.OIO 0.0099 0.954 0.0459 2 0.0999 1. 0000 1. 0000 I.OOOI R o.oooo 0.0000 0.0000 0.0000 Table XXXIV Series IV Comparison Stimulus q Q Q 1 p p P 1 84.80 0.922 0.0002 0.0780 0.018 0.0183 86.48 0.874 0.0014 0.1 149 0.034 0.0330 88.16 0.812 0.0069 0.1514 0.058 0.0552 89.84 0.733 0.0232 0.1753 0.095 0.0849 0.0004 91-52 0.639 0.0560 0.1736 0.147 0.1 186 0.0025 93-20 0.536 O.IOII 0.1425 0.215 0.148 1 0.0106 94.88 0.430 0.1423 0.0937 0.297 0.1612 0.0318 96.56 0.329 0.1623 0.0474 0.393 0.1496 0.0698 98.24 0.239 0.1552 0.0177 0.496 0.1 145 0.1171 99.92 0.165 0.1282 0.0046 0.598 0.0697 0.1561 101.60 0.107 0.0935 0.0008 0.694 0.0326 0.1710 103.28 0.066 0.0615 O.OOOI 0.779 O.OI 12 0.1588 104.96 0.038 0.0370 0.848 0.0027 0.1290 106.64 0.021 0.0206 0.902 0.0004 0.0920 108.32 O.OII 0.0106 0.939 O.OOOI 0.0609 2 1. 0000 1.0000 1. 0001 1. 0000 R 0.0000 0.0000 0.0000 0.0000 Table XXXV Series V Comparison Stimulus q Q Q 1 P P P 1 84.80 0.906 0.0022 0.0944 O.OO9 0.009I 86.90 0.838 0.0125 0.1465 0.022 0.0220 89.00 0.746 0.0437 0.1932 O.O49 0.0472 0.0004 91.10 0.631 O.IOOI 0.2090 O.96 O.0887 0.0041 93.20 0.503 0.1605 0.1775 O.I7I O.I42I 0.0219 95.30 0.375 0.1912 O.I 122 0.274 O.I895 0.0697 97.40 0.259 0.1784 O.O498 O.403 O.2O20 0.1424 99-SO 0.165 0.1364 O.OI45 0.542 O.1624 0.2013 101.60 0.097 0.0884 O.OO26 O.677 O.O928 0.2102 103.70 0.052 0.0501 0.0OO3 O.792 0.035I 0.1711 105.80 0.025 0.0251 O.878 O.O080 0.1 143 107.90 O.OII 0.0113 0.935 O.OOII 0.0646 2 0.9999 I. OOOO 1. 0000 1. 0000 R 0.0002 O.OOOO O.OOOI 0.0000 Table XXXVI Series VI Comparison Stimulus q Q Q 1 p p P l 84.80 87.32 89.84 92.36 94.88 97.40 99.92 102.44 104.96 107.48 110.00 112.52 0.890 0.806 0.692 0.556 0.413 0.280 0.173 0.096 0.048 0.021 0.008 0.003 0.0069 0.0321 0.0895 0.1621 0.2047 0.1929 0.1440 0.0884 0.0462 0.021 1 0.0084 0.0030 O.IIOI 0.1725 0.2208 0.2204 0.1622 0.0821 0.0264 0.0050 0.0005 0.025 0.055 0.109 0.193 0.307 0.444 0.588 0.721 0.829 0.906 0.953 0.979 0.0250 0.0540 0.1004 0.1580 0.2033 0.2038 0.1502 0.0759 0.0244 0.0046 0.0004 0.0008 0.0069 0.0326 0.0914 0.1652 0.2083 0.1956 0.1446 0.0881 0.0458 0.0206 2 R 0.9993 0.0008 1. 0000 0.0000 1. 0000 0.0000 0.9999 0.0000 Table XXXVII Series I'll Comparison Stimulus q Q Q 1 p P P 1 84.80 8774 90.68 93.62 96.56 99.50 102.44 105.38 108.32 111.26 0.954 0.873 0.726 0.523 0.313 0.151 0.058 0.017 0.004 0.00 1 0.0085 0.0615 0.1861 0.2807 0.2451 0.1395 0.0566 0.0171 0.0039 0.0007 0.0459 0.1208 0.2287 0.2886 0.2170 0.0840 0.0141 0.0009 0.009 0.029 0.082 0.164 0.343 0.536 0.720 0.860 0.942 0.081 0.0086 0.0292 0.0785 0.1451 0.2534 0.2601 0.1622 0.0541 0.0083 0.0005 0.0014 0.0159 0.0879 0.2015 0.2656 0.2221 0.1297 0.0567 0.0192 2 R 0.9997 0.0004 1. 0000 0.0000 1. 0000 0.0000 1. 0000 0.0000 Table XXXVIII Series VIII Comparison Stimulus q Q Q 1 p p P 1 84.80 0.938 0.0166 0.0621 O.OII 0.0114 0.0003 88.16 0.839 0.0923 0.1508 0.041 0.0406 0.0065 91-52 0.672 0.2252 0.2582 0.115 0.1092 0.0517 94-88 0.460 0.2855 0.2856 0.254 0.2130 0.1716 98.24 0.259 0.2169 0.1803 0.450 0.2819 0.2806 101.60 0.117 0.1 105 0.0557 0.661 0.2272 0.2623 104.96 0.041 0.0406 0.0070 0.829 0.0968 0.1590 108.32 O.OII O.OII2 0.0003 0.932 0.0185 0.0680 2 O.9988 1. 0000 0.9986 1. 0000 R O.OOII 0.0000 0.0014 0.0000 Table XXXIX Series IX Comparison Stimulus q Q Q l p p P 1 84.80 0.952 0.0287 0.0478 0.004 0.0040 0.0006 88.58 0.840 0.1586 0.1521 0.028 0.0281 0.0200 92.36 0.628 0.3179 0.2980 0.121 0.1 175 0*488 96.14 0.36s 0.2912 0.3188 0.334 0.2837 0.3383 99.92 0.155 0.1463 0.1549 0.622 0.3523 0.3091 103.70 0.046 0.0454 0.0271 0.853 0.1829 0.1408 107.48 0.009 0.0092 0.0013 0.963 0.0304 0.0367 1 1 1.26 O.OOI O.OOI2 0.994 O.OOII 0.0058 2 0.9985 1. 0000 I. OOOO 1. 000 1 R 0.0014 0.0000 0.000 1 0.0000 Table XL 12 SAMUEL W. FERNBERGER Series X Comparison Stimulus q Q Q 1 p p P 1 84.80 89.00 93-20 97.40 101.60 105.80 110.00 0.952 0.828 0.592 0.315 0.116 0.028 0.004 0.0300 0.1978 0.3468 0.2697 0.1 123 0.0280 0.0044 0.0483 0.1634 0.3215 0.3196 0.1 301 0.0166 0.0005 0.014 0.061 0.183 0.400 0.653 0.851 0.954 0.0140 0.0598 0.1699 0.3022 0.2967 0.1340 0.0224 0.0023 0.0366 0.1733 0.3188 0.2817 0.1418 0.0455 2 R 0.9980 0.0020 1. 0000 0.0000 0.9990 O.OOIO 1. 0000 0.0000 Table XLI series there are only two remainders and these are only 0.0001. In the later and shorter series the remainders are very much larger and occur much more frequently. It will also be noticed that there are no remainders for either subject for the just im- perceptible differences both positive and negative. This of course indicates that our series were in every case extended enough so that these imperceptible differences would always fall upon one of the stimuli of our series, within the limit of 0.0001. It would also seem that the remainders for the just perceptible negative differ- ence are on the whole greater than those of the just perceptible positive difference. Subj ect I Subject II Just Just Just Just perceptible perceptible perceptible perceptible negative positive negative positive Series difference difference difference difference I 0.0000 0.0000 O.OOOI 0.0000 IV 0.0000 O.OOOI 0.0000 0.0000 V 0.0000 O.OOOI 0.0002 O.OOOI VI 0.0002 0.0000 0.0008 0.0000 VII 0.0006 0.0000 0.0004 0.0000 VIII 0.0006 0.0014 O.OOII 0.0014 IX 0.0006 O.OOOI 0.0014 O.OOOI X 0.0031 O.OOOI 0.0020 O.OOIO When we have once obtained the probabilities for the different comparison stimuli with which these four differences may occur, we may derive the most probable value of each of the just percepti- ble and just imperceptible differences very easily. Multiplying each value of the comparison stimulus by its probability and adding METHOD OF JUST PERCEPTIBLE DIFFERENCES 73 these products we obtain [xQ], [xQJ, [xP], and [xPJ. These are respectively the most probable values of the just perceptible negative difference, the just imperceptible negative difference, the just perceptible positive difference and the just imperceptible positive difference, calculated on the basis of our experiments by the method of constant stimuli. Tables XLII and XLIII give these values of the four differences for subjects I and II. Op- posite the numbers for the series, which are found in the first columns, we have in the 3d, 5th, 7th, and 9th columns the calcu- lated values for these differences. The observed values of these fundamental differences are found in the other columns of these tables. The observed and calculated values for each series and for every difference, are found directly next to one another so that they may be more easily compared. These observed values are calculated very readily and in the man- ner explained in our description of the experiment, given above. The values for the just perceptible and imperceptible negative and positive differences are ascertained for each group in every series, by picking out each value according to the definitions of these quantities. There were ten groups in each series, five ascending and five descending, so that we will obtain in each case, ten values for each difference. These ten values are averaged and the result is given as the value of the difference. The ease with which the differences are calculated after we have our experimental data is one of the great advantages of the method of just perceptible differences over the other methods of psychophysical measure- ment. In this method, the measure of sensitivity is obtained by the simple solving of four averages, which can be performed in a very short space of time. With the method of constant stimuli, on the other hand, a long and complicated series of calculations must be entered into before these same values are obtained. We now calculate the measure of accuracy that is shown by our observed results of the method of just perceptible differences by means of the probable error. If our threshold is found to be 95 grams and the probable error ± 0.50 grams, it indicates that this threshold will fall within the interval of 94.50 grams and 95.50 grams with the probability of 0.50. This measure of precision is 2 8 .3 U o; XI C c u D cv O 1- CL> -n Q. g u > ^ ■M 3 i — i c H 11 T3 ~ > 4J +5 > x! c •= sr-a WD '-M •— > O a CO 3 a U a o a * LO ' S\ 5 5 8 S § o. -t 00 M- tM i- 00 f^OC - CI 000&00600 On On On On On On On < mOOcjh ^-Of w 't C3 dNO 'tHOin OnOO tJ- ONtXI OnOnOnOnOnOnOnOn TJ- O O CO « CHxi "it: o , O en a xi 0) a a X c -^J —, c v u c ■O cu u a. u u CL> a it: •3 > -i rt M cu C i/) m *-i fl-fu-iNOOOONOOO On i-h WOO i-i <*: c:00 >i > X < METHOD OF JUST PERCEPTIBLE DIFFERENCES 75 calculated only for the thresholds and is obtained by the formula It? P. E. 0.6745 3 n(n-i) In this equation 2v 2 is the sum of the variations of the different thresholds from the average, and n is the number of thresholds considered. Table XLIV gives the probable errors for both sub- Subj ect I Subject II Threshold in Threshold in Threshold in Threshold in direction of direction of direction of direction of Series decrease increase decrease increase I ±0.50 ±0.43 ±0.43 ±0.44 II ±0.88 ±0.82 ±0.87 ±o.75 III ±0.78 ±0.83 ±0.79 ±0.83 IV ±0.96 ±0.72 ±0.83 ±0.61 V ±0.72 ±0.66 ±0.65 ±0.65 VI ±0.27 ±0.64 ±0.62 ±0.52 VII ±0.53 ±0.65 ±0.50 ±0.63 VIII ±0.52 ±0.56 ±0.57 ±0.56 IX ±0.56 ±0.60 ±0.62 ±0.76 X ±0.55 ±0.54 ±0.60 ±0.72 Table XLIV jects for the two thresholds of increase and decrease. It will be noticed that these values are comparatively constant. Considering the size of the values, the precision of which they are measuring, these quantities are small. In no case is the probable error greater than one per cent of the threshold and very often it is close to one-half a per cent or even less. Thus we conclude that the thres- holds in the method of just perceptible differences fall with a high probability within a comparatively limited space. The sums of the products xP, xP 1 , xQ and xQ 1 give the most probable values of the just perceptible and just imperceptible posi- tive and negative differences, but a finite number of observations can not be expected to give exactly these results. The outcome of a series depends on chance influences, which prevent us from obtaining the calculated result exactly, and we must be satisfied with determining the limit inside of which we may expect the results to fall with a given probability. The solution of this prob- lem is given by the theorem of Tchebicheff, which applies to problems of this kind (cfr. Urban, Statistical Methods, pp. 65, and Archiv f. d. ges. Psychologie, Vol. 15, pp. 302-304). We 76 SAMUEL W. FERNBERGER give the formulae for the just perceptible and the just impercep- tible positive difference only, because their form for the two other differences is perfectly obvious. Suppose we make n determina- tions of each one of these quantities with a certain series of com- parison stimuli. We then may expect with a probability exceed- ing i — — that the arithmetical mean of all the observations of the just perceptible difference will be within the limits %xP ± t V2* 2 />— {%xP) 2 and that of the just imperceptible positive difference between the limits SxP 1 ± -i/lx^P 1 — CZxPy . Applying the same theorem to the determination of the upper limit of the interval of uncertainty we find that there exists a proba- bility exceeding i — £. that the actually observed result will be found between the limits The difference 2x 2 P — (2xP) 2 plays a part similar in the theorem of Tchebicheff to that of the product 2spq in the theorem of Bernoulli, since the square roots of both these terms determine the size of the interval inside of which the result may be expected with a given probability. The actual determination of these intervals requires the square root of these differences, but this problem is of less interest than the question as to the comparative size of these intervals in series of different length. For this purpose it is sufficient to give these differences as it is done in Table XLV for Subject I and in Table XL VI for Subject II. These tables also contain the values of the sums of the terms xxP which is needed for the calculation of the differ- ences in question. These quantities also enable us to determine the limits for the result of a series of determinations of the upper * The theorem of Tchebicheff enables us merely to determine the limits inside of which we may expect the result of an actual observation to fall with a given probabilicy. Wirth [Zur erkenntnisstheoretischen und mathema- tischen Begrilndung der Massmethoden fur die U nterschiedsschwelle , Archiv f. d. ges. Psychologie, 191 1, Vol. 20, p. 84] seems to believe that this theorem might enable us to find relations between different values of the psycho- metric functions, but there is no possibility of doing so. Oh X x x a X X m l ts.00 N 00 VO rOC 00 ©i Jn.00 1S.VO 00 O O VD ^d o" ■*(■ "S oi oo' vo K. d K looq oq ■* ■>*• u^ n-\o oo dod po po tt m - <"S 00 K. M oo •* cs «■> rooq in po o Is. PO po -4 in dod fO\d P4 MHMMHHHMf) >d6>fi^H ^od pq m m t^oooOoO^o rv SP4 00VO POM3 " " O 0\ O Ov po 0\ C\ w 0\ C\ N^O O00 PO > in po owq fi o q o 0}S d od n w -4 d od d K. in Oh X X Ph X X o X X w a X X Cfl h n pjq ioo VO t-i "i PC in t^ Tf O^-NOO "*00 — pocg m po P) cs o e> o o o o in in ■* i< -4 d dod w H HH N « OOVO P) C\ m PO00 M O O 00 KTJrodod -*od d\d d oo o) ti- i-i lOKTj-Kinm Q\ tt t-x 0\ w N>n m*o r^ 00 00 00 00 C*00 0\ C\ Ch o 6 oq Nno pooo po aK» pi \d pood oo w ►h QMXTj-r^ONOOOO « 00 txOO OO 00 00 00 00 00 •«J- i-i po\0 tx i^ "* O I^\0 ~~E>>r£:=XX *K£^> > > 78 SAMUEL W. FERNBERGER limit of the interval of uncertainty. In our experiments we have in all the cases n = 5, so that the radicals for the limits of the interval of uncertainty can be found by averaging the corres- ponding values in the tables. Table XL VII gives these values for both subjects. The greater Subject I Subject II Series d 1 d 11 d 1 d" I II Ill IV V VI VII VIII IX X 8.42 10.56 12.26 13,26 I4-30 18.08 18.42 17.00 17.14 21.70 8.30 9.02 12.36 13-47 12.74 18.92 17.82 12.85 20.70 10.66 6.85 9.84 11.48 13.76 I5-IO 16.91 1378 14-54 12.80 19.04 8.22 6.68 1366 14-74 14.58 20.36 19.18 1370 14.82 21.21 Table XLVII the value, the greater will be the interval within which we may expect our determination of the threshold to fall with a given probability and therefore the less accurate that determination will be. These quantities definitely tend to increase in size as the series become shorter, and therefore, the determinations of our thresholds by short series of experiments are not so accurate as those obtained by more extended series. Furthermore, it would definitely appear that the determinations of our thresholds by the graded series I are more accurate than any of the others. We are now in the possession of all the data necessary for the comparison of the two methods under discussion. The thres- holds are the important values in these methods, since they are the basis for the determination of the sensitivity of the subject. It is by the means of the values of the thresholds, then, that we will compare the methods of just perceptible differences and of constant stimuli. Tables XL VIII and XLIX show the values of the thresholds for both subjects, obtained in the different ways that have been described above. The first columns in these tables give the numbers of the series into which the results of the method of just perceptible differences were divided. The values of the threshold in the direction of decrease are found in the next three columns. The first of these are the values obtained from the re- Subject I Threshold in the direction of decrease Threshold in the direction of increase Method of constant stimuli Method of Just Percep- tible Differences Method of constant stimuli Method of Just Percep- tible Differences Series Calculated Observed Calculated Observed I II Ill IV V VI VII VIII IX X 95.02 95.02 94-52 94-8 1 94.20 94.48 94.06 94-74 94-75 93-89 95-96 95-33 94-85 94.99 94-37 94-59 94.24 0483 94-99 94.10 96.81 95.26 93.87 97-40 93-62 93-02 96.71 94-88 96.14 94.88 100.10 99.82 99-77 100.30 99.30 99-93 99.09 99-45 99.82 98.93 99.07 99.26 99-45 99.76 99-13 99-88 99.07 99-33 99-78 98.93 98.58 99-58 98.28 101.26 97.92 99.04 102.58 100.26 101.81 101.18 Table XLVIII Subject II Threshold in the direction of decrease Threshold in the direction of increase Method of constant stimuli Method of Just Percep- tible Differences Method of constant stimuli Method of Just Percep- tible Differences Series Calculated Observed Calculated Observed I II Ill IV V VI VII VIII IX X 93-26 94-44 94.16 9376 93-24 93-34 93-93 94.27 94.20 94-57 94.81 94.89 94-55 94-13 93-63 93-84 94.02 94.40 94-28 94-57 95-74 94.42 94-63 04.88 94.04 95-88 95-53 95-88 93-30 96.56 98.94 98.96 98.19 98.32 98.86 98.37 98.96 99.00 98.34 99.04 98.34 97-94 97-98 98.07 98.64 97-44 98.99 98.85 98.18 98.97 97-25 98.74 98.03 98.24 97-50 100.05 99-79 100.09 99.92 98.87 Table XLIX suits of the method of constant stimuli, that were taken simul- taneously with the different series of the method of just percepti- ble differences. In the next column are the calculated thresholds of the method of just perceptible differences, and in the third column are the observed values for the same method. The second halves of the tables show the same arrangement of values for the threshold in the direction of increase. An examination of these tables shows unsystematic variations between the different results that are to be expected. That is, within a given set, some individual results are greater and some are smaller than the average, but these variations occur in a haphazard manner and not in a regular way. The larger the amount of data that goes to determine any value or set of values, the smaller these chance variations should become. This is borne out by our re- 80 SAMUEL W. FERNBERGER suits, as the random variations are greatest in the observed results of the method of just perceptible differences, which values are determined by smaller experimental data than are the others. But these variations are comparatively small and seem to point to a high degree of similarity between the two methods. There is one observation, however, that may be of significance. If we compare the observed thresholds of the method of constant stimuli and the calculated thresholds of the method of just per- ceptible differences, we find that the variations, in the case of both subjects, are always in one direction. The calculated thres- holds in the direction of decrease by the method of just percepti- ble differences are constantly larger than the corresponding values in the method of constant stimuli; and for the threshold in the direction of increase, they are constantly smaller. Thus the limits of the interval of uncertainty are constantly narrowed in the just perceptible difference method when compared with the same in- terval of the other method. This is a serious fault as it is just this interval that is the measure of sensitivity of our subject. If the values of one method had been constantly greater than those of the other method for both thresholds, the interval of uncer- tainty would have remained the same although the point of sub- jective equality would have changed. It will be noticed that these systematic variations are greatest in the long series of short steps, while in the short series of large steps they become so small that they can be practically disregarded. The variation is greatest in series I, which was performed in an effort to test out a graded approach of the central values of the series. This may be due to the fact that by a carefully graded approach, we emphasize be- fore hand the probability that the differences will fall on the central values. The greater the number of results that go to make up a value, the more nearly should that value coincide with a calculated probability. If this be true, then the values of the series I to IV should be more nearly correct than the series VIII to X. It is a curious fact that the more nearly the experimental arrangement of the method of just perceptible differences ap- proaches that of the method of constant stimuli, the closer do the values under discussion coincide. The experimental arrangement METHOD OF JUST PERCEPTIBLE DIFFERENCES 81 of the method of constant stimuli consists of a small number of pairs of stimuli with comparative large differences of intensity in the steps of the series. This is like the arrangement of series X, and here the values of the thresholds for the two methods prac- tically agree. Series II has quite a different arrangement ; having many steps of a small interval and here the greatest discrepancies are found between the two methods. Thus there seem to be discrepancies that may be due, either to the length of the series, or to the attitude with which the subject approached the two methods. Our form of experimentation tried to eliminate any difference in attitude and was probably almost entirely, but not absolutely, successful. If we had been able to devise a means by which the attitude of the sub- ject was identical for both methods, it seems very probable that our results would have showed a still closer agreement. In practically all of the studies by these methods, it has been found that a higher sensitivity was obtained by the method of just perceptible differences than by the method of constant stimuli. If the difference in our values under discussion are significant, we could then account for the discrepancies in the results of the former studies, as a fundamental difference in the methods themselves. But these differences are not large enough to have us disregard the method of just perceptible differences as a practical method of psychophysical measurement. It seems wisest, however, to disregard the form of a graded approach of the central values, as the method in this form seems to show its greatest discrepan- cies. It would also seem that a short series of large steps is pre- ferable to a long series of small steps. The best form of experi- mentation with this method would be to take a considerable num- ber of series, each series to consist of not more than ten steps and with a considerable difference of weight between the steps. If such a series is used, the results of this study seem to indicate, that any discrepancies between the results obtained in this way and those of the method of constant stimuli taken under identical con- ditions would be so small that they could be disregarded.