The Psychophysics of Movement

James McKeen Cattell and George Fullerton

The writers of this note have recently published a monograph [1] describing some work on which they have been engaged during the past three years. It contains a discussion of more than 20,000 judgments, illustrated with many tables and diagrams. The detailed results are too extended and technical for any but special students of psychophysics. A summary may, however, be of interest to a larger circle of readers.

I. Psychophysical Methods.

The method of the just noticeable difference — in which an observer finds a difference which he can just perceive — is not satisfactory. If the observer simply choose a difference, which he thinks he can always or usually perceive, the result is without objective criterion. Indeed, our experiments show that those who think they can perceive the smallest difference are apt to be the worst observers. If the percentage of mistakes made by the observer be recorded, this method becomes a case of the following. But the" just noticeable difference "is not a convenient difference to use in the method of right and wrong cases. If the percentage of right cases be very large, a single chance variation greatly affects the average. If there be no mistake, we have, indeed, found a difference which can be perceived, but not the difference which can just be perceived, nor any other quantity which can be used as a measure of discrimination. If the just noticeable difference be interpreted by the observer as a difference apparently equal to some other difference, the method is reduced to that of estimated amount of difference.

The method of right and wrong cases—in which two stimuli nearly alike are presented to an observer, and he is required to say which seems the greater—is the most accurate of the methods. It requires a considerable number of experiments—at least 100,—and the number must be the greater the less practised the observer. The method is consequently not well suited for provisional, anthropometric, or clinical purposes. The percentages of right cases obtained do not directly measure the fineness of discrimination. The probable error, that is, the difference with which an observer is right 75% of the times, is the most convenient measure of discrimination. The probability integral may be used to calculate the probable error when the amount of difference is known, and the percentage of right cases is greater or less than 75. It is better not to allow the observer to give doubtful as his decision, but the confidence felt by him in its correctness may be recorded with advantage. If he observer is more apt to be right than wrong, even when he feels little or no confidence in his decision. Some observers are not confident unless they are, in fact, right, while others are often confident when they are wrong.

The method of average error—in which an observer makes one stimulus as nearly as possible like another-is, in many cases, the most convenient of the methods. It is closely related to the preceding, as the probable error can be found either from the average error or from.

( 448) the percentages of right cases. The probable error of the just noticeable difference, or of an estimated amount of difference, may also be determined, and the several methods thus combined. The error obtained by the method of average error is complex, being partly an error of adjustment and partly an error of perception. These errors may be separately determined by requiring the observer to judge the stimuli by the method of right and wrong cases after they have been adjusted. The average error may be analysed into a constant and a variable error. The distribution of the errors tends to follow the probability curve. This method can be used to special advantage when only a few experiments are made, as a result is reached more quickly than by the method ,of right and wrong cases.

The method of estimated amount of difference — in which an observer judges the quantitative relations of stimuli as in making one difference equal to, or double another — gives variable results. The observer probably does not estimate quantitative relations in sensation, but quantitative differences in the stimuli learned by association. It is consequently an open question whether the differences in sensation are qualitative or quantitative.

Great care should be taken in psychological experiments to keep all the conditions constant, except the variable to be investigated. The observer should not know the results of preceding experiments, nor the objective relations of the stimuli. Experiments should not be rejected because they make the averages less accordant. The results of experiment depend on accommodation to the conditions of experiment as well as on differences in senses or faculties, and these factors should be separately studied.

II. The Error of Observation and the Magnitude of the Stimulus.

Weber's law, according to which the least noticeable difference is proportional to the magnitude of the stimulus, does not hold for the extent and force of movement, as the least noticeable difference (or the error of observation) increases more slowly than the stimulus. Fechner's law, according to which the sensation increases as the logarithm of the stimulus, does not hold, as it rests on Weber's law, and on assumptions which are probably incorrect. As there is no logarithmic relation between mental and physical processes, the psychophysical, physiological, and psychological theories put forward to account for it are superfluous.

When amounts of difference in movements are estimated, the stimuli tend to be judged in their objective relations, and not as the logarithm of these. The results obtained by the method of right and wrong cases, and by the method of average error, determine the error of observation. This is a physical quantity. Its correlation with other physical quantities (for example, the magnitude of the stimulus) depends on physiological and mental conditions, and offers an important subject for psychological research. A mental quantity is not, however, directly measured. The error of observation usually increases as the stimulus is taken greater, but more slowly than in direct proportion to the magnitude. If the errors made in observing two stimuli of the same sort be combined, they will not be twice as large as the average error, but will equal the average error multiplied by the square root of two. This results both from theory and from our experiments. Consequently if two magnitudes, say two seconds, be observed continuously, the combined error in observing the two seconds would tend to equal the error in observing one second multiplied by the square root of two, and generally the error in observing a magnitude, extensive or intensive,

( 448) would increase as the square root of the magnitude. The summation of errors in this manner seems to account perfectly for the usual increase of the error of observation ("just noticeable difference ") with larger magnitudes. The error would increase as the square root of the magnitude, if each fraction of the magnitude, physically equal, were, in fact, subject to the same error of observation. In actual perception this would seldom or never be the case, but most of our experiments give an error of observation more nearly proportional to the square root of the magnitude, than directly proportional to the magnitude (Weber's law). We, therefore, substitute for Weber's law the following . The error of observation tends to increase as the square root of the magnitude, the increase being subject to variations whose amount and cause must be determined for each special case.[2]

III. The Extent of Movement

Experiments on the extent of movement were made by measuring the accuracy with which movements of the arm can be adjusted. The four distances chosen were 100, 300, 500, and 700 mm. Time was kept by a seconds pendulum, one second being allowed for each movement, and one second for the interval between the two movements to be compared. Experiments were made by the four psychophysical methods on one observer, and by the method of average error upon two others.

The attempts to mark off a distance just greater and one just less than 500 mm. resulted in, respectively, 539.4 and 477.2 mm. The distance marked off in separate experiments was highly variable. Even for groups of 100 experiments the average, just noticeable difference, varied, for the attempt at just greater, between 60.1 and 21.5 mm., and for the attempt at just less, between 37.7 and 4.8 mm. In striking contrast with these figures was the slight degree of variation in the variable error and its variation. For instance, where in two groups of 100 experiments each, the just noticeable difference was 601 and 215 mm., the corresponding variable error was 9.8 and 8.9. The highly variable character of the just noticeable difference makes it of small value in psychophysical experiment.

By the method of estimated amount of difference three kinds of experiments were made. An attempt was made to halve 500 mm., to double 300 mm., and to find the mean between 300 and 700 mm. The results of these experiments were all contrary to Fechner's law, the attempt to halve 500 mm. resulting in a distance of 305.2, the attempt to double 300 giving one of 560.1, and that to find the mean between 300 and 700 giving 512.4. In these experiments the variable error was, in relation to the whole extent of the movement made, greater than in the experiments by the method of just noticeable difference.

The experiments by the method of average error consisted in attempts to measure off on the scale 100, 300, 500, and 700 mm. The results of the experiments on the first observer showed a marked tendency to over-estimate the shortest distance, while the longest was under-estimated. The variable error for the four movements was, respectively, 5.3, 8.8, 9.5, and 81 ; or about 1/19, 1/23, 1/53/ and 1/79 of the stimuli. The experiments on the other two observers gave the same general results. Their combined variable errors were about 1/7, 1/13, 1/17, and 1/33 of the stimuli. Thus the variable error increases much more slowly than the stimulus, and Weber's law does not obtain. The error increases more nearly as the square root of the stimulus, but more slowly, being actually smaller for 700 than for

( 450) 500 mm. This is probably because the distance was nearly the limit -which could be reached, and the observer was helped by the strain.

In the experiments by the method of right and wrong cases the stimuli used were 500 and 510 mm. When the second stimulus was the greater, 75% of the judgments were right, and when the second was the less, 71.8%. The probable error (the difference which could be distinguished, 75% of the time) is 11 mm. The first movement was slightly under-estimated, the constant error being  .7. The degree of confidence expressed by the observer was a fair index of the objective correctness of his judgment.

IV. The Force of Movement.

A dynamometer may be used to advantage in studying the discrimination of the force of movements, but the clinical dynamometers are too inaccurate for scientific experiment. In making experiments on movement the observer can himself give the first or normal movement as well as the second or judgment movement, and the two movements will thus be made and perceived under like conditions.

"The just noticeable difference" in the force of movement varied greatly, not being proportional to the error of observation, but more accordant results are obtained if the probable error be found by taking into account the number of mistakes made by the observer. The average error of the just noticeable difference may also be used as a measure of discrimination. The just noticeable differences (for two observers) for about 2, 4, 8, and 16 kg. were respectively about  1/5, 1/7, 1/9, and 1/18  of the stimulus. The variable errors of observation were respectively .12, .20, .37, and .41 kg., and the probable errors obtained by taking the percentage of the errors into account were respectively .14,.26, .40, and .45 kg.

Experiments by the method of average error gave (for five observers) variable errors .19, .29,.43, .46, kg. for the magnitudes 2, 4, 8, and 16 kg. respectively. The worst of the five observers had an error about 1/3 larger than the best. Some observers are relatively better with the weaker, some with the stronger movements. There were considerable constant errors varying with different observers. The smallest magnitude was usually under-estimated, and the largest magnitude over-estimated. Neither the just noticeable difference nor the error of observation is a proportional part of the stimulus. Weber's law consequently does not hold for the force of movement. The error of observation is nearly proportional to the square root of the magnitude.

The error, when two movements are made as nearly as possible alike, is partly an error of perception, and partly an error of adjustment, and these two factors may be separated. The error of perception was, on the average, about twice as great as the error of adjustment, but the error of adjustment was relatively the smallest for the best observers. The errors iii making two movements as nearly alike as possible tend to be distributed as required by the probability curve. The combined error obtained by adding algebraically the errors in pairs is nearly equal to the average error multiplied by the square root of two.

Experiments by the method of estimated amount of difference showed that the force of movements tends to be estimated in their objective relations, and not as the logarithm of these. The results are variable, .and subject to large constant errors.

V. The Time of Movement.

Apparatus can be constructed which will measure accurately and conveniently the time either of a slow movement or of a quick blow.

( 451) When movements are discriminated, it is an advantage to let the observer adjust the time of the first as well as of the second movement. The error in judging the time of a movement (50 cm. in extent) with the arm, lasting about 1/8. 1/4, 1/2, or 1 sec., is nearly proportional to the magnitude, being as the average of five observers) about 1/10 thereof. The error with the worst observers was about twice that with the best. With the quicker movements the observer judges chiefly by the force of the blow, and as force is discriminated more accurately than time, this may account for the error increasing more rapidly than the square root of the magnitude.

With 1/2 and 1 sec., when the time of the two movements seemed equal, the second was the slower. When two blows in succession are made as quickly as possible, the second is the quicker, and seems the quicker. Half a second seemed less than half of 1 sec., and more than double 1/4 sec.

The results obtained by analysing the error into an error of perception and an error of adjustment, and from the distribution of errors and summation of errors, were nearly the same as with the force of movement.

The time of the quickest possible blow (50 cm. in extent) varied (with four observers) from .085 to 1.81 sec. While the rate of movement varies considerably with different observers, its average variation under like conditions is small, for a good observer .005 sec. The time was about the same for the right and left hand, and the rate was nearly uniform. The rate of movement should be used in the study of diseases of the nervous system.

Within the limits investigated the extent of movements can be judged better than the force, and the force better than the time.

VI. Lifted Weights.

The probable error in discriminating lifted weights, weighing about 100 grams, varied (for nine observers) from 5 to 8.2 grams, the average being 6.2 grams. This is the difference which could be correctly distinguished three-fourths of the time. The difference which could be correctly given 99 times out of a hundred would be about 21 grams. The probable error is nearly the same, whether calculated from a large difference and large percentage of right eases, or from a small difference and smaller percentage of right cases.

The confidence felt by different observers in the correctness of their judgment varies greatly and is not proportional to their fineness of discrimination. The constant error can be calculated. In these experiments it varied from .5 to 6.8 grams. The second of the two weights seemed relatively the heavier to nearly all the observers. In judging the accuracy of discrimination of an observer, both variable and constant errors should be considered.

The probable error is not greatly altered when the manner of lifting the weights is altered. It becomes larger when the weights are lifted with different hands or up or down only. It is scarcely altered when one weight is lifted four times as high or four times as fast as the other.

VII. Lights.

In our experiments on lights, apparatus was devised to give the observer two sensations of light in succession, each lasting one second and one second apart. The conditions were thus similar to those in the experiments with lifted weights. The lights compared were as 100 to 110, 120, 130, and 140. The probable error (given in hundredths of the intensity of the stimulus) varied for nine observers, from 9.9 to 18.7 with an aver-

( 452) age of 13.9. Reckoning upon this basis, a difference to be correctly given 99 times out of 100 would have to be about 48, or nearly 1/2 the stimulus. This large figure may be due partly to the fact that the illuminated area on the retina was small and the intensity of the lights used not great ; but it was probably chiefly due to the sensations being successive. We consider it an advantage to have the sensations successive, as the conditions can thus be kept constant, and sight can be compared with the other senses, muscular sense, hearing, &c. Different observers differed much in their degree of confidence, in the correctness of their judgment, and their degree of confidence was no indication of the relative fineness of their power of discrimination. For the same observer, however, the degree of confidence corresponded fairly well to the degree of objective accuracy. All the observers showed a tendency to under-estimate the second light, the constant error varying from 1.4 to 16.2. Under the conditions employed the muscular sense is about as again accurate as the sense of sight.

Memory for sensations may be studied by increasing the interval between the two stimuli to be compared, the probable error of an observer measuring his rate of forgetting. Observers remembered lifted weights and lights so well up to nine seconds, that their error of observation was scarcely increased. When the time was from 15 to 61 seconds the error was increased by about one-third. This is contrary to the common view, according to which we are supposed to forget most rapidly at first.



  1. On the perception of Small Differences, with special reference to the Force, Extent, aid Time of Movement . University of Pennsylvania Press, Philadelphia.

  2. One of the writers (G. S. F.) does not fully assent to the subject-matter of this section.

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