Chapter XXI: Section IV: , 1. The experiments of Table IX, C, repeat those of A with (1)

duplication of groups; and D repeats those of C with longer exposure.

The sound for A was that of a small organ-pipe (Ut 4) blown by mouth. As in A of the preceding table the observers did not know in a given experiment with which group the sound would be given, but, as before, it was given the same number of times with the first as with the last. For B the multiplied sound was produced by an electric bell with a wooden gong. This was adopted in preference to metal because of the prompt ceasing of the sound after the stroke,--a very necessary condition when this sound accompanied the first group, that it might be clearly connected with its own group. A metal gong was used for the single sound, that the two might not be too unequal in loudness. Its vibrations were deadened by a rubber band, and each bell was controlled by a floor-button. For C and D a higher sound, from the same pipe unstopped, was used in preference to the former, for the reason that in certain experiments performed just previously the lower sound had been used and was presumably very familiar. So in order that the sound might be brought, if possible, afresh to the attention, the change was made.

TABLE IX

A B

132 _experiments_ 44 _experiments_
_each with_ 3 _each with_ 2
_subjects_. 180 _subjects_. 88
_with_ 1 _subject_ _with_ 1 _subject_

_Exposure_ = 1/25 _sec._ _Exposure_ = 1/25 _sec._

_Sound_\_No _No _Many _One _No
Sound_ tendency- Sounds_ Sound_ tendency_
Subjects 4 1 2

Av.% of
difference
in favor of 5.4 18.2 2.2

C D

44 _experiments each_ 88 _experiments each_

_Exposure_ = 1/25 _sec._ _Exposure_ = 1/4 _sec._

Sound_ _No Sound_ _No_ _Sound_ _No Sound_ _No_
_tendency_ tendency_
Subjects
1 1 2

20.4 4.6 2.2

The results of these experiments may be summarized as follows: (1) The figures give evidence of but two cases out of eleven where sound was influential. (2) Duplication of groups is not effective in developing evidence of the influence of sound. (3) Increased length of exposure works, as in former cases, to lessen the influence of the modifying factor. (4) The introspections are to the effect that the sound seems to be entirely without influence upon the judgment, beyond the distraction it brings in the earlier stages of work. Sometimes it dropped wholly out of consciousness. Sometimes the distraction seemed to last longer. One observer reported, when D was taken, that he felt as if the sound sometimes increased and sometimes decreased the apparent numerousness. In some other experiments not directly upon this point, but later to be reported, a sound was used; and one observer reported that it seemed to become functionally connected with certain gaps in the groups, as though the puff had blown a hole in the group. Here its effect was of course to emphasize negative factors. It appears thus that the sound might function in opposite directions at different times, somewhat in accord with the particular character of the visual presentation. We should expect, then, to have percentages that look insignificant. (5) We shall not have failed to notice the difference between touch and auditory stimuli in the feeling of influence upon the number-judgment. If we seek a cause for the superior influence of touch, we may perhaps find it in the fact that practical experience has trained us to disregard in any case of judgment such simultaneous presentations as were employed for auditory stimuli; while a definite tap upon the brow is a rather unusual experience likely to attract notice to itself in spite of attempts at abstraction. As one observer said, who took part in both kinds of experiments, the touch seemed more "intimate."

3. _The Influence of Kinæsthetic Impression._

The method consisted in the employment of active effort upon a fist dynamometer or a wooden handle during the appearance of one of the groups. The handle was preferable because noiseless. The effort was made with the left hand because the right was used in recording. The amount of it was left to the observer's regulation, with the one instruction that its presence be made decidedly evident but without too great fatigue. The cards of Section IV 1 and 2 were used in the One-Group Apparatus. Similarly again the experiments were repeated with the duplicate-group cards. I present no table here because the figures show practically no influence of the effort. On one subject 176 experiments were made; on a second 132; on a third 88.

It is interesting to note here certain results obtained from one observer when he was in what he described as an active attitude toward the groups, in which he seemed to rouse himself to an unusual pitch of concentration upon the visual situation. This was evidently a condition of increased effort to abstract. Without abstraction he gave 26 to 6 in favor of the effort while with abstraction this tendency had fallen off to 30 to 17. The strength of the tendency is thus strongly indicated. Another observer felt a kind of motor difference between the groups; he expected the effort-group to look larger and felt additionally excited, a scattered activity, while he was passive toward the other group. Perhaps this account puts a little meaning into his small per cent. That his power of abstraction was effective here is hinted by his remark that he felt a difference in the groups even when he judged them equal. The third observer found no subjective evidence that effort modified his judgment.

V. THE "ERRORS" OF EXPERIMENTATION

Throughout the foregoing experiments has been involved the possibility of some one of the three "errors" of experimentation, those of time, space, and distribution, and sometimes all three. Their effect on the results, if it existed, was, to be sure, eliminated in the well-known way, but their existence, if actual, would raise an interesting problem. It was possible, in the case of every group of experiments, to rearrange the tables in such a way as to bring out the evidence for any tendency to overestimate, for instance, the first group as against the second, the right as against the left, or one kind of irregular distribution as against another.

The distribution-error must have a word of explanation. It refers to a tendency to give more wrong judgments in favor of one kind of irregular distribution than of the other kind with which, in a given card, it is mated. In the construction of a set of cards several forms of irregular internal arrangement were used, in order that the judgment might not be one merely of form, and of course on any given card the forms were not the same. Elimination of the effect of these form-differences from the results involved the appearance of any given one as many times in connection with one of the two contrasting factors studied in a given experiment as with the other. Thus two sets of forms were carried through an experimental series--a source of error indeed, but avoidable only by such means as were used to escape the effects of the space-error. Analysis would show which, if either, of the two sets received more judgments in its favor, resulting in further evidence as to the extent to which the judgment of relative number is a function of distribution, and as to the fineness of discrimination for such differences.

Now the tables, when thus rearranged, show that these errors exist to a surprisingly large extent. In many cases their causes, whatever they are, seem to be the controlling factors in the judgment of relative number.

Barring the experiments of Section III, in which the space-error has largely been accounted for, I now propose to gather in one survey all the results of those analyses that have given us the information of the existence of these errors, and all the material of later tables that bears on this point, and to test them by further experimentation. I will begin with the space-error.

TABLE X

_Av. % of _Av. % of _Av. % of
difference in difference in difference in
favor of_ favor of_ favor of_

_Cases_ _Right_ _Cases_ _Left_ _Cases_ _No tendency_

Angier 1 25 6 17.9 9 5.7
Davison 5 26.4 1 10.8 6 6.2
Dunlap 7 18.3 4 4.4
Holt 2 13.6 7 17.6 4 3.2
Hylan 8 23.6 7 6.7
Meakin 2 19.2 1 29.6 8 5.5
Meriam 2 16.5 2 12.9 7 6.8
Moore 2 11 3 16.6 7 3.1
Peterson 1 13.6 1 12.2 9 4.3
Rogers 4 21.9 2 15.9 6 4.5
Rouse 3 15.5 8 5.1
Shaw 3 20.9 5 18 7 6.1
Windate 1 22.8 4 19.5 7 4.5
Yerkes 6 26.6 8 5.2
Henry 1 10 2 16.6 3 8.1
Woods 3 19.3 3 4.8

1. _The Space-Error._

Table X presents to us a summary of the values of the space-error tendency. (1) Taken as a whole they fall into all the three classes that are possible; (_a_) favoring right; (_b_) favoring left; (_c_) no marked tendency. (2) There is no observer that does not at some time show a fairly marked tendency. (3) All the observers fall into (_c_) and all but five into both (_a_) and (_b_). (4) More favor the left than the right,--50 to 35. (5) This survey makes it clear that the observers agree neither with themselves nor with each other in the direction of influence exerted by the causes underlying the space-error.

_a. Special Experiments to establish the Facts._ It might be suspected that irregularities would be more apparent where other factors such as we have been studying enter to complicate the situation from the point of view of pure relative position of the two groups. Table XI presents the answer to this query. The cards used contained groups of gray circles (Gray Darker, Prang) arranged in equal areas of the usual size and shape. The distribution-error was eliminated, though not by duplication, and the small-difference cards were retained. The Two-Group Apparatus was used, with an exposure of 6/5 sec.

TABLE XI

88 _experiments with each_

_Right_ _Left_ _No tendency_

Subjects 4 7 3

Av. % of difference
in favor of 25.6 23.1 3.4

The results give us again our inevitable three classes, and in many cases a difference-value surprisingly large when we reflect on the simplicity of the conditions. That the omission of complicating features was of importance is shown by the fact that more of the observers (11 out of 14) show a marked error than in any other case. Clearly enough the various factors introduced tend to eliminate the space-error, but when in any case it does enter, it is even then capable of rising to as high a degree on the whole as in the uncomplicated series, as is shown by the fact that in but four cases does the new value surpass the best of the old, and in three of these by a trifling amount.

It is interesting to note that the three cases of minimum space-error show a well-defined tendency to be determined by distribution.

_b. Possible Bases of this Error._ The outcome of these special experiments is that the factors found in the groups are at least not directly responsible for the situation that we are considering. The divergence among the observers shows this. In hunting after the cause for this apparent influence of side, we look first for changes in the peripheral, and then in the central, processes that precede the judgment. The material used for the experiments of Table XI seems approximately to have equalized all the objective factors in the two groups. How could there be anything further in the peripheral process whereby group could be differentiated from group? The most evident thing is that the visual stimulus is received in a different way from the two groups. There is a definite peripheral mechanism whose factors seem essentially to be two, however variously they may be combined: (_a_) The relative amount of time given to each group; (_b_) the order in which the groups are viewed.

The observers were instructed and continually reminded to equalize the amount of attention devoted to the group; but as this is not wholly a voluntary matter, the possibility of failure to conform has to be reckoned with. Experiment must therefore be employed to test the influence of these factors before one can fall back upon a central process as the cause for this tendency to favor a side.

_c. Its Relation to Differences between the Groups in Length of Look._ The material was the same used for the experiments of Table XI. The method was the same with the following necessary exceptions: The longer exposure was double the shorter (4/5 sec. to 2/5 sec.), and 2/5 sec. elapsed between the two. Further, the experiments were so arranged as to equalize the influence of the order of exposure with respect to both side and relative length. The means for effecting successive exposure took the earlier form described in the introduction to Section II.

TABLE XII

_88 experiments with each_

_Longer_ _Shorter_ _No tendency_

Subjects 6 7

Av. % of difference
in favor of 18 6.5

These facts are yielded by Table XII: (1) There are but two classes of observers, as no tendency exists to favor the group of longer exposure. (2) The time-error shows a considerably more marked tendency than the length of look, which is indeed somewhat surpassed by the space-error. (3) The persistence of the space-error, even among those that reveal a tendency in length of exposure, shows that the factor of relative difference in length of look cannot account for it. The persistence of it, too, when the order of exposure is controlled, even though the conditions are not wholly adapted to the study of this latter factor, suggest at least that the space-error is independent of even that order; but into this we shall make special enquiry. (4) The judgment of number is independent of the amount of eye-movement devoted to the fixation of the objects in a group. This conclusion, so far as the actual movement is concerned, is established by the fact that so many observers favor the shorter look; and by all the experiments with the One-Group Apparatus where an exposure of 1/25 sec. was used, since that time was too short to admit of movement. That ideated movement is likewise insignificant appears from the fact of marked error arising in the material where the groups were duplicates. Here no motives to different movements could lie in the material.

_d. Its Relation to the Order in which the Groups are viewed._ Table XIII gives us the results of the enquiry. The experimental conditions were not changed except as to the length of exposure. Each group was given 3/5 sec.; and half the experiments were performed in the order right-left and half in the reverse order.

TABLE XIII

_88 experiments with each_

_First_ _Last_ _No tendency_

Subjects 3 9 2

Av. % of difference
in favor of 17.5 28.3 1.7

The results may be thus summarized: (1) The order of exposure is notably influential upon the judgment of relative number, giving the usual three classes, with the tendency to overestimate the last group well in the lead. (2) The persistence of the space-error under these relatively simple conditions shows conclusively that it is not a function of the order of exposure. The two are independent variables.

2. _The Time-Error._

In pursuit of our enquiry we must survey the facts as they are given in the various experiments already reported and later to be reported. These facts are gathered into Table XIV, which furnishes the following items of significance: (1) All the observers, with the exception of Rouse, show at some time a definite tendency. One case only is given for him in this table, but other experiments not included in the tables from which the present is drawn confirm this fact by the ratio 29 to 30. (2) There is a rather striking consistency in the several observers. (3) The predominance of the last group is marked.

TABLE XIV

_Av. % of_ _Av. % of_ _Av. % of_
_difference_ _difference_ _difference_
_in favor of_ _in favor of_ _in favor of_

_Cases_ _First_ _Cases_ _Last_ _Cases_ _No tendency_

Angier 7 19.1 4 4.9
Baldwin 4 20.5 1 3.4
Bell 1 18.2 4 6.3
Davison 2 31.2
Dunlap 1 11.4 1 2.2
Holt 10 19 2 6
Hylan 2 16 5 25.4 5 4.9
Johnston 2 25.2 4 35.5 4 4.2
Meakin 2 39.7
Meriam 1 29.6 1 2.2
Miller 3 17.1 4 5.5
Moore 1 11.4 1 2.2
Olmsted 1 15.2
Peterson 1 17 1 9
Rogers 1 10.2 1 5.6
Rouse 1 1.2
Shaw 6 16.7 5 7
Windate 1 11.4 1 1.2
Yerkes 2 42.7

_a. Relation of the Error to the Absolute Length of the Total Exposure._ Table XV is set to answer this question. It is unsatisfactory in that but two observers took part in both XIII and XV. The material used for judgment consisted of the same cards used in the earlier experiment, but presented now in the One-Group Apparatus. The time of exposure was changed from 3/5 sec. to 1/25 sec. for each group. As the space-error was eliminated, the tendency to a time-error, if present at all, would presumably have freer play.

But the difference-values of the new table are for the most part very small. We have thus the further fact about the time-error that, under the conditions studied, it appears to be independent of the absolute length of exposure, when the groups are equal in this respect. To this we may add another fact drawn from Table XIV, that with the One-Group Apparatus the time-error is greater on the whole where the groups are differentiated by other factors. Thirdly, the values for Table XIII show that with all complicating factors withdrawn, except the differences in position, the error is at a maximum. This may be significant of the effect of space-differences upon that error, or, more probably, be due to the general difference between work by daylight and work in a dark room by artificial light. We shall be better able to consider this later.

TABLE XV

88 experiments with each of two subjects.
176 experiments with one subject.
154 experiments with one subject.
66 experiments with one subject.

Exposure = 1/25 sec.

_First_ _Last_ _No tendency_

Subjects 1 4
Av. % of difference
in favor of 15.2 6.8

3. _The Distribution-Error._

The last of our three "errors" of experimentation is now before us. We may recall once more the meaning the term has had for us in these studies. It points to a tendency discovered by the use of those cards where all objective factors were in the course of a series equalized,--a tendency to mass one's judgments in favor of a particular arrangement of the circles; though each group had been constructed with a view to filling the given area as homogeneously as an irregular arrangement would allow.

As in the two "errors" preceding, so here we must get possession of the facts that gave rise to the present enquiry. Table XVI presents them to us, gathered out of all the tables wherein such a tendency has been technically reckoned with. But first a few words of explanation are needed to make the new table intelligible. Two sets of results are found in its two parts. In each set the particular group-arrangements employed and the frequency of their appearance are exactly the same. The two sets differ, as their headings suggest, in that the material for the second set was formed out of the first by replacing the small-difference cards by those having equal groups. Such a change as this might affect the proportion of judgments given in favor of the two sets of arrangements in a particular series, and these new results are, therefore, no longer fully comparable with the earlier ones. In presenting the directions of tendency in the results, it is impossible here, as in all the similar cases throughout the tables, to name a factor as a standard in whose favor all the judgments in the plus column should be understood as given,--impossible for this reason that, because the very method by which the circles were distributed in the groups, the experimenter was unable to satisfy himself as to the significant differences in the arrangements. All the results, however, when analyzed on this basis, were recorded consistently, so that consistencies and agreements among the observers might be readily apparent. We can now understand in part what Table XVI has to say to us.

TABLE XVI

A
_No_
_Cases_ _Class 1_ _Cases_ _Class 2_ _Cases_ _tendency_

Angier 4 19.1 2 6.3
Davison 3 23.1
Dunlap 2 13.7 1 2.2
Holt 3 15.5 1 25 2 4.5
Hylan 1 11.4 2 14 3 6.7
Johnston 5 30.4
Meakin 3 34.9
Meriam 1 16 2 6.8
Miller 5 32.1
Moore 1 39.8 2 5.7
Olmsted 1 27.2
Peterson 3 42.1
Rogers 1 11.4 2 5.7
Rouse 2 14.2
Shaw 6 29.1
Windate 3 30
Yerkes 3 17.8

B
_No_
_Cases_ _Class 1_ _Cases_ _Class 2_ _Cases_ _tendency_

Angier 3 2.9
Davison
Dunlap
Holt 1 22 1 21.6 2 3.1
Hylan 2 50.8 1 11.4 1 8.4
Johnston
Meakin
Meriam
Miller
Moore
Olmsted
Peterson
Rogers
Rouse
Shaw 2 26 1 0
Windate
Yerkes

Here as elsewhere the per cents recorded indicate the average per
cent of difference in favor of a given class.

Here are the facts, first of A: (1) The only lapses from consistency are confined to two observers; and in both these cases there is but a single break in a uniform trend. (2) With three exceptions all agree in the trend of their difference-values. Of these three--Holt, Hylan, and Moore--the last furnishes but one significant value, and so must be left out of the reckoning on this point. (3) Of the 64 cases, 50 rise above 10%, some far beyond, showing the importance for the judgment of relative number of this factor of distribution. (4) Of the 50 cases, 45 agree in tendency. (5) That with this surprising agreement we have still a few exceptions, adds another item to the growing array of evidence on behalf of the importance of some subjective factor for the number-judgment. As to the nature of this factor we are yet in the dark. (6) To these facts B of this same table adds the further information that the observers inconsistent in the old are not consistent in the new, while the consistent still maintain their record.

_a. Analysis of the Experimental Conditions of Distribution._ At once we are interested to enquire for the factors underlying these results. To put ourselves upon the right track we must first consider what factors are involved in any such arrangement of objects as we have used in the material for these studies, and then, more precisely, we may ask in what way such arrangements could differ significantly. Finally, by an experimental trial-and-error process, we may solve our problem.

The groups of objects in our material were arranged in an area marked out in each corner by a circle. Within this area the circles were set irregularly, with the result that the group, as a mass of objects distinguished from a homogeneous background, had a more or less irregular outline whose irregularity varied with different internal arrangements. Within its outlines this area presented a mixed pattern of bright and dark. While the total enclosure marked off by the corner circles was always the same and theoretically the relative amounts of brightness and darkness in equal groups was likewise the same, yet practically differences, more or less slight, might enter through the changing character of the rude outlines whose ideal completeness could scarcely be brought out of a black background by the uninitiated. The amount of this difference is sometimes surprising to one whose chief thought of the group has been as vignetted in process of construction. As the objects are pushed toward the edges the central spaces open out; as they are withdrawn toward the interior gaps appear in the margin.

It is not a very easy task to fill an area with objects in irregular arrangement in such a way that no sections of vacancy or filling stand out by contrast against the remainder of the same element. To succeed in this is to fill the area homogeneously. But the chances are good that some vacant patch will get slightly the better of its neighbors or some section of circles will gather a little more closely than the surrounding circles; or perhaps a gap in the outline will be unexpectedly intrusive. Now in a given area the circles of one part cannot become more thickly massed without a corresponding enlargement of the vacancies of the other parts, and of course the converse is as true; but this theoretical situation may be quite out of ken at the moment when the group is seen. Either member of this pair of complements may stand out vividly in the field and its fellow quite escape perception. The very nicety with which in practical affairs we have to make a reliable comparison of this sort shows what suspicion of accuracy the off-hand judgment has bred. And further, the widening of a gap or thickening of the filling in one small part of a group may give a complementary loss to the rest of the group small enough to be unperceived when distributed throughout the larger section.

Two factors must therefore be considered as possibly significant in moving the judgment,--vacancies and filling; and with the former must be reckoned indrawing of the outline. Psychologically, increase in the prominence of either of these factors would be all one with their objective increase. With respect to the direction of their influence upon the judgment of number the increase of vacancies must signify the waning, and the increase of filling the waxing, of the objective number in the group.

It is in advance altogether probable that the results gathered into Table XVI were brought about by these two factors, at least in large part. And we have also in these factors the possibility of two types; for as we saw above, increased vacancies in one part involves increase of filling in another, and conversely. So the interesting question turns upon the altogether disproportional representation of types. Which is the type of the majority?

_b. Experimental Test of Hypotheses._ The question was put to the test of experiment. This was done by using groups in which now vacancies and now filling were objectively emphasized in contrast with the usual homogeneous group. First the vacancies. A set of cards was prepared after the method previously used to eliminate the distribution-error without duplication of groups on any one card. (See Section II.) In the present case, however, the two sets of arrangements were definitely differentiated as already indicated. One set had a homogeneously filled area, the other a prominent vacancy within or gap in the edge. The size of these variations was kept pretty close to the limit of noticeableness, that the increase in compactness of the other portions might be as slight as possible. It was experimentally necessary to free the material as far as might be from ambiguity, and practically important to avoid rousing the suspicions of the observers and the resulting reflections. It seemed very likely that the strength of the tendency shown by the distribution-error was due to its appearance in situations where the observers knew that other factors were being tested.

The general method already described was used in preparing the groups that gave objective prominence to compacted parts of the filling. To fulfil the conditions outlined above was here even more difficult than in the first set; and the cause will appear in the sequel. The small-difference cards were omitted and the One-Group Apparatus used.

A further attempt was made to head off reflection by a subterfuge. It had been found that, among the factors whose influence on the judgment had been studied, hearing had been as little effective as any. So the small stopped pipe used for those experiments was again brought into service and the error resulting eliminated in the usual way. Incidentally our new tables will thus give us further information about the effect of this factor, though of course under conditions that are theoretically highly unfavorable, since we are forcing upon the attention of the observers other factors that experience has shown them only too ready to seize upon. So if a tendency traceable to the factor of hearing should appear, we ought perhaps to give it somewhat more than its face value.

TABLE XVII

A.

_Exposure = 1/25 sec._

_Homogeneous_ _Vacant_ _No tendency_

Angier [1]50
Baldwin [2]53.4
Bell [1]52.2
Holt [2]44.4
Hylan [2]51.2
Johnston [1]56.8
Miller [2]4.2
Shaw [1]29.6

B.

_Exposure = 1/4 sec._

_Homogeneous_ _Vacant_ _No tendency_

Angier [2]51.2
Baldwin [2]55.6
Bell
Holt [1]13.6
Hylan [1]52.2
Johnston [2]62.6
Miller [1]25
Shaw [2]14.8

C.

_88 experiments each_
_Exposure = 1/25 sec._

_Homogeneous_ _Compact_ _No tendency_

Angier 39.6
Baldwin 35.2
Bell 3.4
Holt 27.2
Hylan 39.6
Johnston 44.4
Miller 16
Shaw 2.2

[1] 44 Experiments.
[2] 88 Experiments.

The per cents recorded indicate the average per cent of difference in
favor of a given factor.

Now we are ready to inspect the results. Table XVII, A is the outcome of the attempt to emphasize vacancies. Its experiments with 1/25 sec. exposure were repeated with one of 1/4 sec. as Table XVII, B, shows. In Table XVII, C, the emphasis of compactness is concerned.

For convenience we may again resort to a summary outline in extracting the meaning from these tables. First Table XVII, A. (1) All the observers but one agree in favoring the homogeneous, most of them with very high difference-values. (2) Miller alone gives no tendency, and his notes show a conflict between the increased vacancy and the increased compactness. In other words, his discrimination was too keen for the material. Under the circumstances he constitutes no exception to the conclusion that the vacancy objectively emphasized was the cause for an underestimation of its group.

From Table XVII, B, we learn the following: (1) All the observers save one favor the homogeneous group, in most cases by large values. (2) The difference in the length of exposure seems to have no significance for this tendency, since, while Holt and Shaw decline, Miller rises in the scale.

Table XVII, C, gives us these facts: (1) The difference-values have noticeably fallen off. (2) We have again the customary three classes, but with homogeneous leading as in the earlier tables. (3) By his present favoring of the compact, Miller has now appeared in all three classes, while Holt has developed the preference for the compact that was budding in XVII, B. (4) The presence of four well-marked preferences for the homogeneous shows that the vacancies in the compact group were more significant for the number-judgment than was the increased compactness of the filling, and that in spite of the experimental effort to the contrary. (5) The decrease of this tendency and the growth of the opposing, indicates that the judgment is determined in either case by the more vivid factor.

The conclusions to be drawn from these facts lie close at hand. (_a_) The results in Table XVI, with their disproportionate division into classes, were evidently due to the tendency of three observers to note the filling and of the rest to be concerned with the vacancies. (_b_) The judgment of relative number under these conditions is primarily a judgment of vacancies. (_c_) The subjective factor of vividness determines the direction of error, and may attach to either vacancies or filling, though it usually attaches to the former.

It may not be out of place here to speculate a bit as to the probable cause for so close a dependence of the number-judgment upon what has no number, so to say; upon an object that has no standing in the official conclusion. The situation seems to be fundamentally based upon the conditions that determine contrast. In a homogeneous field no part stands out. Introduce a small object quite different in brightness or complementary in color and the attention is drawn instantly to it, but internal differences in its content are quite lost in the common quality by which it differs from the ground. A case somewhat analogous is furnished by our material, particularly in the One- and Two-Group Apparatus. The small group is so unified by its contrast with the field that internal differences must be made out with relative effort. Now internal differences are necessary to the numerical character demanded of it, and they can be brought out in no way save by attending to the vacancies and so isolating parts in the threatening unity, each in a kind of space-matrix. The most careful observer could not do better on his way to truth; and that is why the error was so much larger when the factor of space-differences was studied.

That group is normally the more numerous in which the vacancies are less completely developed under observation. We say "normally" here by virtue of the speculation just completed as to the best method of attaining a judgment objectively true. For a man thus proceeding, our proposition is a sound statement of fact, to which the following results of our experiments bear witness. (_a_) The experiments recorded in Table XII on Relative Difference in Length of Look shows no exception of a value equal to 10% to the general statement that all tendencies, when any existed, were in the direction of favoring the shorter group. The shorter the time of exposure the less completely would the vacancies develop. (_b_) Table IV, E, shows that without exception the darker group tends to be judged the more numerous. (_c_) Table XXI shows for each subject that in a shorter exposure the absolute number seems considerably greater than in a longer exposure.

No comment seems necessary to concentrate the force of such evidence. If we carry our proposition to the detailed results of our separate studies in factors of distribution, we shall find that it helps us to understand those few exceptions to the general trend of observers as they appear in Tables II and XVII. The exceptions there favored the groups in which compactness of parts went along with certain large vacancies. Possibly enough they refused to fall in with the objective analysis, and, disregarding the prominent vacancies, devoted themselves to a development of the vacancies within the compacted parts.

_c. The Factor of Hearing._ The time-error analyses of the experiments of Table XVII have already contributed their facts to the special section dealing with that error. But one or two interesting facts have remained unnoticed in the sound-analysis. In the experiments of Table XVII, A, there is a single case of marked tendency to favor the sound group. With the lengthened exposure of B, this tendency, as usual, disappears; but returns in C to some extent and two other observers share it. A fourth markedly favors the group without sound. So the experiments of this last table present as marked external evidence as we have for the influence of hearing upon the judgment. These facts are presented in Table XVIII.

TABLE XVIII

A
_44 experiments with_
_each of 4 subjects,_
_88 with each of 3._

_Exposure = 1/25 sec._

_No-_ _No_
_Sound_ _Sound_ _tendency_
Subjects 1 6

Av. % of
difference
in favor of 27.2 4.1

B
_88 experiments with_
_each of 3 subjects,_
_44 with each of 3._

_Exposure = 1/4 sec._

_No-_ _No_
_Sound_ _Sound_ _tendency_
Subjects 6

Av. % of
difference
in favor of 5.9

C
_88 experiments each_

_Exposure = 1/25 sec._

_No-_ _No_
_Sound_ _Sound_ _tendency_
Subjects 3 1 3

Av. % of
difference
in favor of 12.9 20.4 4.5

It is a further curious fact, well sustained by these same experiments, that where there is some confusion, each of the factors present has a better chance to determine the judgment. The values for both time-error and sound rise higher for the majority in C than in A or B.

VI. THE INFLUENCE OF FACTORS IN THE SAME SENSE-FIELD UPON THE JUDGMENT OF ABSOLUTE NUMBER

The nature of the enquiry that we have been pursuing through so many pages is such that it may be raised exactly as well in the case of absolute as in that of relative number. There appears to be no reason why in this new field the results should not be exactly comparable with those in the old, to be taken indeed as a kind of test for the interpretation to be put upon the old. Without a single exception, unless it were imposed by a technical difficulty, all the earlier factors could be studied with the new purpose. Our practical interest to go to such lengths would depend pretty largely upon the results of first attempts. If wholly confirmatory, these would probably suffice.

The experimental conditions were of the simplest. The 3-8 in. steel balls of Section III were again pressed into service as objects for the number-judgment. They were thrown loosely into a fixed black frame, 20 cm. square. To avoid suggestive noises, its undersurface was made of a thick piece of felt covered with black cloth; and the whole rested of course on a black-topped table. The exposures were 2 sec. long, timed by watch-ticks. Between experiments the observer held a cardboard screen between him and the objects. When conditions were ready for a new judgment, closing his eyes he lowered the screen, opening his eyes again at the word of command and shutting them at the close of the experiment.

Of course the observers felt that their judgments were for the most part extremely vague. With small numbers they had a greater feeling of confidence. Yet altogether it was surprising with what readiness an absolute number-judgment would spring up in the presence of any given collection whatever within the limits set by the experimental series. Sometimes the observers thought that they made rough calculations on the basis of the filling in a unit of area. So far as this held it would tend to cut off the more astonishing departures from correctness, and it would probably advantage the smaller groups more than the large. Still it was entirely too rough a method to prevent the influence of the factors introduced, as the results will show. There was no time for systematic counting, which, in any case, the observers knew to be forbidden.

The figures in which the observers reported their judgments of absolute number have a value that is chiefly qualitative. The marked inconsistencies and disagreements are our guarantee for this statement. With all the observers there was but the loosest association between group-appearance and number-name. The innumerable variations in internal space-relations were of course responsible. For one observer a particular name probably had a quantitative significance far in excess of its value for another observer in this respect. To one man 100 might have meant about the same as 60, for example, to his neighbor. On the whole they were parsimonious; but Baldwin decidedly not.

A more or less constant influence was exerted on any given judgment by the comparison of the presented group with the traces of the preceding still in mind. The observers felt, however, that the judgment was largely independent of such comparison, and its fluctuations give some credence to this feeling.

The numbers chosen ranged by fives, from 25 to 100. In four cases a number was immediately repeated that rough suggestions as to the definiteness of the judgment and its dependence upon the actual number might be gained. These were indeed but rough suggestions, since, with certain exceptions to be noticed later, the arrangement was disturbed between times; but they made possible a closer watch upon the flickering of the judgment than could be kept by a mere repetition of the series. In the latter case it might be unstable and yet relatively firm in the other. A standard series is here recorded. Its order was determined by drawing the numbers out of a heap, but the repetitions were inserted arbitrarily.

1. 95
2. 25
3. 35
4. 65
5. No change
6. 30
7. 90
8. 85
9. 45
10. 100
11. 50
12. No change
13. 60
14. 40
15. No change
16. 70
17. 80
18. 55
19. 75
20. No change

1. _Absolute Number under Standard Conditions._

An indispensable preliminary for the present study is the establishment of a standard. Unless we know something in advance about the characteristics of the judgment of absolute number in relatively simple conditions, we shall be unable to tell what influence, if any, to attribute to the modifying factor in later experiments. Having then decided as to the general conditions under which we will study the problem we must make these the standard conditions of our work; and having discovered the nature of the judgments given under them, measure up to these results in all that is to follow. These standard conditions have already been set forth in the introduction to this section. The results are recorded in Tables XIX and XX.

TABLE XIX

KEY: St = Standard
Sc = Scattered
Co = Compact

_Subject_ = _Baldwin_ _Subject_ = _Miller_

_Trials with_
_each number_ 6 3 3 4 2 3

_Original_
_Numbers_ _St_ _Sc_ _Co_ _St_ _Sc_ _Co_

25 1 0 -6 -10 -6 -4
30 3 5 -5 -12 -7 -6
35 10 7 -3 -14 -4 -7
40 10 13 -8 -15 -8 -11
40 10 8 -8 -14 -10 -12
45 10 18 -3 -21 -13 -10
50 17 23 2 -19 -15 -17
50 15 22 2 -17 -15 -14
55 27 40 0 -18 -3 -12
60 19 33 5 -20 -23 -17
65 27 53 2 -18 -3 -7
65 24 57 5 -13 -13 2
70 38 77 0 -19 8 -8
75 31 73 0 -24 10 -20
75 28 78 -5 -24 8 -8
80 36 83 -5 -14 5 -18
85 54 85 5 -19 -8 0
90 54 90 5 -13 8 -3
95 48 73 5 -30 20 -15
100 61 87 7 -13 10 -7

The figures recorded are the average of the algebraic sums of other
figures that represent the difference between the actual and the
estimated number. Fractions are replaced by an added unit, if the
value is 1/2 or over.

Baldwin never underestimated the scattered group, and only once the
standard; but 31 times the compact. Miller only once overestimated
the standard group, and but 6 times the compact; but 13 times the
scattered.

Turning to these tables we notice at once, as characteristic of all the observers, the following facts: (1) Wide variation from objective correctness. (2) A far wider discrepancy with the larger numbers than with the smaller. Miller does not wholly agree here. His judgments by series show inconstancy, tending at first to follow the rule, but in the last two series to a maximum error near the middle. Certain remarks of this observer suggest that possibly in the latter case reflection as to the convenience of certain actual numbers for manipulation may have had influence. The three earlier series of Hutchison conform to the rule. The remainder, on the contrary, show no definite progression in tendency. It should be noted here that both Miller and Hutchison were more inclined than the other two observers to rough calculation. The effect of its adoption or of increased practice in it is shown by the disappearance of the characteristics of the earlier series. We have thus in these two cases a doubleness of standard that we must not fail to consider in our later comparisons. (3) There is a pronounced instability of judgment, as shown by the fluctuations for the same number in different series, and especially in successive judgments, of the same in any given series. (4) There is a general tendency to judge in multiples of five. That there should be any splitting of fives, particularly in the large numbers, might be regarded as mere caprice. Not so did it seem to the observers. They were conscious of an apparent absurdity in it where judgments were necessarily so vague; but they insisted that this stood for a kind of qualitative shading in the perception which threw out the choice of the round numbers just above and below. (5) The number is on the whole underestimated, three observers agreeing in this respect; but the fourth shows a very large and consistent tendency in the opposite direction.

In spite of the manifold special inconstancies and disagreements, these general tendencies are decidedly well-featured in the results. We may say that we have found a kind of standard illusion that will serve us for a guide through our later studies.

2. _The Influence of Distribution._

TABLE XX

_Subject_ = _Hutchison_ _Subject_ = _Olmsted_
_Trials_
_with_
_each_
_number_ 3 2 4
_First_ _Second_
_Original_ _Standard_ _Standard_ _Mixed_ _Small_
_numbers_ _Series_ _Series_ _Sizes_ _Sizes_
25 -5 -2 -3 -1
30 -4 0 -5 0
35 -6 -5 10 1
40 -11 -5 -5 -13
40 -12 -9 -13 -11
45 -7 -4 -15 -10
50 -14 -10 -20 -12
50 -13 -8 -15 -15
55 -10 -7 -20 -11
60 -19 -18 -25 -20
65 -15 -7 -20 -5
65 -15 -10 -10 -18
70 -14 -3 -15 -20
75 -21 -17 -28 -23
75 -25 -20 -20 -24
80 -24 -17 -30 -19
85 -17 -13 -35 -13
90 -13 -7 -25 -20
95 -25 -13 -40 -23
100 -22 -13 -35 -24

6 2 3
_Mixed _Small_
_Standard_ _Sizes_ _Sizes_
-8 -8 -12
-11 -14 -16
-8 -5 -14
-18 -17 -20
-13 -18 -22
-18 -20 -18
-19 -18 -23
-22 -15 -27
-24 -28 -23
-22 -23 -27
-28 -28 -33
-22 -28 -23
-27 -30 -28
-32 -23 -27
-33 -25 -32
-27 -23 -33
-27 -38 -38
-33 -40 -37
-39 -55 -40
-36 -15 -37

For the meaning of these figures see under Table XIX. Hutchison
overestimated the standard group only 5 times, never the mixed-size
group, and 9 times the small-size group. Olmsted never overestimates
at any time.

The first of the modifying factors to be considered has to do with the arrangement of the objects. Hitherto they had been thrown loosely into the frame. Now in successive studies they were, first, well scattered over the surface and, second, brought together into several compact nuclei. The last arrangement was adopted in preference to that of a single mass as being less open to comparison with preceding judgments and to judgment on the basis of form and size of group.

The results are shown in Table XIX: (1) The effect of scattering the objects is very markedly to raise the apparent number. Baldwin's preceding overestimations soar still higher; while Miller's former tendency to underestimation is checked to such an extent that 13 overestimations appear. (2) The effect of compacting the objects is just as markedly in the opposite direction. Baldwin gives 31 underestimations, and Miller reverts in a measure to his former type. (3) When similar arrangements were up for study in relative number we found two classes of observers, one favoring the compact, the other the scattered. The present results of Baldwin and Miller put them into the latter class.

3. _The Influence of Complexity of Group-Content._

This new factor of complexity in the content of the group was realized experimentally by making up the collection out of steel balls of two sizes, 1/8 in. and 3/8 in. The former looked almost infinitesimal beside the latter. The same objective numbers were still maintained and divided between the two sizes except where in so doing a five must be broken. In such a case the extra five went to the larger balls.

The results are found in Table XX. Olmsted shows no definite influence of the new factor. Hutchison, however, shows a very evident decrease in his estimations, when comparison is made with his second standard series. With the earlier series the new results rather closely correspond. That the latter are not simply a vacillating reversion seems fairly clear from this observer's account of his method. The small balls, he says, did not distinctly come in visually. To his judgment of the large he added an amount based on a very insecure estimate of the small. The number of the latter seemed from time to time pretty constant.

This situation corresponds very fully to that in the investigation of the same factor by use of a group of mixed colors, where relative number was in question. (Section II.) The tendency there discovered was to neglect the other colors in favor of one which thus surpassed the others in vividness. There as here the mixed group seemed smaller.

4. _The Influence of Size of Objects._

A study of this factor was made possible by substituting for the usual objects steel balls of a smaller size, 1-4 in. The results are contained in Table XX. They are not so striking as those obtained in our study of distribution. Still the influence of this new factor is evident, in the reduction of the apparent number. Olmsted shows this more generally for the smaller numbers. We find it in Hutchison when we compare the new results with the second standard series. This tendency to underestimation increases in the two final series of the present set. At the beginning of these two he remarked that he thought he had been overestimating the group. This tendency of smaller size to reduce apparent number was found true for the majority of observers in our earlier study of relative number.

5. _The Influence of the Length of Exposure._

I found that in relative number the shorter the look the more marked was the influence of certain factors. Reports of the observers making this seem highly probable happened in this way: When working with the One-Group Apparatus in relative number the shutter of the camera would occasionally stick, leaving a group exposed beyond its usual time. The effect of this upon some but not all the observers was to cause a noticeable shrinking in numerousness. Of those questioned, the only one failing to notice this effect is included among the observers in this new study.

To test this possibility resort was had to the One-Group Apparatus as affording a more satisfactory means for getting different lengths of exposure of small absolute magnitude. Cards were prepared containing a single group of larger area (67 × 82 mm.) than had been used for relative number. The objects were the usual white circles. Each corner was marked as usual; and, by reason of the number involved, the outline of the area was more regular than had been true in the earlier work. The number of circles on each card varied by steps of two from 16 to 30, giving eight cards in all. The series was arranged irregularly as before, and two of the cards repeated immediately upon their first presentation, making ten experiments in one set. The order of the series follows:

1. 24
2. 22
3. 26
4. No change
5. 18
6. 28
7. 16
8. 20
9. 30
10. No change

Two time-magnitudes were used for comparison,--1-25 sec. and 1 sec. The latter was managed with bulb exposure. All the experiments with the shorter time were made before those with the longer had been begun. The results are given in Table XXI. So far as the material is comparable, we may include in our comparison the standard experiments of Tables XIX and XX with 2 sec. exposure.

TABLE XXI

KEY: A = 1/25 sec.
B = 1 sec.

_Baldwin_ _Miller_ _Hutchison_ _Olmsted_
_Actual_
_Numbers_ A B A B A B A B
_Number of_
_trials with_
_each number_ 5 6 5 5 5 4 4 4
16 6 9 -2 -3 11 -1 2 -5
18 8 10 -4 -1 12 -1 -3 -5
20 34 19 1 3 19 1 1 1
22 37 17 6 6 14 6 4 -2
24 49 26 12 5 14 10 5 0
26 70 33 15 8 12 9 5 4
26 79 37 21 9 14 4 9 4
28 93 42 29 11 18 7 8 7
30 106 52 38 12 18 5 19 10
30 103 50 45 11 20 4 19 4

For the meaning of these figures see note under Table XIX.

The outcome may be thus summarized: (1) The apparent number is inversely proportional to the length of exposure. The tables show a perfectly clear progression from 2 sec. to 1-25 sec. All those that formerly underestimated are brought into the opposite class. (2) The results of the earlier experiments are confirmed on the whole with respect to the occurrence of greater errors with the larger numbers. (3) Baldwin's overestimation reaches astonishing heights. (4) These new facts for absolute number are quite in accord with Table XII, where, under the conditions of interpretation laid down, the tendencies were wholly in favor of the shorter look.