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 = 125 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 | B | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cases | Class 1 | Cases | Class 2 | Cases | No tendency | Cases | Class 1 | Cases | Class 2 | Cases | No tendency | |
| Angier | 4 | 19.1 | 2 | 6.3 | 3 | 2.9 | ||||||
| Davison | 3 | 23.1 | ||||||||||
| Dunlap | 2 | 13.7 | 1 | 2.2 | ||||||||
| Holt | 3 | 15.5 | 1 | 25 | 2 | 4.5 | 1 | 22 | 1 | 21.6 | 2 | 3.1 |
| Hylan | 1 | 11.4 | 2 | 14 | 3 | 6.7 | 2 | 50.8 | 1 | 11.4 | 1 | 8.4 |
| 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 | 2 | 26 | 1 | 0 | ||||||
| Windate | 3 | 30 | ||||||||||
| Yerkes | 3 | 17.8 | ||||||||||
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. | B. | C. 88 experiments each | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Exposure = 125 sec. | Exposure = 14 sec. | Exposure = 125 sec. | |||||||
| Homogeneous | Vacant | No tendency | Homogeneous | Vacant | No tendency | Homogeneous | Vacant | No tendency | |
| Angier | [1]50 | [2]51.2 | 39.6 | ||||||
| Baldwin | [2]53.4 | [2]55.6 | 35.2 | ||||||
| Bell | [1]52.2 | 3.4 | |||||||
| Holt | [2]44.4 | [1]13.6 | 27.2 | ||||||
| Hylan | [2]51.2 | [1]52.2 | 39.6 | ||||||
| Johnston | [1]56.8 | [2]62.6 | 44.4 | ||||||
| Miller | [2]4.2 | [1]25 | 16 | ||||||
| Shaw | [1]29.6 | [2]14.8 | 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 125 sec. exposure were repeated with one of 14 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.