influence exerted on the judgment of a given factor by other factors
presented at the same time. The object of judgment in these studies is
visual number, which is to be submitted under varying conditions of the
objects whose number is in question, for example, varying conditions
of form, size, distribution, with the intent to discover whether
this judgment is a function of these other factors as well as of the
numerical. The scope of the enquiry includes both relative and absolute
number.
The objects chosen as a basis for the number-judgment were bits of
paper pasted in two well-defined groups side by side upon a background
of black cardboard. This card fitted into an upright frame where it was
held in place by a pivoted spring, which allowed easy adjustment and
removal of the card. The opening of the frame, 15×20 cm. was concealed
at will from the observer by a black wooden screen that played up and
down on guiding posts, when released by a cord and lever from the
catch that held it in place before the card. It fell by gravity upon a
cushion that deadened the sound; and it was restored to its position
by the operator's thrusting his fingers beneath and lifting it till
the catch above caught and held. The entire apparatus, as well as the
operator's movements, was concealed from the observer by a large black
cardboard screen resting upon a black-covered table. The one opening in
this screen was just large enough to allow a full view of the card when
the inner wooden screen fell from sight.
This apparatus which we will call the Two-Group Apparatus, admitted of
simultaneous exposure of the two groups of objects, and that only. At
first, to make successive exposure possible, a light wooden frame was
constructed in whose grooves two leaves of black cardboard ran like
sliding doors. By means of rods fastened to their outer edges these
leaves could be pulled apart or thrust together till their inner edges
met. When this apparatus was placed between the outer screen and the
frame bearing the card, and the inner wooden screen had been dropped
out of the way, this substitute divided screen was sufficient roughly
to accomplish the end in view.
With this apparatus the illumination was daylight, coming through a
very large window at the back of the observers. By means of a curtain,
marked variations in the light could be prevented.
For the length of simultaneous exposure of the groups the following
rule was adopted: Each observer was to be allowed time enough to get a
satisfactory feeling of relative number, but not time enough to admit
of counting. This time was kept constant during the work of any one
sitting. As the weeks went on, it was found possible, under the rule
laid down above, to shorten the time for some of the observers, and
to use with all the same length of exposure that had sufficed for the
speediest. The range of variation was from 1.2 sec. to 1.6 sec. Time
was measured by the ticks of a watch. Later tests showed for the time
studied that, where effective at all, the longer exposure diminished a
given tendency. Often it had no apparent effect.
The method of control already described is not only rather rough but
does not exclude the possibility of a space error. This possibility
proved actual by experiment. So an apparatus was contrived that should
present the groups in succession at approximately the same place and
should shorten the exposure, if desirable, to a small fraction of a
second.
This new apparatus, which we will call the One-Group Apparatus,
required artificial light and a dark room. By means of a 125 cp.
incandescent electric lamp, images of the groups of objects were
reflected through the lens of a camera and came to a focus upon its
ground-glass screen. A second screen of ground glass was placed in
front of the first and as close to it as possible, that an even
distribution of light might be obtained. The cards containing
the objects were of the same general character as in the earlier
experiments. They were held in a moveable slide whereby each group in
succession could be brought before the lens. When the slide was drawn
to the limit in one direction a single circle appeared in a black
field. This circle was used as a signal and a means for directing
the eye in the dark to that region where the groups were to appear.
The exposures were made with a camera bulb, the shutter being set
for instantaneous movement, with diaphragm 22 and length of exposure
1/26 sec. A shorter time was thought on trial to make perception too
difficult. The apparatus rested upon a table of special construction
and was enclosed as far as the glass screen with a wooden frame covered
with denim. Double curtains of this material formed this enclosure on
one side and made possible an easy adjustment of the cards between
exposures, as well as the admission of the operator's hand during a
given experiment for the adjustment of the shutter. This had to be
set, of course, before each of the three exposures constituting one
experiment. During its progress the hand was not removed at all, the
curtains falling about the arm in such a way that little light escaped.
The other hand managed the moveable slide from behind the enclosure.
Time was measured by watch-ticks. The three exposures--dot-signal,
Group 1, Group 2--were separated from each other by intervals of 1.6
sec. This was fixed upon as the minimum for convenient operation of the
apparatus.
In much of the experimentation on relative number two observers were
employed at once. Their chairs were placed closely side by side on
a line about 150 cm. from the plane in which the groups appeared.
These groups were not very far from being on a level with the eye.
Each observer recorded his own judgment, against the number of that
experiment. There were three possible kinds of judgments,--equality or
either group larger. If the judgment was of difference it was recorded
in terms of the larger.
When the dark room was used, special arrangements were required, for
convenience of the observers in making their record. After several
schemes were tested the following was adopted as least trying to their
eyes: A large, black-topped table was placed before them, bearing an
electric lamp enclosed in a black box with a small aperture that could
be closed at pleasure; or, if left open, did not let enough light
escape to disturb the perception of the groups.
The absolute number of objects in the groups was determined, first, by
the character of the problem, and then by convenience. If we are to
learn anything about the influence exerted upon the number-judgment
by other factors than the numerical, we must eliminate all influence
of the latter. Correct judgments may be determined by this factor
alone; erroneous judgments must have been otherwise conditioned; and
these conditions it is the task of our method to isolate and study, as
modifying factors. From correct judgments we learn nothing definite
about our problem, but from erroneous everything. Other things being
equal, it is preferable to eliminate from the results the influence
of this numerical factor, just as one handles any other disturbing,
unavoidable element, by equalizing the numbers in the two groups.
What may be called the standard number of objects in each is twenty.
This choice was governed by the purpose of using a number large enough
to make counting impossible in a brief time and yet not so large as
unnecessarily to increase the labor of preparation and the difficulty,
for the observer, of getting an idea of the groups as a whole. To the
cards containing equal groups, 20 to 20, were added others, 20 to 19,
19 to 20, for the purpose of easy variation in arrangement, by omitting
one object from a group, without making the actual numerical difference
easily perceivable. In later work these small objective differences
were dropped. Yet other cards, 23 to 17, 17 to 23, were added, to the
end that the observers might find unmistakeable number-differences, and
so not be bothered by the suspicion that the groups were all equal.
The reversal of the number-relations, as indicated above, was in the
interest of equalizing the influence of the actual numerical factor in
the two groups.
The following proportion was kept among the numbers of observations
made upon each kind of card: 1/2 upon groups objectively equal; 5/12
upon those differing by one from each other, where half each went to
(20 to 19) and (19 to 20); 1/12 to those showing the maximum objective
difference of six, where again half went to (17 to 23) and half to (23
to 17). Of course the observations upon cards of this last sort are
excluded from the tables.
As to the number of cards employed for each series of experiments,
it was found at first convenient to use seven,--3 (20 to 20), 1 (20
to 19), 1 (19 to 20), 1 (17 to 23), 1 (23 to 17). In each group the
arrangement of objects was irregular. The use of three of the first
sort was to encourage freshness of judgment, each having its particular
irregularity. Cards were but rarely remembered, practically never
except in the case of groups differing widely in number. So far as the
observers could tell, judgment was formed afresh in all these cases.
In later experiments eight cards were used. This number was in the
interest of avoiding the distribution-error. At first it was thought
sufficient that all the groups should be merely irregular. Later it
became evident that discrimination was very fine here and so that this
factor must be eliminated by the usual precise method.
The space- and time-errors, where likely to be present, were eliminated
in the usual way by performing an equal number of experiments with the
groups in reversed arrangement. Several methods of doing this were at
first tried; but these were all abandoned in favor of the following:
The experiments were arranged in sets of 24, in each of which the
proportion of kinds of cards was kept as indicated above. Each set
with one space- or time-order of the groups was repeated with that
arrangement reversed.
A word must be added as to the arrangement of results in the tables.
Judgments of equality upon objectively unequal groups are entered
as overestimations of the smaller groups. The per cent of correct
judgments is equally divided between the two other classes, and for
this reason that interest centres, not in correctness at all, but in
the difference between the tendency of error in one direction and that
in the other direction. No doubtful judgments were admitted, but in
such cases another trial was allowed later, usually when the observer
was not aware that he was being given a new chance. The subjects are
divided into three classes according as the results show a tendency to
favor one or the other group or no tendency either way. A difference of
10% is arbitrarily taken as significant.
1. _The Influence of Group-Area._ The Two-Group Apparatus was employed.
The four sets of experiments carried out with this factor differed
primarily in the material upon which the observer's judgment was based,
and secondarily in certain matters of method. The attempt in them all
was to approximate more completely to the isolation of the factor
under investigation. They are numbered in the order of approximation.
As marked results were obtained from each, they have all been offered
for consideration in the four parts of Table I. A description of the
material used in each case follows.
_A._ Squares (1 cm.) Neutral Gray no. 1. (Bradley), arranged
irregularly in two groups with irregular outlines on a background of
black cardboard. One group was large in area, the other small, the
attempt being made to fill each space homogeneously. Groups were not
proportional in shape of area.
_B._ As above, save that circles (11 mm. approx. in diameter) were
substituted for squares, in the interest of distinctness for the
several objects.
_C._ The area of the groups was oblong and regular, and the sides
were proportional. (Compact 72.5 mm.: 58 mm.; scattered 110 mm.: 88
mm. These relations were determined by the size of the frame that had
already been used and by the desire to make the difference in area
as marked as other necessary conditions would admit.) Each area was
marked by a circle in each corner. The color of the compact group
was the deepest shade of normal gray (Prang Normal Gray Darker); of
scattered group the next higher shade (Normal Gray Dark). These dark
grays were used in order to reduce to a minimum the tendency to produce
after-images. The difference in the shades of the two groups was in the
interest of avoiding the greater brightness due to the mass-effect of
the compact group.
_D._ As in C, except that India ink outline circles (1/3 to 1/2 mm.
line) were used on a background of granite cardboard. This change was
made to avoid, as far as possible, the greater mass-stimulation due to
the reënforcing effect of the compact arrangement. The size of circles
remained as before.
TABLE I
KEY: NS = Number of Subjects
AV% = Av. % of difference in favor of
A B C D
274 248 120 132
_experiments_ _experiments_ _experiments_ _experiments_
_with each_ _with each_ _with each_ _with each_
_subject_ _subject_ _subject_ _subject_
_NS_ _AV%_ _NS_ _AV%_ _NS_ _AV%_ _NS_ _AV%_
Small 5 31.5 10 34.6 7 44.1 10 46
Large 3 35.1 4 44.1 5 41.5 3 26.3
No tendency 1 8.8 2 4.2 4 5.7 3 6.6
The per cent recorded in the no-tendency class is an average of _all_
per cents below 10, whether in favor of the one or the other of the
two remaining classes. This is true for all the following tables.
The following facts are presented by the several parts of Table I: (1)
The large per cents of difference show that area is to a large extent a
determinant of the judgment of relative number. (2) Different subjects
show opposite tendencies. (3) A comparison of the results of individual
subjects through the four series shows that this opposition in tendency
occurs in the same subject at different times. The introspective notes
of one of these subjects show the internal process of change from one
tendency to the other. It consists in a gradual increase of analytic
activity toward the compact group. At first glance the composition of
the scattered group was more evident; but when attention was fairly
turned toward the compact, the inability to isolate objects made them
seem very numerous. The importance of a coöperating subjective factor
is here evident. (4) Out of a possible 57 cases there are but 10
showing no tendency.
2. _The Influence of the Internal Arrangement._ As before, the
Two-Group Apparatus was employed; and the factor was studied in three
aspects.
_A._ The material consisted of two groups of gray circles (Normal Gray
Darker, Prang) covering equal areas. In one group this area was filled
homogeneously, in the other the circles were gathered into nuclei. In
order that there might be exactly the same relation of parts when the
cards were reversed, each group was so arranged on a diagonal axis of
symmetry from upper left to lower right corner that each half repeated
the other in reverse order. Otherwise the arrangement of circles was
irregular.
Six cards only were used,--four (20 to 20), one (17 to 23), and one
(23 to 17). Slight differences among them occurred in the arrangement
of the equality-cards, which might help to counteract any incipient
reasoning from sameness of appearance to sameness of number. The large
increase in the difference-values is accounted for in part by the fact
that the cards (19 to 20) and (20 to 19) were omitted, and for this
reason: that when an observer tended largely to favor a particular
group, the introduction of a card in which that group was objectively
greater would mean an increase in the number of correct judgments;
whereas the introduction of two objectively equal groups for the others
would increase considerably the number of erroneous judgments.
_B._ The numerical character of the cards here used shows a return to
the usual. The material was like that of _A_, except for a new internal
arrangement. Here the area of one group was filled homogeneously, while
that of the other contained a pattern of this sort: An ellipse just
contained within the boundaries of the normal area; a circle in each of
the four corners of that area; and in the centre a diamond formed of
four circles. Numerical changes were always confined to the ellipse, as
less open to counting than the rest of the figure.
_C._ Material and method repeat _B_ but with another internal
arrangement. One group, as before, showed an area homogeneously filled,
and irregularly, as usual. The other group carries to an extreme the
distinction of open and filled space made prominent in the other groups
of this table by massing the circles in an outline completely enclosing
the area and in a diagonal from upper left to lower right corner. The
outline did not show even spacing; more circles were crowded in one
part than in others, that counting might be more difficult.
That there might be no attempt to remember cards, in all cases where
there were twenty objects in the homogeneous group the same kind of
irregular arrangement was repeated. This is a different method from
that employed in _A_. Since the length of exposure was so short and the
arrangement in the group irregular, either one is probably as good as
the other.
_D._ The material used for _B_ and _C_ had a kind of regularity,
since definite patterns were used. The introspection of the observers
showed, however, that the patterns as such were not in question in
the judgment, but rather the vacancies and the crowding. With the
Two-Group Apparatus an arrangement in parallel lines could rather
easily be counted, but with the One-Group Apparatus and its means for
instantaneous exposure this difficulty was to some extent overcome. The
arrangement of the objects in parallel lines was therefore adopted
and matched against the irregularity of an accompanying group. The
same size of group-area was kept, but the small difference-cards were
omitted. There was no other change beyond those made necessary by the
apparatus and already indicated on an earlier page. The length of
exposure was 1/25 sec. The bearing of this time-factor on the results
will be considered in a later section.
TABLE II
KEY: H = Homogeneous
N = Nucleated
P = Pattern
O = Outlined
R = Regular
I = Irregular
NT = No tendency
A B C D
132 132 132 88
_Experiments_ _experiments_ _experiments_ _experiments_
_with each_ _with each_ _with each_ _with each_
_subject_ _subject_ _subject_ _subject_
H N NT H P NT H O NT R I NT
Number of
subjects 10 2 2 10 1 3 7 5 2 2 1
Av. % of
difference
in favor of 53.23 1.5 6.1 35.7 52.8 5.3 42.7 34.8 7.9 40.3 1.2
The several parts of Table II give us the following facts: (1) The
judgment of relative number is very markedly a function of the internal
arrangement. (2) The marked tendency among the observers to favor
the homogeneous in _A_ and _B_ meets a check in _C_. Recalling the
direction of difference between _C_ and the other sets, that in _C_
the gradually increasing contrast between the inner vacancies and the
filling reaches a maximum, we may suspect that these vacancies begin
to seem no longer a part of the group-situation, while the compactness
of the filling, where it does occur, is thrust prominently forward.
The notes of the observers confirm this suspicion. (3) The results of
the different subjects show that the shifting of tendencies occurs
as before. (4) As to the way in which regularity functions in the
judgment, the notes of one observer are very clear. The blank spaces
in the irregular are noticeable, he says, which is not true of the
regular, where, on the contrary, one has a feeling of compactness
of figure. I am able to confirm this character of the spaces by my
experience outside this experiment. A simple pattern is very easily
apprehended and irrelevancies of the background dismissed. Increase its
complexity to a maximum, as in the case of an irregular group, and I
am almost at a halt to isolate objects from their fellows and maintain
them apart, yet together. The background is hardly to be shut out. This
is probably due to the absence of a centrally excited image of the
group. The object and the not-object run together. (5) The position
of the single observer in the no-tendency class of _D_ was marked
subjectively by great difficulty in forming a judgment. The groups
seemed incomparable, the vividness of form excluding the perception of
number.
3. _The Influence of Complexity in Group-Composition._
Complexity of group-content was attained by introducing objects of
different colors; so there was not a clean isolation of factors. By
comparing these results with those recorded in Table IV, A, we shall be
able somewhat roughly to make allowance for the factor of mere color.
Sets of 132 experiments each from sixteen observers were obtained for
each of these factors. Unfortunately the distribution-error was not
eliminated. Later experiments showed the importance of this factor,
and, in consequence, the impossibility of interpreting the results
under consideration. So new experiments were performed under the proper
conditions, but at a time when only a few of the first observers could
be used. Their results from the earlier series are given in Table III,
A. The exclusion of the small-difference cards from the later series
(Table III, B) and the consequent increase of the number of experiments
on objective equality prevent comparison.
The material in _A_ consisted of two groups of circles of the usual
size, one Normal Gray (Prang), the other of three colors--Red, Yellow
Orange Shade 2 (Bradley), Light Blue Blue Green (Prang). The intent
was to equalize the two groups in brightness. When the observers were
questioned about the relative brightness, supporters were found for all
three possible opinions. So it seems probable that the groups did not
differ widely in this respect. As nearly as the number-condition would
allow, the three colors were represented equally in the group; and they
were distributed so as to make the whole as homogeneous as possible.
In _B_ the changes were the correction for distribution as described
in the introduction to this section, and the substitution of
equality-cards for those of slight numerical difference. In addition,
three other colors replaced those of _A_, in the interest of regulated
brightness and more pleasing æsthetic effect. These were, in the
Bradley system of broken spectrum scales, A-Red, medium; A-Yellow
Orange, dark; A-Blue Green, dark. With these exceptions _B_ was like
_A_. The observers were all inclined to consider the gray brighter than
the mixed. The Two-Group Apparatus was used.
TABLE III
KEY: G = Gray
MC = Mixed Colors
NT = No Tendency
A B
132 132
_experiments_ _experiments_
_with each_ _with each_
_subject_ _subject_
G MC NT G MC NT
Number of
subjects 3 1 3 1
Av. % of
difference
in favor of 17.7 2.2 19.9 6.8
The following facts may be gathered from Table III: (1) The tendency
to overestimate the gray is due in part at least to the additional
factor of complexity in the other group, as is shown by the markedly
changed tendencies in Table IV, A, where a solid color takes the place
of the mixed colors. The actual colors involved in the two cases are
different, to be sure, and necessarily so, and this difference may of
course be invoked as the cause, as well as a possible difference in
brightness between the gray and mixed in III, B. The introspective
notes help us here. One observer felt that he favored the gray
primarily because there was a tendency to consider but one color in the
mixed. Another was drawn toward the gray because it seemed definite and
consistent. For both of these observers æsthetic elements were involved
in favor of the gray. The latter found also that the greater brightness
of the gray gave it a larger area. With a third subject the fact of
variety was felt as decidedly important; but his notes show a conflict
between this factor and that of distribution which was the conscious
basis for his normal judgment.
4. _The Influence of Differences in the Kind of Objects._
_A. The Factor of Color._ The material in A 1 consisted of two groups
of circles of the usual size, one Normal Gray (Prang), the other
Red (Bradley). The attempt was made by this choice to equalize the
brightness. The size and shape of the group-area were those of the
smaller area of Table I, C and D. In A 2 the only changes were the
correction for distribution, as described in the introduction to this
section, and the substitution of equality-cards for those of slight
numerical difference. The Two-Group Apparatus was used.
The following results appear in Table IV, A 1 and A 2: (1) While
complexity seemed on the whole to diminish apparent number, red
noticeably increases it. Some of the observers report that group as
more vivid and interesting. One observer compensated by emphasizing the
gray in attention, as his results showed. (2) If one ask how the color
red functioned in the judgment, the reply must apparently be, by its
brightness and vividness. The mixed group functioned in a double way,
as vivid and so more numerous, as fragmentary and so fewer.
TABLE IV
A1 A2
132 _experiments_ 132 _experiments_
_with each subject_ _with each subject_
_Gray_ _Red_ _No tendency_ _Gray_ _Red_ _No tendency_
Number of
subjects 3 1 3 1
Av.% of
difference
in favor of 18.3 5.4 30.3 9
B C
132 _experiments_ 132 _experiments_
_with each subject_ _with each subject_
_Large_ _Small_ _No tendency_ _Circles_ _Squares_ _No tendency_
Number of
subjects 8 3 5 8 4 4
Av.% of
difference
in favor of 33.6 27.5 6 28.4 18.8 3.5
D1 D2
88 _experiments_ 132 _experiments each_
_with two subjects_
_44 experiments_
_with two subjects_
_exposure_ = 1/25 _sec._ _exposure_ = 1/4 _sec._
_Simple_ _Complex_ _No tendency_ _Simple_ _Complex_ _No tendency_
Number of
subjects 3 1 2
Av.% of
difference
in favor of 30.7 5.6 21.2
E
88 _experiments_
_with each subject_
_exposure_ = 1/25 _sec._
_Bright_ _Dark_ _No tendency_
Number of
subjects 3
Av.% of
difference
in favor of 47.4
_B. The Factor of Size._ The Two-Group Apparatus was used, the material
consisting of India ink circles (1/3 to 1/2 mm. line) on a background
of granite paper. This paper was chosen here and for the experiments
of Table I, D, to get a suitable mean between too sharp contrast and
sufficient distinctness. The circles in the one group were ten mm.
in diameter; in the other seven mm. The two areas were approximately
equal, and of the same size as that of the more compact group in Table
I. This is in fact the standard size throughout these studies in
Relative Number, wherever area is not in question. The small-difference
cards were included.
Two sources of possible complication must be considered. It is
unavoidable that the factor of differences in compactness should
enter and that clean results on the basis of object-size be denied.
Our interpretation must not fail to consider this fact. Because of
this, it seems unlikely that a distribution-error should arise; so
the usual precaution to eliminate it was omitted both here and in the
study of area (Table I). Distribution affects the appearance of the
vacant spaces. When differences in the _amount_ are by the conditions
inevitably prominent, differences in the _conformation_ may be safely
regarded as of minimal vividness.
The following results appear in Table IV, B: (1) The illusion of
numerical inequality is marked for many subjects. (2) The judgment
is quite possibly a function of the two factors--object-size and
group-vacancies. If we recall the fact that the small-object group
is more scattered than the other, we shall note that the leading
class here is like the leading class in Table I, and we may fairly
reckon this factor as of importance in the issue. Of the incomplete
introspective notes on this question, those of only one observer speak
clearly for the size. He says: "There is an overpowering feeling of
predominance in case of the large and I must judge for them. The large
space covered seems an important factor. The longer I reflect upon
the relative numbers the more numerous seem the larger, that is, they
appear to increase over the small after the exposure. It is hard to
give judgments of equal in most cases."
_C. The Factor of Form._ The material consisted of a group of circles,
each of the same size as in former material; and a group of squares,
each approximately equal to a circle of the other group. These were
made of Prang's gray paper (Normal Gray Darker) and pasted upon a
black background. The areas of the two groups were approximately equal.
The squares were set irregularly except for those in the corners, where
the edges were placed parallel to the edges of the card. The Two-Group
Apparatus was used and the small-difference cards included.
In this material, again, the formal elimination of the
distribution-error was not attempted. The striking difference in
the conformation of the vacancies through the form-differences of
the objects probably makes the repetition of the exact positions
insignificant. Still the fact must be noted.
The results appear in Table IV, C. (1) The illusion of numerical
inequality is here again marked for many observers. (2) The
introspective notes are not on the whole very illuminating as to the
basis of judgment. One observer, who favored circles, found that the
appearance of more orderly arrangement in squares made them seem
few. Another, who favored squares, found, on the contrary, the more
regular the more numerous, and thought that the squares may have seemed
more regular. A third, who favored circles, found the squares better
individualized, with whom a fourth agreed in both respects, who also
was influenced by the apparently greater bulkiness of the squares.
Fewer could go into a given area. A fifth, on the other hand, who
found the circles better individualized, still favored them. So we
have these observers apparently doing the same thing under opposite
conditions, and the opposite thing under the same conditions. Here
indeed is a situation for any theory. So far as we can learn from the
foregoing, the form may influence the judgment merely through its
space-characteristics, but possibly also through the vividness of
intrinsic interest.
_D. The Factor of Complexity._ The One-Group Apparatus was used in this
work and results were obtained for two different lengths of exposure,
1/25 sec. and 1/4 sec. The material differed, in that to the centres
of the circles of one group were added small Red (Bradley) circles (6
mm.). With the apparatus used, the color was not very effective, the
brightness contrast between dark centres and white periphery being
chiefly prominent. The total group-brightness was of course diminished
by those centres. The small-difference cards were omitted.
The results are recorded in Table IV, D 1 and D 2. (1) The illusion
is apparently strong. (2) The amounts of the difference-values show
that the shorter exposure is more favorable to the illusion. (3) The
introspective notes indicate that both brightness and complexity
functioned in the judgment. Two observers, both of whom show large
tendencies, were not conscious of any influence of complexity. One of
these did find differences in brightness important; and in favoring
the darker group his results exactly coincide with those of Table IV,
E, where this factor is under direct consideration. A third found the
complex group interesting. With a fourth the complex group developed
in number amazingly during the few moments after exposure and had an
appearance of great intricacy, often seeming to be in active movement.
A fifth observer too felt that its numerical character depended on its
complexity.
_E. The Factor of Brightness._ Hitherto the absolute arrangement of
the objects in any two groups compared, where this factor has not been
the object of enquiry, has been in the two cases different, though
with respect to irregularity alike. This course was governed by a
desire to avoid the substitution of a form-judgment for one on number,
through recognition of the fact that both groups had identical forms.
The resulting distribution-error I tried to eliminate in the usual
way. Tests toward the end of these studies showed that there was no
danger from this source. Errors seemed about as frequent as before.
No observer made any comment on the fact, except one who through his
official connection with the laboratory work knew that the test would
be made sometime, but not exactly when. During many of the experiments
he did not perceive the likeness of form; and when he did the numerical
judgment arose without connection with that factor, as was shown by
the feeling that the two groups were unequal in number. He called the
relative fewness of the first group a case of "perspective effect."
This must have significance for any account of the time-error; but by
no means carries with it its own interpretation.
One welcome result of these tests was their assurance that I might
without fear further simplify the experimental conditions by avoiding
the possibility of a distribution-error. The material for these
experiments on brightness therefore profited by this possibility. Each
card had a different specific irregularity, but always in duplicate. In
choosing the degree of brightness-difference Prang's brightest shade of
normal gray was found as dark as could be conveniently perceived with
the artificial light of the One-Group Apparatus. The contrast between
this shade and white was quite evident enough for the purpose. The
small-difference cards were omitted.
Table IV, E, shows the decisive character of the results. The observers
fall all into one class in favoring the darker group, and by a large
difference-value. The following introspection of one observer shows
the extent to which the factors of brightness and number fuse: "I
frequently lose sight of time-order. It is a question of number and
not one of light-intensity, and if called upon to state which group
came first I might not be able to answer. In equality-judgments the
difference of light comes out distinctly."
5. _The Influence of Complexity of Environment._
The material prepared for these experiments certainly lays stress
upon _relative_, not _absolute_, complexity; for the conditions were
satisfied by placing 5 mm. strips of white paper, equal in length
to the width of a group, a few millimetres off at the top and the
bottom of the groups that were on one side of the cards. The One-Group
Apparatus was used and the small-difference cards omitted.
TABLE V
44 experiments with two subjects.
88 experiments with two subjects.
Exposure = 1/25 sec.
_Simple_ _Complex_ _No_
_environment_ _environment_ _tendency_
Number of subjects 2 2
Av. % of difference
in favor of 15.9 8.5
The results are recorded in Table V. (1) The drift of tendency is
toward the group with the more complex environment. No one markedly
favors the other group. (2) The notes of the observers indicate that
the added strips functioned through their effect upon the apparent area
of their group. The observers all found the dimensions increased; but
with some, apparently by contrast, the added height brought out sharply
the narrowness. One observer found this true in general; another,
when the barred group came first. The latter says: "The unbarred
group, coming first, appears to reflect its compact character on the
barred one, when it comes, so that it does not look so attenuated and
strange." Here the image brought over to the second took the width of
the second somewhat out of relation to its illusory height, whereas in
the reverse order the contrast relation was fully maintained.
III. THE INFLUENCE OF FACTORS PRESENTED IN OTHER SENSE-FIELDS BY THE
OBJECTS WHOSE NUMBER IS IN QUESTION
A very simple apparatus was employed. The objects whose number was in
question were bright steel balls (3/8 inch) thrown loosely into square
black frames, 13 cm. inside, placed side by side on a black-topped
table. The experiments were performed in series of 30. The groups were
kept equal numerically, with this exception, that into each series were
introduced four experiments where the groups were so unequal that the
observer could have no question as to the correctness of his judgment
and the existence of objective differences. This numerical superiority
was given to each group alternately, and judgments on it were, of
course, excluded from the results. The actual numbers employed in a
series varied between 35 and 60 in accordance with the following scheme:
1. 50 each
2. 45 "
3. 50 "
4. 55 "
5. 60 to 40
6. 50 each
7. 45 "
8. 40 "
9. 45 "
10. 50 "
11. 55 "
12. 60 "
13. 40 to 60
14. 50 each
15. 45 "
16. 40 "
17. 35 "
18. 40 "
19. 45 "
20. 50 "
21. 55 "
22. 60 "
23. 60 to 45
24. 50 each
25. 45 "
26. 50 "
27. 60 "
28. 55 "
29. 50 "
30. 45 to 60
The time of a single exposure--in this case two groups at once--was
3 sec. measured by a stop-watch. As to the arrangement of the balls,
care was taken that they should not be massed in one place, but
scattered somewhat homogeneously over the space within the frames. The
illumination was daylight, so managed that shadows cast by the balls
were reduced to a minimum. The observer sat close to the table with
the groups directly in front of him. He either kept his eyes closed
between experiments or held a small screen before them. Sometimes he
merely turned away. The operator worked from the opposite side of the
table, taking care to make the necessary noises as little suggestive
as possible. The observers agreed that they were not consciously
influenced by the manipulation.
The progress of these experiments disclosed an astonishing space-error.
So far as was conveniently possible the usual technique of elimination
was employed.
1. _The Influence of Active Pressure._
In this study the groups were differentiated in this way: With one hand
the observer rolled the balls of one group under his fingers, while the
other group was presented to vision only. The method of observation
consisted in rapidly and lightly rolling the balls under the fingers a
few times and then surveying both groups visually for the remainder of
the exposure, judgment being given on the visual number.
Evidently there is much that is rough about this procedure. Pressure
and kinæsthetic factors are lumped off together; the length of the
touch-stimulus was not exactly determined; and there is the possibility
that the visual stimulation from the group touched is weakened. To be
sure the method prevents any great difference in the latter respect;
and if we are guarded in our interpretation, something of interest may
be learned.
There appeared to be no convenient way to eliminate the space-error.
The right hand was used with the right group and the left with the
left. So here again interpretation must be circumspect.
TABLE VI
A B
52 _experiments with_ 260 _experiments with_
_each subject_ _one and_ 208 _with the_
_other subject_
_No_ _No_ _No_
_Touch_ _touch_ _tendency_ _Uneven_ _Even_ _tendency_
Subjects 2 1 1
Av. % of
difference
in favor of 7.6 10 1
SPACE-ERROR
_No_ _No_
_Right_ _Left_ _tendency_ _Right_ _Left_ _tendency_
Subjects 1 1 1 1
Av. % of
difference
in favor of 69.2 23 30.8 20.2
C
145 _experiments with_
_one and_ 260 _with the_
_other subject_
_No Weight_ _No weight_ _No tendency_
Subjects 2
Av. % of difference
in favor of 2.4
SPACE-ERROR
_Right_ _Left_ _No tendency_
1 1
Av. % of difference
in favor of 32 4.4
Turning to the results in Table VI, A, we find the following: (1) The
influence of the pressure-kinæsthetic complex practically does not
appear; while the space-error shows a marked tendency that, for the
two observers, is in opposite directions. (2) On the other hand, the
notes of one observer show that in his case at least the face value of
the table is erroneous. To this effect he says in substance that he can
make a more accurate estimate of the number in the group touched. He
tries to ignore these sensations of touch, but with ill success in the
case of the left hand, where clumsiness not only makes it difficult to
touch the balls gently but also to keep them under the fingers, which
often feel the ground-space. For this cause the group seems small in
number. Clearly enough, then, it is the space-error that tells the
story of the effect of the added stimuli on this observer, only it
must not be interpreted as space-error. The pressure functioned in the
judgment through its numerical aspect. But the positive effect with the
right hand was turned to a negative with the left through its emphasis
of vacancies. The high difference-value in the space-column becomes
thus a striking evidence of the effect of pressure, and the results are
accounted for without reference to the kinæsthetic factor. The other
observer felt that the active pressure was relatively indifferent. (3)
The entire absence of correct judgments on the objectively equal groups
shows to what a surprising extent other factors have modified the
numerical.
2. _The Influence of Unevenness in Active Pressure-Feeling._
In the experiments of this section the groups differed in this way,
that one rested on the smooth table-top while the other had for its
bottom a coarse wire mesh covered with black cloth. The balls of the
one rolled smoothly beneath the fingers while the other balls moved
lumpily over their mesh. Both hands were used--each for the group on
its side; and the method of observation and length of exposure agreed
with those conditions in the preceding section, except that the balls
were rolled a little more vigorously that the factor studied might
come clearly into consciousness. The groups were interchanged for half
the number of series. This could not of course completely eliminate
the space-error, since kinæsthetic differences in the limbs remained
uncompensated. In general the criticism in the preceding section is
again applicable.
The results appear in Table VI, B. (1) One observer shows a tendency
to favor the smoothly rolling group, while the other again shows no
tendency. Both have large space-errors of the same character as in A of
this table. (2) The introspection of the observer showing no tendency
is to the effect that touch plays little or no conscious part in the
situation. The other's notes give no hint that the factor studied
here was influential; but to the effect on the judgment of touch in
general, especially with the right hand, they give clear witness. The
touch-sensations, he says, were difficult to ignore. Those from the
right hand were more vivid than those from the left; and the right
hand seemed more sensitive. Judgment was based on a general feeling of
"moreishness" which came promptly. There is nothing to contradict the
evidence of the earlier experiments that touch is again influential
through its numerical character. (3) Both observers regard factors of
distribution as of fundamental importance, though one was inclined at
first to insist that there was nothing but number in his judgment. The
significance of this unanalyzed feeling will appear in a later section.
(4) These results agree with the preceding in the approximate exclusion
of correct judgments.
3. _The Influence of Active Weight._
The variation here in question consisted in lifting one of the groups
during judgment of the relative number in the two groups. The apparatus
was made by transforming into trays the frames containing the balls,
by putting into these frames wire-mesh bottoms covered with black
cloth. They were set each upon four small wooden pillars so that the
hand could be easily thrust under the tray. At the signal a given tray
was several times raised a little way and lowered, and the judgment
formed on the same factor as before. Here again the space-error was not
entirely eliminated. Each hand was used with the group on its side, but
kinæsthetic differences peculiar to each of the limbs remained. There
was always some motion among the balls in the lifted tray, though the
gentleness of the lifting prevented the existence of much. This is a
radical defect, but one not easily avoided with maintenance of other
desirable conditions. Even more serious, as the issue proved, was the
failure to control the lifting impulse; yet, as it happens, we are not
prevented from getting an experimental answer to our question.
The results are recorded in Table VI, C. (1) They show no apparent
effect of the weight, and with one observer the further unusual fact
of no space-error. This error is marked enough in the case of the
other, and, conforming in direction to that of the preceding sections
of this table, allows us in so far to adopt the same interpretation of
his results. (2) The introspection of one observer was to the effect
that he felt a tendency toward a modification of the number-judgment
by weight. It was especially strong when the group was lighter or
heavier than was anticipated, the light group seeming less numerous,
and the heavy group more. Occasionally he caught himself weighing the
second group mentally; and sometimes he had to recover himself from a
tendency to make judgments on a wrong basis, presumably that of mere
weight. With such a conflict of tendencies the character of the results
is not surprising. Particularly important are the opposing tendencies
lying in the factor of weight itself. The other observer reported that
a very heavy weight exerted an influence that it was hard but not
impossible to ignore, while a lighter weight did not effectively enter
the situation at all. His earlier inclination to say that there was
nothing but number in his judgment inclines one to believe that fusion
of factors may have passed beyond the stage of ready analysis. (3)
Our analysis has given us reason to believe that active weight has a
definite tendency to modify the judgment of relative number.
4. _The Influence of Muscular Strain in Observation._
The study was made from the point of view of more than one set of
experimental conditions, viz.:
(1) Equal strain (minimum).
(_a_) Right--left.
(_b_) Up--down.
(2) Equal strain (maximum) eyes turned.
(3) Strain _vs._ ease.
(_a_) Head and eyes turned.
(_b_) Eyes turned.
The conditions of (1) (_a_) were exactly those of the earlier
experiments with the exception that the groups were undistinguished
save by position. In (1) (_b_) one group was so placed between the
other and the observer that there might be as little increased effort
as possible in viewing the farther. In the up-down movement more
muscles are involved in the lift than in the fall of the eye. So really
we have here a case of (3) but not so marked. In (2) the groups were
put to the right and left at such distances that, when sitting between,
the observer could just take each one in without turning his head. This
brought a decided strain upon the eye-muscles. In (3) (_a_) the groups
were separated by the length of the table--a distance of 90 cm.; and
the observer placed in alternate series before each; as he was in (3)
(_b_) where the farther group was carried to the limit of vision to be
reached without turning the head. Here the strain was like that in (2),
but for one group only.
An incompleteness in experimental analysis lies in the impossibility of
separating the factors of distance and strain.
TABLE VII
A - Baldwin 46 experiments
Hutchinson 78 experiments
Equal strain (minimum)
B - 52 experiments
with each subject
Equal strain (minimum)
C - 52 experiments
Equal strain (maximum)
Eyes Turned
D - 104 experiments
with each subject
Head and eyes turned
E - 52 experiments
A B C D E
Right Left Lower Upper R. L. Ease Strain E. St.
Subjects Baldwin Hutchison 2 Bald. 2 Bald.
Av.% of
difference
in favor of 30.4 28.2 68.3 71.2 52.4 80.8
Here are the facts of chief interest: (1) The following tabulation
gives us a ready view of the character of the results in Table VII; and
shows the extent to which they are consistent:
A B C D E
Baldwin favors right {upper left {farther {nearer
{strain {strain {no-strain
left left
Hutchison favors left {upper {farther
{strain {strain
left
(2) The only inconsistency in the strain-distance complex is with
Baldwin in E. He reported that the more distant group appeared rather
as an undifferentiated mass whose number was not so well obtained,
while in the near the individuals were significant. He seemed to
be in the midst of these. The case seems analogous to that of the
observer whose introspection was reported under Table I, and who at
first accepted what we may call the objective analysis, by which
the scattered group gave up more distinct objects than the compact;
but later attempting voluntarily to disintegrate the compact, found
a bewildering confusion in the task that made this group seem very
numerous, and brought about in the end an exact reversal of tendency.
(3) Can we now separate in the results between the influences of strain
and of distance? So far we have regarded them as one complex. But the
introspections speak merely of the space-characters of the objects,
Hutchison agreeing with Baldwin that the more remote group is judged
as an area rather than as a collection of definite objects. (4) The
almost entire absence of correct judgments in these experiments adds
new evidence to that of the immediately preceding experiments in proof
of the insignificance of the actual numerical relation for the judgment
of relative number.
IV. THE INFLUENCE OF FACTORS OUTSIDE OF THE OBJECTS AND IN OTHER
SENSE-FIELDS
The One-Group Apparatus was employed, and cards in general
corresponding to those where area was not in question,--white-circle
groups equal in size and irregular in inner distribution, which was not
duplicated on the same card, though the resulting distribution-error
was formally eliminated in the usual way. The usual care was taken
to fill the group-area homogeneously. The small-difference cards
were retained at first; but on later discovering the possibility of
duplication a few supplementary experiments were added.
1. _The Influence of Touch._
The apparatus employed to give the touch-stimulus consisted in a long
lever attached to the armature of a small electro-magnet. In the end
of the lever was inserted at right angles a wooden peg, cork-tipped.
In view of the other conditions of the experiment a convenient spot
for the application of the stimulus was found to be the forehead where
it curves backward above the right eye. The apparatus was supported
by rods and clamps upon a long upright steel rod set in an iron base
and placed behind the chairs of the observers. The same rod carried a
head-rest, designed not as a support but merely to show the observer
that he had returned to the original position after he had bent forward
to record judgment. Where two observers were used at once two sets of
this apparatus were employed, with the magnets in a single circuit
governed by a floor-button. The touch-stimulus was made to coincide as
closely as possible with the appearance of a given group.
In view of the practical remoteness of this factor from the object of
judgment the experimentation here took two forms,--one in which the
observer was passive toward the touch-stimulus; the other in which the
effort was made closely to associate the touch with the visual group by
imagining the group to be responsible for the touch. For the passive
method the touch was given irregularly now on the first and now on the
last, but as many times on one as on the other.
For the active method, it was given always on the last group. This
constancy was held to favor the active association of touch and
particular group. The constant time-error was guarded against by
experiments in which no modifying factor was introduced. A and B of
Table VIII present the results of the passive and active methods
respectively. C and D repeat A with duplication of groups,--C with the
usual (1/25 sec.), D with a longer, exposure. These last sets were
taken that the factor of touch might be studied when the objective
conditions of the strong distribution influence should have been
removed. It might prove that a factor swamped in the former situation
might emerge into effectiveness.
The following summary gathers the chief facts of Table VIII: (1)
Touch appears practically without effect in A. (2) In B, the
results for touch seem again insignificant; but comparison with the
control-results, to isolate touch from time-order, while it shows no
marked change for Angier, does show for the others that touch was
effective in determining the direction of error by difference-values,
in the two cases of 10.2 and 14 per cent. The active method seems to be
slightly more favorable to the influence of touch. (3) The duplication
of the groups in C gives a large increase to the apparent effectiveness
of touch, which is considerably diminished but not destroyed by
the lengthening of the exposure in D. (4) The introspection for A
indicates that touch under these experimental conditions has little
subjective importance for the judgment of number. It is sometimes quite
unnoticed. Angier made a possible exception in its favor in cases of
great hesitancy where it added "importance" to the group with which
it occurred. Usually he felt little doubt. With Shaw the touch was at
first distracting but later indifferent. Johnston's notes indicate
rather more effect. The touch prevented strict attention to the figure
impression whereby the space-intervals in that group lost in value.
Later it lost its confusing effect. Here seems to be subjective
tendency, but not enough to predominate in results.
TABLE VIII
A
88 _experiments
with each of two
subjects._ 132 _experiments
with one
subject._
B
198 _experiments
with each of two
subjects._ 110 _experiments
with one
subject._
C
44 _experiments
with each subject._
D
88 _experiments
with one subject
and_ 44 _with the
other_
_Exposure_ = 1/4 _sec._
A B
_No _No
_Touch_ _No touch_ tendency_ _Touch_ No touch_ tendency_
Subjects 3 3
Av.% of
difference
in favor of 3.2 5.7
C D
_No _No
_Touch_ _No touch_ tendency_ _Touch_ _No touch_ tendency_
Subjects 2 2
Av.% of
difference
in favor of 26.1 13.7
Results in B for the subjects separately were as follows: Angier 4%,
Johnston 8.2%, Shaw 5%, all, so far as they went, in favor of the
touch-group. Control experiments to determine the time-error gave
the following results: Angier 6.8% in favor of the group last seen,
Johnston 2.2%, and Shaw 9% in favor of the first group.
Some further introspective evidence appears in connection with the
active method of B. Angier confirms his earlier account exactly.
Usually the factors of distribution practically associated with number
determine the judgment promptly; but in cases of doubt the touch is
felt to add to its group something that appears as number-value.
Johnston's subjective situation seems a little complicated. I may
summarize thus: (_a_) The connection between the touch and its group
being established, that group seems smaller, as being, together with
the touch, somewhere nearly equal to the first. (_b_) The connection
established and touch failing to come, that group seems smaller. (_c_)
The connection not established and attention being concentrated on
the visual impression, the touch-group feels much larger. The curious
attitude in (_a_) results in a discounting in advance of the actual
number. This done, the touch adds numerical value to its group. In
(_c_) the effort at abstraction appears to emphasize the second (touch)
group. Later, he reported similarly that the touch-stimulus seemed
to add to the number of circles in its group even when the judgment
favored the other group; and that "any outside stimulus connected
with the one of two exposures tends to lose its own significance and
be translated into number of dots to help the accompanying exposure
to equal or exceed the first." The touch-group is felt to have more
significance through association with an idea of superior energy or
greater motor impulse.
Of the character of the influence exerted by the touch, Shaw reported
that there seemed to be a diminution in the size of the first group
and something extra in the second. More specifically, this effect
appeared at times as an added circle at the right of the second (touch)
group. He thought that this effect was overruled by the real bases of
number-judgment which he summarized as "size, regularity, density, etc."
These notes show a definite tendency on the part of the touch-stimulus
to break in upon the course of the number-judgment ordinarily
determined by the practical association of a specific group of factors
with number. That this result gets no more marked registration in
the percentages is apparently due to the strength of these customary
associations.
(5) The extent to which the distribution-error complicates the
present study is shown by the prompt increase in effectiveness of the
touch-stimulus when the groups were duplicated, as in C and D.
2. _The Influence of Hearing._
The scheme of the experimentation upon this factor conformed in
general to that of Section IV, 1. But a new sort of differentiation
was possible in the auditory field, and one more readily suggestive
of numerousness, perhaps, in that by use of an electric bell a rapid
succession of sounds could be given with one group while with the other
a single sound could be produced. An actual numerical difference in the
auditory field might fuse with the factor of relative visual number
and determine the judgment to its direction. These results are set
down in B of Table IX. The same set of cards was used in the One-Group
Apparatus for these experiments as for those of Section IV, 1. In these
two sections of Table IX the observers did not know on which group
the sound or the particular sound would be given; but any possible
disturbing effect of this irregularity was formally eliminated as in
Section IV, 1. The experiments of Table IX, C, repeat those of A with
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.