Chapter XXII: Section IV: , 1. The experiments of Table IX, C, repeat those of A with (2)

The issue of these tentative experiments in absolute number confirms the teaching of our studies in the related field. Absolute number, like relative, has been found largely subject to a modifying influence of certain factors. In the new field, too, distribution has asserted its supremacy among these, and similar effects of shortening exposure have been observed. There has been variation among the observers and some shifting of tendency, both of which point as before to the coöperation of some subjective factor in our results. Indeed the whole situation, as opened by these preliminary studies, indicates a theoretical interpretation that for both fields is at bottom one. So to an attempt to reach such an interpretation the next section will be devoted.

VII. THEORETICAL DISCUSSION

1. _The Fact of Modification._

That such an influence upon the judgment of number should have been exercised by the factors considered seems in many cases to receive an adequate account on the principle of association. Our practical experience in the simultaneous variability of number and certain other characteristics of a group of objects has been such as to lead us into illusions when the two no longer vary together. In such a case, when we have no time to count, we are actually led to _see_ a group as smaller or larger in accordance with the variations perceived in the associated factor. This interpretation is supported by the fact that on the whole the space-factors were more markedly influential in creating illusions than were any others. For those cases, however, in which the modification was effected by a factor unconnected with number, as color, or the simultaneous stimulation of other senses by irrelevant objects, it appears that the mere occurrence of greater total stimulation during the appearance of one group is sufficient to create illusion, either through failure of the observer to discriminate between the relevant and the irrelevant, or because he is led through fear of disturbance to overemphasize the other group.

2. _The Direction of Modification._

The foregoing account of the general fact throws no light upon the _direction_ of the influence. Why should a given factor make a group seem more numerous and not less? Why should it affect one man in one way and his neighbor in another? Why should it vary with the same man at different times? Appearances no less contradictory than these are what we must face in carrying a theoretical account to completion. The following propositions with appended commentary are offered in satisfaction of these requirements.

_a._ Differences in vividness among the factors determine differences in number.

Our study of the factor of distribution in Table XVII, where it was possible in a measure to control the vividness, furnishes evidence for this proposition. Introspective reports in other cases confirm this view by showing that the direction of the attention, the popular way of stating our proposition, was the determining feature. This will receive further support in our discussion of the following proposition.

_b._ If the vivid factor or complex be positive, _i. e._, associated in experience with the numerous, or if it be neutral, its group will seem the more numerous. If negative, _i. e._, associated in experience with the few, its group will seem the less numerous.

The experiments upon the effect of distribution support this proposition, especially as set out in Table XVII. When the vacancies in a given group were made vivid, the other group seemed more numerous; when its filling surpassed in vividness, the judgment was given for it. We have other confirmation in the fact that lengthening the time of exposure reduced the absolute number. Take also this note of one observer on the material in Table II, C:

"I noticed that I had set the open spaces in the outlined group over against the lack of them in the homogeneous, without paying much attention to the nearness together of the spots in the lines of the outlined. Then for a time my attitude was quite vacillating. I found my attention drawn to the nearness together of the spots in part of the outlined group so strongly that if I did not turn it voluntarily to the fact that the other was filled without any large open spaces, I was led to call almost any outlined group the larger. Toward the end of the experiment I got back into my original attitude, in which the outlined group seemed to have its spots hardly more thickly arranged in any part than the homogeneous, and to have also the bare spots and so to be the fewer."

That the vividness of a neutral factor or complex increases the apparent number was suggested by comments of the observers. One observer reported of the material in Table IV: "The greater brightness of red gave it more importance. The natural thing seemed to be to give the red the judgment. The gray fought more for recognition." And again: "The red seems a vitalized space and the dots more omnipresent, also the red lasts longer in memory and is there more vivid, so that often in cases of doubt, where the decisive comparison was made in memory, the red may have been given the vote. Often there was an immediate unanalyzed feeling that if the groups had been both of the same color, the judgment would have been for the gray." In both cases his results showed this tendency. Another observer, whose results agree with the former, found that his eye was directed involuntarily toward the red.

This fact was put to a special test. In the material of Table II, B, a card, in which the pattern group had an appearance strikingly different from the normal, was introduced, the two groups being objectively equal. With three observers the effect was overestimation of this group, and with a fourth, the suppression to equality of a previous overestimation of the opposite group. This fact, together with the observers' comments, seems to justify the conclusion that the vividness of a neutral factor or complex was the determining condition of the judgment. That the observers did not all show positive results in this experiment may be set down to the difficulty in controlling the subjective conditions of vividness. Of course the space-relations within the new pattern were different from those in the old. The only justification for taking no account of these is the character of the introspections themselves.

It should be said of the red group that beside its vividness it had characters mentioned by other observers that might independently have made its number seem greater. It was called "dazzling," "blurred," and its area seemed increased. In this respect the effect of the color should be discussed as a special case of distribution or object-size.

The vividness tested in this special way seems due to contrast, in the one case with surroundings, in the other case with the expected. Such a judgment is very far removed from the normal bases, rather more so, it would seem, than even those where a group had sound or touch accompaniment; for in the latter case there could be no question about the "moreness"; the only doubt could be about its legitimacy. Of the precise extent to which this cause of vividness has operated throughout our studies, even where spatial differences have been concerned, we cannot be sure. The patterns of the materials in B and C of Table II seem to offer that possibility. That it should enter anywhere opens, indeed, the entire field.

_c._ The observers fall into the following classes on the basis of the character of the association:

(1) Relatively fixed association,

(_a_) involving correct adjustment to objects (vividness of relevant
factors);

(_b_) involving incorrect adjustment to objects (vividness of
irrelevant factors).

(2) Relatively unstable association.

It will be remembered that in the case of nearly every factor studied under Relative Number, we found three classes of observers,--those favoring one group, those favoring the other group, and a so-called "no-tendency" class. The bases of classification were, first, the relative constancy in the character of the error, and, secondly, its direction. In this third or "no-tendency" class were really lumped off two kinds of observers, not separated at the time because our special interest did not demand it. Along with those that gave large errors in both directions was a much smaller class that gave a relatively large proportion of correct judgments; but could never claim any one observer all the time. In the new classification of Proposition _c_ this mixed composition is recognized by dividing it between (1) (_a_) and (2). The prime condition of correct judgment is asserted to be one in principle with that of the illusion,--namely, vividness, but in this case vividness of relevant factors. Our original "tendency" classes both fall under (1) (_b_).

Proposition _c_ is merely an attempt to apply the principles of association and vividness to an organization of our results. The types in question have no hard-and-fast connection with any particular observer; they rather represent a kind of ideal fixation of opposite tendencies playing through all.

3. _The Time-Error._

So far the time-error has been left without interpretation. The chief facts to be considered were: (_a_) Divergence of error and general trend in favor of last. (_b_) Individual inconsistencies. (_c_) Occasional absence.

We are in a position now to invoke at once the principle of vividness to account for the existence of the error and the vividness of recency to account for the predominant tendency to favor the last of the two groups exposed. In this respect this error may be classed with the effect of red. That is to say, a factor or complex, directly through its vividness and not indirectly through its association with the numerous or the few, draws the judgment after it. Here the content of the group is the effective thing, not the character of the vacancies.

But the observers do not all agree in the direction of the time-error nor are they always consistent. Here we shall get help from a proposition offered in the discussion of the distribution-error in which it is asserted that the group seems the more numerous in which the vacancies are less developed under observation. We have already noted a decided tendency to depend in judging upon the vacancies. Let us suppose that the two groups presented in succession differ with respect to the success of the observer in developing these vacancies. If this be true, that difference may well depend upon the occurrence of maximal attention during the exposure of but one of the groups. The tendency of the majority to overestimate the second group suggests that the attention is likely to be at a maximum when the experiment begins. If, on the other hand, it ripen later, or if the observer seek to rescue the second group from relative unclearness, then we should have the first group overestimated. The time-error would disappear for those that could attend alike to both. Clearly enough this account is decidedly hypothetical.

4. _The Space-Error._

Our attempt to reduce this error to one of time in some form was proven a failure. The facts brought out indicate that at the bottom is some subjective factor thus far not isolated. This factor is not a preference going directly with right- or left-handedness because on the surface at least it runs in the observers independent of such asymmetry. A single bit of available introspection would seem, however, to point to some relation of that sort; for one observer, who favored the left, felt that a group on that side gained an importance that was somehow due to the greater absolute value of a weight in the left than in the right hand. Even if this be decisive for him it will still be inapplicable to errors in the opposite direction unless we assume that with variations in bodily energy the emphasis is cast now in one, now in the other, direction, after the analogy of those two types of man to be found in our social experience, for whom respectively mountains are molehills and molehills mountains. Such successive differences in type in a single individual would then find an intelligible account in the shifting tides of that bodily energy. It is to be noted that the observer just quoted once, but only once, made a decisive reversal of his error from left to right.

It may occur to some one that the use of two observers sitting side by side may have given them a preference for one position of the groups. In the first place care was taken that both groups should be as readily seen from one point of view as from the other. Secondly and chiefly, there is no regularity among the observers in this respect.

It is not unlikely that a chance aspect of a particular group develops an emphasis that gives the mechanism of subjective adjustment a particular bent that for a time is relatively independent of the objective situation. Still the fact that there are some cases of persistence in type is rather damaging to this assumption and speaks rather for the earlier one. That one, if true, seems indeed adequate to account for the situation. As an hypothesis it accords with analogous physiological facts; but its weakness lies in imposing the burden of a strong tendency upon asymmetrical differences that may be in comparison relatively slight. Finally, these studies furnish no proof that the bodily condition of an observer of a particular type corresponds to the demands of the hypothesis.

_Summary:_ 1. The estimation of relative number in the visual field is modified by group-area, internal distribution, order, and complexity in group-composition; by the size, form, color, brightness, and complexity of the individual members; and by the character of the environment. It is further modified by factors contributed by the objects through other senses, as in active pressure, special differences in pressure character, active weight, and that complex of muscular and spatial factors arising when a group is observed under the condition of eye-muscle strain. The judgment is also influenced by factors outside the group in the field of touch, but not in that of kinæsthetic impressions.

2. On the whole the most influential factors were those lying in the space-characters of the groups; while those of least moment were contributed by other objects in other fields of sensation. Hearing was very nearly ineffectual.

3. In very many cases the observers fell into three groups, one of no-tendency, and a second and third showing opposite tendencies with respect to the factor investigated.

4. With a majority of observers there is a tendency to underestimation in the judgment of absolute number, though with a single observer the tendency is directly the reverse. Scattering the objects increases, and compacting diminishes, the apparent number. The smaller the size of the objects the fewer, under conditions, do they appear; while heterogeneity in group-composition lessens the number for one observer and has no apparent effect upon the other.

5. The apparent absolute number of objects is inversely proportional to the length of exposure of a group; and in relative number the influence of a factor was on the whole greater for shorter exposures.

6. The marked tendency to a space-error was found to be independent of differences between the groups in the length of look, and of the order in which they were viewed.

7. The distribution-error is grounded in a fundamental tendency to base the judgment of relative number upon the character of the vacancies in a group; though a secondary tendency to depend upon the filling was shown to exist. The subjective factor of vividness, attaching now to one and now to the other of the foregoing factors, determines which shall be operative, though it usually is joined to the first. The ground for the primary tendency may very well be the necessities imposed upon discrimination by the material. The contrast effect between the large black background and the brighter objects tends to unify the latter, which, to be discriminated as a number, must be split up by an emphasis of the vacancies.

8. The time-error is possibly due to differences in power to dismember the groups exposed in succession in one experiment, while its variations in direction seem adequately accounted for by differences in the time at which attention becomes maximal during the progress of a single test.

9. The ground for these facts of modification is found in the strength of the association between these several factors and the elements that signify number.

10. The basis for the different tendencies found among the observers is the differing vividness among several factors. If the vivid factor is associated with the idea of numerousness, or is in this respect neutral, its group will seem more numerous. If it has been associated with the idea of fewness, its group will seem less numerous. The difference between the two classes of "tendency" and "no-tendency" lies in the fact that for the latter either only correct clues are vivid, or else there is so frequent an alternation in vividness of opposing incorrect clues that through any given series no tendency appears, while for the "tendency" class misleading clues are without shifting in the ascendant.

At the time when these experiments were completed, no work precisely upon this problem had been published. Since then, however, Dr. J. F. Messenger has issued a monograph entitled The Perception of Number (Psych. Rev., Mono. Supp., vol. 5, no. 5), of which certain parts fall within the scope of the present studies. He was concerned with the estimation of absolute number and was primarily interested to discover the nature of the number-judgment. The reader of both articles will find agreement between the results and interpretations here recorded and such part of Messenger's work as has been a common object of study,--viz., the factors of distribution and size.

FOOTNOTES:

[Footnote 123: 44 experiments.]

[Footnote 124: 88 experiments.]

TIME-ESTIMATION IN ITS RELATIONS TO SEX, AGE, AND PHYSIOLOGICAL RHYTHMS

BY ROBERT M. YERKES AND F. M. URBAN

The desirability of a statistical study of time-estimation was suggested to us by a note concerning "sex-differences in the sense of time" which was published in Science recently by Prof. Robert MacDougall.[125] By comparing the time-estimates of groups of men and women consisting of fifteen individuals each, MacDougall discovered that for intervals of from one quarter of a minute to a minute and a half, the women exhibited a far stronger tendency to overestimate than did the men, and were at the same time markedly less accurate. The nature and extent of the overestimation discovered by MacDougall are indicated by the results presented in the accompanying table from Science. The numerals, 1, 2, 3, and 4 refer to different fillings of the intervals (listening to reading, marking letters, etc.), the signs + and - to over- and under-estimation respectively.

Period, One Minute.

Sex 1 2 3 4
Men +29" + 1.3" +22" - 3.5"
Women +66 +22.0 +80 +24.0

These apparent sex-differences in time-estimation demand further attention, first, because the number of individuals studied by MacDougall, as he recognized, is too small to establish the fact of the existence of such differences, and, second, because if the differences really do exist they should be studied in their relations to age and the fundamental physiological rhythms.[126]

It seemed probable that further investigation of this subject might reveal some important facts concerning the development of the ability to estimate time in the individual, the significance of various conditions for time-estimation, the psychology of sex, and the relations of rhythms to personal affinities, antipathies, and motor capacities.

In this report the results of a statistical study of the sex-differences in time-estimation are discussed, and in later papers we shall present the results of investigations of the relations of time-estimation to age and to individual and sex rhythms, and attempt to work out a convenient and serviceable rhythm-formula. The need of such a formula for expressing individual rhythms is obvious, as is also the need of comparative studies of individual and sex rhythms.

TIME ESTIMATION

Name Place
Age Date

ORDER OF TESTS TIME IN SECONDS

_Time of_ _Male_ _Female_
_Intervals._ (_17 years_) (_17 years_)
No. 1 No. 9

1. Idleness _108̋_ _70̋_ _120̋_
2. Reading _36_ _30_ _118_
3. Writing _72_ _36_ _60_
4. Estimating _18_ _15_ _30_
5. Reading _108_ _90_ _68_
6. Idleness _36_ _35_ _60_
7. Writing _18_ _10_ _10_
8. Estimating _108_ _100_ _125_
9. Reading _72_ _100_ _66_
10. Idleness _72_ _75_ _58_
11. Writing _36_ _25_ _22_
12. Estimating _72_ _60_ _60_
13. Reading _18_ _14_ _15_
14. Writing _108_ _130_ _59_
15. Estimating _36_ _30_ _41_
16. Idleness _18_ _10_ _18_

How did you estimate the interval when you were asked to estimate it
as accurately as you could?

n o f e y m i q r s a d r g d e s t k n w e r a x u p x z y o n d f n
o d c a e h p m a l g s r w y t b c k p s o n q a r v q c o m p v r i
c p k t o s n q z r l x m i h u v o q g P p f u t o i c n g s c a r n
o t c d a a o b i a r s a d e r w o a i e r g l c r t h f s o r a e n
s i o c r b x g r z b h o w l t s

Number of letters counted 85 88
Pulse-rate 72 81

The experimental data now to be considered were obtained as follows. Record-sheets of the form reproduced above were printed, with blanks for age and name of subject, place, date, for sixteen judgments of time-intervals (numerals 1 to 16), for a statement of the subject's method of estimating time, for the number of letters counted in thirty seconds, and for the pulse-rate. Four intervals were used, 18, 36, 72, and 108 seconds, and for each of these intervals judgments were taken under the four conditions designated on the record-sheet as idleness, reading, writing, and estimating. In the experiments the intervals were not given in order of regular increase or decrease of the length of interval, nor were all the judgments for any one interval taken together, but instead, for the purpose of avoiding the influence of expectation of a particular interval or filling, they were arranged irregularly in the order of column two of the record-sheet. This column, as also columns three and four, which are specimen series of judgments for a male and a female respectively, of course were not printed on the record-sheets which were supplied to the subjects.

The experimental procedure was as follows:

(1) Each subject was given a record-sheet.

(2) The experimenter was provided with a record-sheet on which the time of the intervals numbered from 1 to 16 was given. Care was taken that the subjects should not know the length of the intervals before the experiments.

(3) The beginning of each interval was indicated to the subjects by the word "start" uttered distinctly by the experimenter; the end, by the word "stop."

(4) Before beginning the sixteen tests the experimenter gave a thirty-second interval as a standard of judgment. The experiment then proceeded with only sufficient pause between judgments to allow of the recording of estimates by the subjects.

(5) During the filling called "idleness" the subject did not pay special attention to the estimation of the time, but instead permitted his attention to wander.

(6) During "reading" the experimenter read aloud to the subjects.

(7) During "writing" the subjects wrote from the dictation of the experimenter.

(8) During "estimating" the subjects judged the interval as accurately as they could, by whatever method they chose except the use of a time-piece.

(9) Each subject recorded his judgment of the length of an interval in seconds at the appropriate place on the record-sheet as soon as the interval was ended.

(10) The question following judgment number 16 on the sheet was answered as soon as the sixteen judgments had been recorded.

(11) The number of letters counted in thirty seconds was determined by the use of the lines of letters at the bottom of the sheet. The subjects began at the left of each line and counted singly as many letters as they could between the "start" and "stop" signals of the experimenter. They then marked the last letter counted and immediately recorded, in the place provided on the record-sheet, the number of letters counted.

(12) The pulse was counted by the experimenter immediately after the experiment when possible and the rate recorded on the sheet.

(13) The experimenter avoided delays, interruptions, or other irregularities in the course of the series of experiments.

The materials for our discussion of the sex-differences in time-estimation consist of the judgments of 251 males and 274 females. The majority of the males were students in Harvard College, the majority of the females, in Radcliffe and Smith Colleges. The remainder of the records were obtained in Ohio State University, Pomona College, and West Chester State Normal School. The authors gratefully acknowledge their indebtedness for assistance in the obtaining of records to Professors A. H. Pierce, T. H. Haines, W. H. Scott, D. R. Major, A. M. Smith, H. A. Miller, and B. T. Baldwin. The males ranged in age from 17 to 23 years, the females from 17 to 20. The total number of judgments, the distribution of which among the various ages is shown in Table 1, is 4014 for the males, 4375 for the females.

Despite the fact that our experiments are open to the criticisms of all work done under variable conditions and by different experimenters, it cannot be doubted that the results indicate certain sex-differences in time-estimation which suggest additional problems. For the present we refrain from interpretations for the most part and state merely the statistical results of the investigation.

Previous studies of the "time-sense" and the conditions which influence time-estimation suggested to us the desirability of examining our data with reference to (1) sex-differences in estimates of intervals, (2) age-differences, (3) the influence of different fillings, and (4) differences dependent upon the length of the interval. The results have been studied, therefore, with reference to the significance of sex, age, filling, and length of interval, but as no marked age-differences appeared, the detailed tables which were constructed to exhibit the results for the subjects of each year of age have not been printed.

In all the tables the results for males and females are presented separately. The judgments for the sixteen intervals are arranged with reference to the length of the interval, not in the order in which they were taken; all the 18̋ intervals, for example, are grouped (Table 2). The letters I, E, R, W, refer to the fillings of the intervals.

TABLE 1

NUMBER OF SUBJECTS, AGE, SEX, AND NUMBER OF JUDGMENTS

_Males_

_Age_ _No. of subjects_ _No. of judgments_

17 yrs. 16 256
18 27 432
19 40 639
20 67 1071
21 50 800
22 35 560
23 16 256

Totals 251 4014

_Interval_ _No. of judgments_

18" 1004
36" 1003
72" 1003
108" 1004

Total 4014

_Females_

_Age_ _No. of subjects_ _No. of judgments_

17 yrs. 73 1160
18 57 911
19 64 1024
20 80 1280

Totals 274 4375

_Interval_ _No. of judgments_

18" 1092
36" 1094
72" 1096
108" 1093

Total 4375

A general survey of the individual records, all of which for any one year and sex were tabulated, for convenience of examination, on a single large sheet of coördinate paper, showed that the judgments vary within a wide range and are very inexact. Table 2 exhibits the number of correct judgments for each sex, interval, and filling. Of the 4014 male judgments only 96 (2.39%) were correct; of the 4375 female judgments only 46 (1.05%) were correct. The number of correct judgments decreases as the length of the interval increases. For the 18-second intervals there were 7.37% for the males, 2.48% for the females, while for the 108-second intervals there were only 0.10% and 0.37% respectively.

TABLE 2

FREQUENCY OF OCCURRENCE OF CORRECT JUDGMENTS

_Males_

18̋ 36̋ 72̋ 108̋
I E R W I E R W I E R W I E R W Σ %
29 26 11 8 5 6 3 0 3 3 0 1 0 1 0 0 96 2.39
Totals 74 = 7.37% 14 = 1.40% 7 = 0.70% 1 =0.10% 96 2.39

_Females_

7 15 1 4 2 4 1 0 2 5 0 1 2 1 0 1 46 1.05
Totals 27 = 2.48% 7 = 0.62% 8 = 0.73% 4 = 0.37% 46 1.05

_List of abbreviations which occur in the tables._

Σ always designates the sum of the results of the column which it
heads.

I, E, R, W refer respectively to the intervals of idleness,
estimating, reading, and writing.

The % sign refers to the value of the result in question in terms of
the total number of judgments.

C refers to the results of the letter-counting test.

The male judgments for the 108-second intervals range from 11 to 300 seconds. If random guesses be made within these limits the probability of the occurrence of right guesses (108") would be 1 in 290; therefore among 1004 guesses (the number of male judgments for 108-second intervals) 3.5 would be right. In the experiment only one judgment of the 1004 was correct. For the other intervals, with the exception of 18 seconds, the number of correct judgments is only slightly greater than random guessing would have given. Both males and females, however, show considerably more correct judgments for 18-second intervals than the number of probable right guesses. Within the range of the male judgments and for their number 16.9 right guesses might be expected, for the females 10.9. In contrast with these numbers the experiments furnished 74 and 27 correct judgments respectively.

It is noteworthy that for those intervals which are most frequently correctly judged, not only is the number of correct judgments greater for the males than for the females (the ratio of the percentages is about 3 to 1), but the ratio of the number of correct judgments to the probable number of right guesses is also greater for the males.

The female judgments vary within a wider range and are less often correct than the male. For the latter the total number of correct judgments is more than twice that for the former.

Another interesting fact concerning the judgments of the time-intervals is that certain numerals occur in the last place of a judgment more frequently than we should expect if their occurrence depended on random guessing. Tables 3 and 4 exhibit the results of an analysis of the data made for the purpose of studying this fact. In Table 3 the frequency of occurrence of the digits 0, 1, 2, 3, etc., in the last place of the male judgments is given for each filling under the four intervals. For example, the digit 0 occurred in the last place of the male judgments for the interval reading 36 seconds 98 times, as we learn by referring to the first line and third row of the second column of Table 3.

Examination of Tables 3 and 4 shows at once the marked preference of the subjects for 0 and 5. The percentage of male judgments which end in 0 is 41.50; of female 58.51. Similarly the percentages of occurrence of the digit 5 for the males is 24.41, and for the females 23.11. Only two of the other digits (2 and 8) occur with a frequency of over 5%.

Among the 4014 male judgments 0 occurred as a final digit 1666 times, 5, 980 times. Among the 4375 female judgments 0 occurred 2560 times, 5, 1011 times. In the male judgments 0 occurred about four times as often as it would in random guessing; in the female, almost six times as often.

Comparison of Tables 3 and 4 indicates that the occurrence of 0 is 17.10% greater for the females, while that of 5 is 1.30% greater for the males. The sum of the percentages of occurrence of 0 and 5 for the males is 65.91, therefore the probability that a male judgment ends in one of these digits is almost twice that in favor of any other digit. For the females the sum of the same percentages is 81.62, and the probability of occurrence of 0 or 5 is therefore more than four times that of the other eight digits.

Tables 3 and 4 show that even numbers occur more frequently than uneven as final digits. Of the total number of judgments 2461 (3063)[127] end with even digits and 1553 (1312) with uneven.

TABLE 3. FREQUENCY OF OCCURRENCE OF THE NUMERALS (0 TO 9) AS FINAL DIGIT OF THE JUDGMENTS COMBINED RESULTS FOR ALL MALES

18̋ 36̋ 72̋
I E R W I E R W I E R W

0 70 57 72 76 131 77 98 90 124 74 120 125
1 5 11 9 9 10 9 8 7 1 18 5 2
2 20 15 21 18 5 19 16 15 11 20 5 14
3 9 17 14 10 8 11 10 8 6 14 12 4
4 8 12 14 9 3 14 12 3 4 14 8 7
5 67 58 56 63 62 54 60 85 72 59 62 73
6 13 17 12 13 9 14 11 6 10 9 7 7
7 15 22 14 12 6 15 11 10 5 14 13 4
8 36 30 29 31 15 19 16 20 11 17 11 11
9 8 12 10 10 2 18 9 7 7 12 8 3

108̋
I E R W Σ % C

0 151 121 127 153 1666 41.504 21.92
1 4 11 10 2 121 3.014 7.57
2 9 12 9 8 217 5.406 13.15
3 3 12 14 3 155 3.861 7.17
4 6 18 9 3 144 3.587 7.57
5 54 45 52 58 980 24.414 10.36
6 8 7 6 4 153 3.812 8.76
7 4 7 10 8 170 4.235 7.17
8 8 13 5 9 281 7.004 8.76
9 4 5 9 3 127 3.164 7.57

TABLE 4. FREQUENCY OF OCCURRENCE OF THE NUMERALS (0 TO 9) AS FINAL DIGIT OF THE JUDGMENTS COMBINED RESULTS FOR ALL FEMALES

18̋ 36̋ 72̋
I E R W I E R W I E R W

0 120 117 106 126 169 124 182 151 176 145 175 194
1 4 8 3 2 6 7 1 3 2 3 2 1
2 17 18 12 13 9 22 7 11 7 11 3 4
3 4 9 9 11 5 9 3 7 6 15 10 4
4 2 11 13 10 1 6 4 2 9 14 3 1
5 89 63 91 85 62 61 60 79 56 55 65 63
6 9 12 11 6 5 13 7 5 4 6 3 3
7 5 5 7 7 4 9 5 3 2 7 3 3
8 16 22 17 12 10 13 5 10 10 9 6 1
9 5 9 5 1 3 9 0 2 2 9 4 0

108̋
I E R W Σ % C

0 226 197 196 186 2560 58.515 26.14
1 1 5 2 0 50 1.143 9.09
2 4 11 6 3 158 3.611 5.30
3 3 7 4 7 113 2.583 5.68
4 2 4 2 2 86 1.966 11.36
5 32 44 49 57 1011 23.109 14.39
6 0 9 3 1 97 2.217 6.44
7 0 5 2 5 72 1.646 6.44
8 5 11 8 7 162 3.703 10.61
9 1 10 2 4 66 1.508 4.55

In order that the probability of the occurrence of even and uneven numbers may be calculated, those judgments which end in 0 and 5 must be subtracted from the total number of judgments, for the occurrence of these two digits is apparently due to a constant influence. The problem may be formulated thus. First, what is the probability that a judgment is determined by the constant influence in favor of 0 and 5? Second, what is the probability of even and uneven numbers, when the influence in favor of 0 and 5 is eliminated? The calculated probability of 0 or 5 is 0.65919 (0.81622) and therefore the probability that a given judgment is not determined by this influence is 0.34081 (0.18378). The probable limits of these numbers are 0.00505 (0.00395).

After the subtraction of those judgments which end in 0 or 5, there remain 1368 (804), of which 795 (503) are even and 573 (301) uneven. The probability of an even number is 0.58115 (0.62562) and the inverse probability of an uneven number is 0.41885 (0.37438). The probable limits of these numbers are 0.00900 (0.01027). There are therefore even chances that the percentage of occurrence of even numbers is between the limits 57.215 and 59.015 (61.535 and 63.589), or outside these limits.[128]

Statistical studies have already proved that in random guessing even numbers occur somewhat more frequently than uneven. It is therefore worthy of notice that in these results the frequencies of even numbers are not uniformly greater than that of uneven; for with the exception of the digit 6 in the female judgments, the digits next to 0 and 5, _i. e._, 9 and 1, 4 and 6, occur with least frequency.

In the case of the number of letters counted in a half minute, also, it appears (see last column (C) of Tables 3 and 4) that 0 and 5 occur more frequently in the last place than chance would lead us to expect. In contrast with the results for the time-judgments, in the same tables, the percentages of occurrence of the various digits in counting present less marked differences. For the males 3 and 7 occur least frequently, for the females 2 and 9.

To sum up the results of our examination of the materials with reference to the occurrence of digits in the final place of the judgments, the order of decreasing frequency of the various digits is 0, 5, 8, and 2. Of the others 3 and 7 occur more frequently than 4 and 6, with one exception. The statement that even numbers in general occur more frequently than odd must be modified by the statement that in these results the digits next to 0 and 5, namely, 9, 1, 4, and 6 occur with least frequency. These statements hold for both males and females, but for the latter the frequency of occurrence of 0 is far greater than for the males.

These results clearly indicate that the judgments are not random guesses. In seeking further for some explanation of the surprising frequency of occurrence of judgments which end in 0 or 5, we discovered that certain numbers occur very frequently, namely, the multiples of 15, 30, and 60. In order to exhibit this tendency quantitatively Tables 5 and 6 have been constructed.

In these tables will be found tabulated the number of times 15 and multiples of it which are not also multiples of 30 or 60 occur for any given interval and filling. Likewise are tabulated the frequencies of occurrence of 30 and multiples of it which are not multiples of 60, and finally, of 60 and its multiples. The numbers as they occurred in the three categories run as follows:

15 30 60
45 90 120
75 150 180
105 210 240
135 270 300
165 330 360
195 390 420

Fifteen and its multiples, as given above, are arranged in one division of the tables, thirty and sixty each in its own separate division. The line at the bottom of the tables marked Σ gives the frequency of occurrence of these three groups of numbers for all the subjects and for each interval and filling.

As is shown by the percentage of frequency columns of the tables, in no instance do the multiples of 15 constitute less than 19.52% of the male judgments and 23.81% of the female judgments. The lowest frequency for any of the four intervals is 20.32% of the total number of judgments. The maximum frequency for the males (43.03%) and for the females (56.57%) is for the interval idleness 108 seconds. That the male and female maxima should fall on the same interval is interesting.

TABLE 5

FREQUENCY OF OCCURRENCE OF 15, 30, 60, AND THEIR MULTIPLES

_Males_

15 30 60 Σ % _Average_

{I 48 5 0 53 21.12
{E 42 9 0 51 20.32
18̋ {R 43 8 0 51 20.32 20.32
{W 45 4 0 49 19.52

{I 29 60 7 96 38.25
{E 17 41 5 63 25.20
36̋ {R 22 41 6 69 27.41 29.57
{W 36 28 5 69 27.41

{I 29 18 46 93 37.05
{E 25 7 34 66 26.29
72̋ {R 28 26 33 87 34.66 34.70
{W 28 42 32 102 40.80

{I 25 31 52 108 43.03
{E 16 23 20 59 23.51
108̋ {R 22 30 39 91 36.25 36.16
{W 25 37 43 105 41.83

Σ 480 410 322 1212
% 11.96 10.21 8.02 30.12

TABLE 6

FREQUENCY OF OCCURRENCE OF 15, 30, 60, AND THEIR MULTIPLES

_Females_

15 30 60 Σ % _Average_

{I 58 30 0 88 32.47
{E 30 35 8 73 26.64
18̋ {R 56 26 3 85 31.02 28.48
{W 41 24 0 65 23.81

{I 31 55 39 125 45.62
{E 21 36 17 74 27.21
36̋ {R 31 63 33 127 46.35 38.86
{W 41 45 13 99 36.26

{I 29 28 60 117 42.70
{E 17 18 52 87 31.75
72̋ {R 26 37 52 115 41.97 41.60
{W 30 51 56 137 50.00

{I 12 45 98 155 56.57
{E 21 27 58 106 38.83
108̋ {R 23 36 84 143 52.19 47.83
{W 24 31 64 119 43.75

Σ 491 587 637 1715
% 11.22 13.42 14.56 39.22

If no influence worked in favor of the multiples of 15 in these experiments only one judgment in thirty would be 15 (3.33%) and only one in sixty, 30 or 60 (1.67%). For the probability of the occurrence of 15, 30, and 60 is 1/60 + 1/60 + 1/30 = 1/15, as is obvious from the fact that among sixty consecutive numbers (1 to 60) there are four which are multiples of 15. According to probability we should expect multiples of 15, 30, and 60 to occur 268 times among the 4014 male judgments and 292 times among the 4375 female judgments. As a matter of fact there are 1212 such judgments for the males, 1715 for the females. The probability that a male judgment is a multiple of 15, 30, or 60 is 0.3012 (probable error 0.0049); for a female judgment the probability is 0.39199 (probable error 0.0050).

These statistics indicate that the subjects are constantly and strongly influenced in favor of judgments which are simple fractions of a minute. Closer inspection of the tables gives some suggestion of the nature of this influence.

Comparison of the four intervals (Tables 5 and 6) with respect to the occurrence of simple fractions of a minute shows that the frequency of such numbers increases rapidly as the length of the interval increases. The various percentages of frequency for males and females and for the four intervals are again presented here for convenience of comparison.

18̋ 36̋ 72̋ 108̋

_Males_ 20.32% 29.57% 34.70% 36.16%
_Females_ 28.48 38.86 41.60 47.83

As is obvious from these figures both frequency and rate of increase are far higher for the females than for the males.

Examination of the percentages (totals) at the bottom of Tables 5 and 6 reveals another remarkable sex-difference; for the frequency of occurrence of 15 and its multiples regularly decreases for males from the 15 to the 60 class, whereas for females it regularly increases.

Undoubtedly the time-judgments of these experiments were strongly influenced by thought of the conventional time-unit, the minute, for in all quantitative work there are errors in favor of the standard of measurement and simple fractions thereof. In the present instance this tendency to favor the unit was strengthened, perhaps, by the giving of a half-minute interval as a standard for comparison at the beginning of the tests.

Two explanations of the sex-differences above mentioned are suggested by our study of the data. One is the fact that the females are less exact than the males; the other that they generally overestimate the intervals, whereas the males often underestimate them. One's estimate of an interval is determined partly by confidence of accuracy. The longer an interval the less we feel able to estimate it accurately, and, as a consequence, the more frequently it is judged as the same as the time-unit or a simple fraction of that unit. The females are less exact in their estimates than the males, and less exact for long than for short intervals, and as an accompaniment of their inexactitude we find the frequent occurrence of multiples of 15, 30, and 60.

But confidence of ability to estimate accurately must be considered in connection with the fact which suggests our second explanation, namely, that the female estimates are higher than the male. Tables 7 and 8 show that the females almost invariably overestimate the intervals rather largely, while the males sometimes underestimate considerably. The range of the male judgments is from 1 to 300, of the female from 1 to 400. Obviously the chance of occurrence of 15, 30, 60, and their multiples varies with the range. The greater the range the greater the probable frequency of 30 and 60 in comparison with 15. In random guessing the probabilities of the occurrence of 15, 30, and 60 for long and short intervals is the same, but our results show that this is not true in the case of these time-estimation judgments. It seems possible, therefore, that the sex-differences referred to are due to the fact that the intervals seem longer to the females, and that, therefore, a feeling of greater inexactitude than would be felt for shorter intervals leads to the choice of simple fractions of a minute more frequently than in the male judgments and more frequently for the long than for the short intervals.

It is of interest in this connection to note that the length of a second is usually underestimated by females, overestimated by males. The average number of seconds counted in half a minute by twenty men and twenty women was as follows:

_Men._ M. 30.4, M.V. 8.7, R.V. 34.94.
_Women._ M. 38.9, M.V. 10.6, R.V. 36.70

These figures would seem to indicate that the overestimation of the intervals of these experiments by the females is due to the use of a time-unit which is shorter than that of the males (although presumably of the same length).

We cannot with certainty say whether inaccuracy of judgment stands in the relation of condition or consequence of the occurrence of simple fractions of a minute, but it would appear that the female tendency to overestimate is responsible for the sex-differences already noted. For whatever be the facts concerning longer intervals the second as judged by the female is considerably shorter than that of the male.

Since a complicated periodicity of frequency in the distribution of the judgments is exhibited in the results of Tables 3-6 it is obvious that the distribution-curve will have a tertiary mode for each number ending in 0 or 5, a secondary mode for 15, 30, 60, and their multiples, and a primary mode which may or may not coincide with one of the secondary or tertiary modes. Extreme irregularity is characteristic of the distribution-curve. Different groups of judgments, as, for example, those for the two sexes, those for the different intervals, etc., give somewhat different forms of distribution, for the frequency of occurrence of 0 and 5, as well as of the multiples of 15, is variable.

These facts are important in connection with the selection of an interval for the construction of the distribution-curves, in that they indicate how large the interval or class of the distribution-curves and tables should be.

It is clear from the results of Tables 3 and 4 that the smallest interval which can be of value is 10 seconds, for a smaller interval would necessarily exhibit irregularities due to the greater frequency of 0 than of 5. The question is whether the interval can be so enlarged, without the loss of all details of the nature of the distribution, that every class will represent the influence of the same conditions. For this purpose only three intervals are possible: 10, 30, and 60 seconds. Of these 30 and 60 are undesirable because the interval 60 gives classes which are so large that all details of the distribution are lost, while 30 exhibits only a few details without doing away with the periodicity due to the preference for multiples of 60.

The further question remains, with which digit should the interval end, in order that uniformity of conditions for the various classes may be gained? Theoretically there are ten possibilities, but of these all except two, 0 and 5, are excluded by reason of the unequal frequency of the various digits already discussed. In favor of 0 is the fact that all the classes thus formed are of equal size, _i. e._ 1-10, 11-20, etc., whereas for 5 the first class would differ from the others in being only half as large, 1-5. This, however, is only a slight disadvantage, for there are very few judgments which fall in this class. On the other hand, since 0 is the final digit of most frequent occurrence, classes ending in 5 have the advantage of placing the value of greatest weight in the middle. On the whole it seemed desirable to arrange the judgments in 10 second classes, beginning with the class 1-10. But for purposes of comparison the male judgments have been distributed in classes of 10 seconds, which end in 5, 1-5, 6-15, 16-25, etc.[129]

TABLE 7

DISTRIBUTION OF MALE JUDGMENTS IN 10" CLASSES

_Classes_ 18̋ 36̋ 72̋ 108̋ C
I E R W I E R W I E R W I E R W

1 - 10" 31 16 69 136 3 1 4 18 1 1 1 4 0 0 0 0 0
11 - 20 174 179 142 108 40 34 45 97 4 2 5 23 1 2 1 7 0
21 - 30 34 47 35 7 99 106 98 82 15 9 26 65 5 0 6 23 0
31 - 40 7 7 3 56 69 69 28 25 16 34 47 10 2 6 31 1
41 - 50 4 2 2 37 23 22 18 41 43 54 33 20 9 19 28 0
51 - 60 1 9 13 9 6 64 68 53 35 31 17 27 32 7
61 - 70 5 3 0 0 37 42 28 18 27 26 22 19 8
71 - 80 0 1 3 1 23 34 16 9 33 34 35 20 34
81 - 90 1 1 1 15 15 15 8 30 40 43 29 58
91 -100 0 9 8 7 1 23 35 25 17 65
101 -110 0 2 6 3 3 10 26 17 6 33
111 -120 0 8 5 5 2 29 16 17 16 29
121 -130 0 3 1 2 0 5 15 7 8 10
131 -140 0 1 1 1 1 6 9 5 0 4
141 -150 1 0 1 0 5 5 8 4 1
151 -160 1 0 2 3 2 2 1
161 -170 1 0 2 1 0 2
171 -180 1 1 5 4 5 2
181 -190 0 1 2 1
191 -200 3 2 2 2
201 -210 0 1 1 0
211 -220 0 0 0 0
221 -230 1 1 0 0
231 -240 2 0 1 0
241 -250 1 1 1
251 -260 0 0
261 -270 0 0
271 -280 0 0
281 -290 0 0
291 -300 1 1

Totals
251 251 251 251 251 250 251 251 251 251 251 250 251 251 251 251 251

As a result of these groupings of the male judgments it appeared that the former method gives a far more regular distribution than the latter. In view of this result and the above considerations, Tables 7 and 8 were constructed by the use of 10-second classes, beginning with 1-10. In these tables (column C) the distribution of the letter-counting results has been included for convenience of comparison of the two kinds of judgments as to form of distribution.

As instances of the general form of distribution of the judgments the curves have been plotted for letter-counting, Fig. 1 _A_. (Males ----, Females, ... ,) for idleness 36 seconds, Fig. 1 _B_, and for idleness 108 seconds, Fig. 2. The distribution of the letter-counting judgments in classes of 10 is very regular in comparison with that of the several time-estimation judgments. For idleness 36 seconds there are several fairly distinct modes, and for idleness 108 seconds the modes are still more numerous and more marked.

TABLE 8

DISTRIBUTION OF FEMALE JUDGMENTS IN 10̋ CLASSES

18̋ 36̋ 72̋ 108̋
_Classes_
I E R W I E R W I E R W I E R W C