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

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

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 18ʹʹ 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

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

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

TABLE 4.

FREQUENCY OF OCCURRENCE OF THE NUMERALS (0 to 9) AS FINAL DIGIT OF THE JUDGMENTS COMBINED RESULTS FOR ALL FEMALES

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:

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

TABLE 6

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

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 160 + 160 + 130 = 115, 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.