in sixty, 30 or 60 (1.67 %). For the probability of the occurrence of 15, 30, and 60 is ++3=1, 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 compari

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 timeunit 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 o 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 o 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 o 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, o and 5, are excluded by reason of the unequal frequency of the various digits already discussed. In favor of o 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 o 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.1

1 The judgments might be grouped in 10 second classes beginning with the lowest number in the experiments, but this would have the disadvantage of ren

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

dering the distribution tables or curves for different groups of judgments incomparable.

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