TABLE 4.
FREQUENCY OF OCCURRENCE OF THE NUMERALS (0 to 9) AS FINAL DIGIT OF THE JUDGMENTS COMBINED RESULTS FOR ALL FEMALES
| 18" | 36" | 72" | 108" | ||||||||||||||||
| I | E | R | W | I | E | R | W | I | E | R | W | I | E | R | W | Σ | % | C | |
| 0 | 120 | 117 | 106 | 126 | 169 | 124 | 182 | 151 | 176 | 145 | 175 | 194 | 226 | 197 | 196 | 186 | 2560 | 58.515 | 26.14 |
| 1 | 4 | 8 | 3 | 2 | 6 | 7 | 1 | 3 | 2 | 3 | 2 | 1 | 1 | 5 | 2 | 0 | 50 | 1.143 | 9.09 |
| 2 | 17 | 18 | 12 | 13 | 9 | 22 | 7 | 11 | 7 | 11 | 3 | 4 | 4 | 11 | 6 | 3 | 158 | 3.611 | 5.30 |
| 3 | 4 | 9 | 9 | 11 | 5 | 9 | 3 | 7 | 6 | 15 | 10 | 4 | 3 | 7 | 4 | 7 | 113 | 2.583 | 5.68 |
| 4 | 2 | 11 | 13 | 10 | 1 | 6 | 4 | 2 | 9 | 14 | 3 | 1 | 2 | 4 | 2 | 2 | 86 | 1.966 | 11.36 |
| 5 | 89 | 63 | 91 | 85 | 62 | 61 | 60 | 79 | 56 | 55 | 65 | 63 | 32 | 44 | 49 | 57 | 1011 | 23.109 | 14.39 |
| 6 | 9 | 12 | 11 | 6 | 5 | 13 | 7 | 5 | 4 | 6 | 3 | 3 | 0 | 9 | 3 | 1 | 97 | 2.217 | 6.44 |
| 7 | 5 | 5 | 7 | 7 | 4 | 9 | 5 | 3 | 2 | 7 | 3 | 3 | 0 | 5 | 2 | 5 | 72 | 1.646 | 6.44 |
| 8 | 16 | 22 | 17 | 12 | 10 | 13 | 5 | 10 | 10 | 9 | 6 | 1 | 5 | 11 | 8 | 7 | 162 | 3.703 | 10.61 |
| 9 | 5 | 9 | 5 | 1 | 3 | 9 | 0 | 2 | 2 | 9 | 4 | 0 | 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.