Although the number of the tests was so small, the accordance between the calculation by theory and the actual count of deviations is in all these cases so close that the usefulness of the mean values will be admitted, the wide limits of error being, of course, taken into consideration.
Section 18. Grouping of the Results of the Separate Series
The previously mentioned hypotheses concerning the grouping of the times necessary for learning the separate series were naturally not merely theoretical suppositions, but had already been confirmed by the groupings actually found. The two large series of tests mentioned above, one consisting of 92 tests of eight single series each, and the other of 84 tests of 6 single series each, thus giving 736 and 504 separate values respectively, afford a sufficiently broad basis for judgment. Both groups of numbers, and both in the same way, show the following peculiarities:
1. The distribution of the arithmetical values above their mean is considerably looser and extends farther than below the mean. The most extreme values above lie 2 times and 1.8 times, respectively, as far from the mean as the most distant of those below.
2. As a result of this dominance by the higher numbers the mean is displaced upward from the region of the densest distribution, and as a result the deviations below get the preponderance in number. There occur respectively 404 and 266 deviations below as against 329 and 230 above.
3. The number of deviations from the region of densest distribution towards both limits does not decrease uniformly—as one would be very much inclined to expect from the relatively large numbers combined—but several maxima and minima of density are distinctly noticeable. Therefore constant sources of error were at work in the production of the separate values—i.e., in the memorisation of the separate series. These resulted on the one hand in an unsymmetrical distribution of the numbers, and on the other hand in an accumulation of them in certain regions. In accordance with the investigations already presented in this chapter, it can only be supposed that these influences compensated each other when the values of several series learned in succession were combined.
