Table V. tells us that the percentage of whorls in the right ring-finger is 45, and in the left ring-finger 31. Table VIa tells us that the percentage of the double event of a whorl occurring on both the ring-fingers of the same person is 26. It is required to express the relationship between the right and left ring-fingers on a centesimal scale, in which 0° shall stand for no relationship at all, and 100° for the closest possible relationship.
If no relationship should exist, there would nevertheless be a certain percentage of instances, due to pure chance, of the double event of whorls occurring in both ring-fingers, and it is easy to calculate their frequency from the above data. The number of possible combinations of 100 right ring-fingers with 100 left ones is 100 × 100, and of these 45 × 31 would be double events as above (call these for brevity “double whorls”). Consequently the chance of a double whorl in any single couplet is 45 × 31⁄100 × 100, and their average frequency in 100 couplets,—in other words, their average percentage is 45 × 31⁄100 = 13·95, say 14. If, then, the observed percentage of double whorls should be only 14, it would be a proof that the A. L. W. classes of patterns on the right and left ring-fingers were quite independent; so their relationship, as expressed on the centesimal scale, would be 0°. There could never be less than 14 double whorls under the given conditions, except through some statistical irregularity.
Now consider the opposite extreme of the closest possible relationship, subject however, and this is the weak point, to the paramount condition that the average frequencies of the A. L. W. classes may be taken as pre-established. As there are 45 per cent of whorls on the right ring-finger, and only 31 on the left, the tendency to form double whorls, however stringent it may be, can only be satisfied in 31 cases. There remains a superfluity of 14 per cent cases in the right ring-finger which perforce must have for their partners either arches or loops. Hence the percentage of frequency that indicates the closest feasible relationship under the pre-established conditions, would be 31.
The range of all possible relationships in respect to whorls, would consequently lie between a percentage frequency of the minimum 14 and the maximum 31, while the observed frequency is of the intermediate value of 26. Subtracting the 14 from these three values, we have the series of 0, 12, 17. These terms can be converted into their equivalents in a centesimal scale that reaches from 0° to 100° instead of from 0° to 17°, by the ordinary rule of three, 12:x::17:100; x=70 or 71, whence the value x of the observed relationship on the centesimal scale would be 70° or 71°, neglecting decimals.
This method of obtaining the value of 100° is open to grave objection in the present example. We have no right to consider that the 45 per cent of whorls on the right ring-finger, and the 31 on the left, can be due to pre-established conditions, which would exercise a paramount effect even though the whorls were due entirely to causes common to both fingers. There is some self-contradiction in such a supposition. Neither are we at liberty to assume that the respective effects of the special causes X and Z are equal in average amount; if they were, the percentage of whorls on the right and on the left finger would invariably be equal.
In this particular example the difficulty of determining correctly the scale value of 100° is exceptionally great; elsewhere, the percentages of frequency in the two members of each couplet are more alike. In the two fore-fingers, and again in the two middle fingers, they are closely alike. Therefore, in these latter cases, it is not unreasonable to pass over the objection that X and Z have not been proved to be equal, but we must accept the results in all other cases with great caution.
When the digits are of different names,—as the thumb and the fore-finger,—whether the digits be on the same or on opposite hands, there are two cases to be worked out; namely, such as (1) right thumb and left fore-finger, and (2) left thumb and right fore-finger. Each accounts for 50 per cent of the observed cases; therefore the mean of the two percentages is the correct percentage. The relationships calculated in the following table do not include arches, except in two instances mentioned in a subsequent paragraph, as the arches are elsewhere too rare to furnish useful results.
It did not seem necessary to repeat the calculation for couplets of digits of different names, situated on opposite hands, as those that were calculated on closely the same data for similar couplets situated on the same hands, suffice for both. It is evident from the irregularity in the run of the figures that the units in the several entries cannot be more than vaguely approximate. They have, however, been retained, as being possibly better than nothing at all.
Table VIII.