The numbers for the larger families are as yet too small to be worth giving, but they show the same tendency. It will be seen that in every case the observed central values are less than the calculated; and that the observed extreme values are much greater than the calculated. The results seem to suggest (so far) that a family cannot be likened to a chance drawing of the requisite number from one bag. A better analogy would be to suppose two bags, one with M.s in excess and the other with F.s in less excess, and that some persons draw from one and some from the other. But fuller statistics are needed.
It will be observed that the total excess of male births is large. This may arise from undue omission of females; but I have carefully confined myself to the two or three last generations, in each pedigree, for greater security.
[1] Essay on Probabilities, p. 114.
[2] Doubts have been expressed about the truly random character of the digits in this case (v.
De Morgan, Budget of Paradoxes, p. 291), and Jevons has gone so far as to ask (Principles of Science, p. 529), “Why should the value of π, when expressed to a great number of figures, contain the digit 7 much less frequently than any other digit!” I do not quite understand what this means. If such a question were asked in relation to any unusual divergence from the à priori chance in a case of throwing dice, say, we should probably substitute for it the following, as being more appropriate to our science:—Assign the degree of improbability of the event in question; i.e.
its statistical rarity. And we should then proceed to judge, in the way indicated in the text, whether this improbability gave rise to any grounds of suspicion.
The calculation is simple. The actual number of 7's, in the 708 digits, is 53: whilst the fair average would be 71. The question is, What is the chance of such a departure from the average in 708 turns? By the usual methods of calculation (v.
Galloway on Probability) the chances against an excess or defect of 18 are about 44 : 1, in respect of any specified digit. But of course what we want to decide are the chances against some one of the ten showing this divergence. This I estimate as being approximately determined by the fraction (44/45)10, viz.
0.8. This represents odds of only about 4 : 1 against such an occurrence, which is nothing remarkable. As a matter of fact several digits in the two other magnitudes which Mr Shanks had calculated to the same length, viz.