NOTE ON THE PROPORTIONS OF THE SEXES.
The following remarks were rather too long for convenient insertion on [p. 259], and are therefore appended here.
The ‘random’ character of male and female births has generally been rested almost entirely on statistics of place and time. But what is more wanted, surely, is the proportion displayed when we compare a number of families. This seems so obvious that I cannot but suppose that the investigation must have been already made somewhere, though I have not found any trace of it in the most likely quarters. Thus Prof.
Lexis (Massenerscheinungen) when supporting his view that the proportion between the sexes at birth is almost the only instance known to him, in natural phenomena, of true normal dispersion about a mean, rests his conclusions on the ordinary statistics of the registers of different countries.
It certainly needs proof that the same characteristics will hold good when the family is taken as the unit, especially as some theories (e.g.
that of Sadler) would imply that ‘runs’ of boys or girls would be proportionally commoner than pure chance would assign. Lexis has shown that this is most markedly the case with twins: i.e., to use an obviously intelligible notation, (M for male, F for female), that M.M. and F.F. are very much commoner in proportion than M.F.
I have collected statistics including over 13,000 male and female births, arranged in families of four and upwards. They were taken from the pedigrees in the Herald's Visitations, and therefore represent as a rule a somewhat select class, viz.
the families of the eldest sons of English country gentlemen in the sixteenth century. They are not sufficiently extensive yet for publication, but I give a summary of the results to indicate their tendency so far. The upper line of figures in each case gives the observed results: i.e.
in the case of a family of four, the numbers which had four male, three male and one female, two male and two female, and so on. The lower line gives the calculated results; i.e.
the corresponding numbers which would have been obtained had batches of M.s and F.s been drawn from a bag in which they were mixed in the ratio assigned by the total observed numbers for those families.
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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.
Tan−1 1/5 and Tan−1 1/239, show the same divergencies (v.
Proc.
Roy.
Soc.
xxi. 319).
I may call attention here to a point which should have been noticed in the chapter on Randomness. We must be cautious when we decide upon the random character by mere inspection. It is very instructive here to compare the digits in π with those within the ‘period’ of a circulating decimal of very long period. That of 1 ÷ 7699, which yields the full period of 7698 figures, was calculated some years ago by two Cambridge graduates (Mr Lunn and Mr Suffield), and privately printed. If we confine our examination to a portion of the succession the random character seems plausible; i.e.
the digits, and their various combinations, come out in nearly, but not exactly, equal numbers. So if we take batches of 10; the averages hover nicely about 45. But if we took the whole period which ‘circulates,’ we should find these characteristics overdone, and the random character would disappear. That is, instead of a merely ultimate approximation to equality we should have (as far as this is possible) an absolute attainment of it.
[3] Of course this conventional estimate is nothing different in kind from that which may attach to any order or succession. Ten heads in succession is intrinsically or objectively indistinguishable in character from alternate heads and tails, or seven heads and three tails, &c. Its distinction only consists in its almost universal acceptance as remarkable.
[4] Our Inheritance in the Great Pyramid, Ed. III. 1877.
[5] Made in Nature (Jan. 24, 1878) by Mr J. G. Jackson. It must be remarked that Mr Smyth's alternative statement of his case leads up to that explanation:—“The vertical height of the great pyramid is the radius of a theoretical circle the length of whose curved circumference is exactly equal to the sum of the lengths of the four straight sides of the actual and practical square base.” As regards the alternatives of chance and design, here, it must be remembered in justice to Mr Smyth's argument that the antithesis he admits to chance is not human, but divine design.
[6] See Cournot, Essai sur les fondements de nos connaissances. Vol. I.
p. 71.
[7] It deserves notice that considerations of this kind have found their way into the Law Courts though of course without any attempt at numerical valuation. Thus, in the celebrated De Ros trial, in so far as the evidence was indirect, one main ground of suspicion seems to have been that Lord De Ros, when dealing at whist, obtained far more court cards than chance could be expected to assign him; and that in consequence his average gains for several years in succession were unusually large. The counsel for the defence urged that still larger gains had been secured by other players without suspicion of unfairness,—(I cannot find that it was explained over how large an area of experience these instances had been sought; nor how far the magnitude of the stakes, as distinguished from the number of successes, accounted for that of the actual gains),—and that large allowance must be made for skill where the actual gains were computed. (See the Times’ report, Feb. 11, 1837.)
[8] Metretike. At the end of this volume will be found a useful list of a number of other publications by the same author on allied topics.
[9] That is, if we look simply to statistical results, as Arbuthnott did, and as we should do if we were examining the tosses of a penny. If the remarkable theory of Dr Düsing (Die Regulierung des Geschlechts-verhältnisses… Jena, 1884) be confirmed, the matter would assume a somewhat different aspect. He attempts to show, both on physiological grounds, and by analysis of statistics referring to men and animals, that there is a decidedly compensatory process at work. That is, if for any cause either sex attains a preponderance, agencies are at once set in motion which tend to redress the balance. This is a modification and improvement of the older theory, that the relative age of the parents has something to do with the sex of the offspring.
Quetelet (Letters, p. 61) has attempted to prove a proposition about the succession of male and female births by certain experiments supposed to be tried upon an urn with black and white balls in it. But this is going too far. (See the note at the end of this chapter.)
[10] It is precisely analogous to the conclusion that the flowers of the daisies (as distinguished from the plants, v.
[p. 109]) are not distributed at random, but have a tendency to go in groups of two or more. Mere observation shows this: and then, from our knowledge of the growth of plants we may infer that these little groups spring from the same root.
[11] In this discussion, writers often speak of the probability of a “physical connection” between these double stars. The phrase seems misleading, for on the usual hypothesis of universal gravitation all stars are physically connected, by gravitation. It is therefore better, as above, to make it simply a question of relative proximity, and to leave it to astronomy to infer what follows from unusual proximity.
[12] Professor Forbes in the paper in the Philosophical Magazine already referred to (Ch. VII.
§ 18) gave several diagrams to show what were the actual arrangements of a random distribution. He scattered peas over a chess-board, and then counted the number which rested on each square. His figures seem to show that the general appearance of the stars is much the same as that produced by such a plan of scattering.
Some recent investigations by Mr R. A. Proctor seem to show, however, that there are at least two exceptions to this tolerably uniform distribution. (1) He has ascertained that the stars are decidedly more thickly aggregated in the Milky Way than elsewhere. So far as this is to be relied on the argument is the same as in the case of the double stars; it tends to prove that the proximity of the stars in the Milky Way is not merely apparent, but actual. (2) He has ascertained that there are two large areas, in the North and South hemispheres, in which the stars are much more thickly aggregated than elsewhere. Here, it seems to me, Probability proves nothing: we are simply denying that the distribution is uniform. What may follow in the way of inferences as to the physical process of causation by which the stars have been disposed is a question for the Astronomer. See Mr Proctor's Essays on Astronomy, p. 297. Also a series of Essays in The Universe and the coming Transits.