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differ little from mediocrity is so preponderant that it is more frequently the case that an exceptional man is the somewhat exceptional son of rather mediocre parents, than the average son of very exceptional parents. It appears from the very same table of observations by which the value of the filial regression was determined when it is read in a different way, namely, in vertical columns instead of in horizontal lines, that the most probable mid-parentage of a man is one that deviates only one-third as much as the man does. There is a great difference between this value of and the numerical converse mentioned above of; it is four and a half times smaller, since 4, or, being multiplied into, is equal to.

It will be gathered from what has been said, that a mid-parental deviate of one unit implies a mid-grandparental deviate of, a midancestral unit in the next generation of, and so on. I reckon from these and other data, by methods that I cannot stop now to explain, but will do so in the Appendix, that the heritage derived on an average from the mid-parental deviate, independently of what it may imply, or of what may be known concerning the previous ancestry, is only. Consequently, that similarly derived from a single parent is only 4, and that from a single grandparent is only

Let it not be supposed for a moment that any of these statements invalidate the general doctrine that the children of a gifted pair are much more likely to be gifted than the children of a mediocre pair. What they assert is that the ablest child of one gifted pair is not likely to be as gifted as the ablest of all the children of very many mediocre pairs. However, as, notwithstanding this explanation, some suspicion may remain of a paradox lurking in my strongly contrasted results, I will call attention to the form in which the table of data (Table I) was drawn up, and give an anecdote connected with it.

It is deduced from a large sheet on which I entered every child's height, opposite to its mid-parental height, and in every case each was entered to the nearest tenth of an inch. Then I counted the number of entries in each square inch, and copied them out as they appear in the table. The meaning of the table is best understood by examples. Thus, out of a total of 928 children who were born to the 205 mid-parents on my list, there were 18 of the height of 69.2 inches (counting to the nearest inch), who were born to mid-parents of the height of 70.5 inches (also counting to the nearest inch). So again there were 25 children of 70.2 inches born to mid-parents of 69.5 inches. I found it hard at first to catch the full significance of the entries in the table, which had curious relations that were very interesting to investigate. They came out distinctly when I "smoothed" the entries by writing at each intersection of a horizontal column with a vertical one, the sum of the entries in the four adjacent squares, and using these to work upon. I then noticed (see Plate X) that lines drawn through entries of the same value formed a series of concentric and similar

ellipses. Their common centre lay at the intersection of the vertical and horizontal lines, that corresponded to 68 inches. Their axes were similarly inclined. The points where each ellipse in succession was touched by a horizontal tangent, lay in a straight line inclined to the vertical in the ratio of; those where they were touched by a vertical tangent lay in a straight line inclined to the horizontal in the ration of. These ratios confirm the values of average regression already obtained by a different method, of from mid-parent to offspring, and of from offspring to mid-parent, because it will be obvious on studying Plate X that the point where each horizontal line in succession is touched by an ellipse, the greatest value in that line must occur at the point of contact. The same is true in respect to the vertical lines. These and other relations were evidently a subject for mathematical analysis and verification. They were all clearly dependent on three elementary data, supposing the law of frequency of error to be applicable throughout; these data being (1) the measure of racial variability, whence that of the mid-parentages may be inferred as has already been explained, (2) that of co-family variability (counting the offspring of like mid-parentages as members of the same co-family), and (3) the average ratio of regression. I noted these values, and phrased the problem in abstract terms such as a competent mathematician could deal with, disentangled from all reference to heredity, and in that shape submitted it to Mr. J. Hamilton Dickson, of St. Peter's College, Cambridge. I asked him kindly to investigate for me the surface of frequency of error that would result from these three data, and the various particulars of its sections, one of which would form the ellipses to which I have alluded.

I may be permitted to say that I never felt such a glow of loyalty and respect towards the sovereignty and magnificent sway of mathematical analysis as when his answer reached me, confirming, by purely mathematical reasoning, my various and laborious statistical conclusions with far more minuteness than I had dared to hope, for the original data ran somewhat roughly, and I had to smooth them with tender caution. His calculation corrected my 1 6 observed value of mid-parental regression from to the 3 17.6' relation between the major and minor axis of the ellipses was changed 3 per cent. (it should be as 7:2), their inclination was changed less than 2° (it should be to an angle whose tangent is). It is obvious, then, that the law of error holds throughout the investigation with sufficient precision to be of real service, and that the various results of my statistics are not casual and disconnected determinations, but strictly interdependent.

In the lecture at the Royal Institution to which I have referred, I pointed out the remarkable way in which one generation was succeeded by another that proved to be its statistical counterpart. I there had to discuss the various agencies of the survival of the fittest, of relative fertility, and so forth; but the selection of

256

Anthropological Miscellanea,

human stature as the subject of investigation

now enables me to

get rid of all these complications and to discuss this very curious
successive generation there proves to be the same number of men
question under its simplest form. How is it, I ask, that in each
per thousand, who range between any limits of stature we please
to specify, although the tall men are rarely descended from equally
tall parents, or the short men from equally short? How is the
balance from other sources so nicely made up? The answer is
that the process comprises two opposite sets of actions, one con-
centrative and the other dispersive, and of such a character that
they necessarily neutralise one another, and fall into a state of
stable equilibrium (see Table IV). By the first set, a system of
scattered elements is replaced by another system which is less
scattered; by the second set, each of these new elements becomes
a centre whence a third system of elements are dispersed.

The details are as follows:-In the first of these two stages we
start from the population generally, in the first generation; then
the units of the population group themselves, as it were by chance,
into married couples, whence the more compact system of mid-
parentages is derived, and then by a regression of the values of
the mid-parentages the still more compact system of the generants
is derived.
the offspring diverge upwards and downwards to form the second
In the second stage each generant is a centre whence
generation.
tendencies is due to the regression being proportionate to the
deviation. It acts like a spring against a weight; the spring
The stability of the balance between the opposed
stretches until its resilient force balances the weight, then the two
forces of spring and weight are in stable equilibrium; for if the
weight be lifted by the hand, it will obviously fall down again
when the hand is withdrawn, and, if it be depressed by the hand,
the resilience of the spring will be thereby increased, so that the
weight will rise when the hand is withdrawn.

variability, of

A simple equation connects the three data of race
the ratio of regression, and of co-family variability, whence, if any
two are given, the third may be found. My observations give
separate measures of all three, and their values fit well into the
equation, which is of the simple form-

where v, p=17, f=1.5.

v2 p2

+ ƒ2=p3, 2

It will therefore be understood that the complete table of midparental and filial heights may be calculated from two simple numbers, and that the most elementary data upon which it admits of being constructed are (1) the ratio between the mid-parental and the rest of the ancestral influences, and (2) the measure of the co-family variability.

The mean regression in stature of a population is easily ascertained; I do not see much use in knowing it, but will give the work merely as a simple example. It has already been stated that half

[graphic]

TABLE IV.

PROCESS THROUGH WHICH THE DISTRIBUTION OF STATURES, IN SUCCESSIVE GENERATIONS OF THE SAME PEOPLE, REMAINS

UNCHANGED.

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above

NOTE.-The cases are symmetrically disposed and below the common mean value of 68 inches. The Upper and Lower Quartiles are the values that in each case divide the number of cases above the Median or mean value, and those below it, respectively into equal parts. Thus in each column there are (1) 25 cases per cent. above the Upper Quartile, (2) 25 cases between the Upper Quartile and the Median, (3) 25 cases between the Median and the Lower Quartile, (4) 25 cases below the Lower Quartile. The difference between either Quartile and the Median is technically called the "Probable" deviation.

the population vary less than 1.7 inch from mediocrity, this being what is technically known as the "probable "deviation. The mean deviation is, by a well-known theory, 1.18 times that of the probable deviation, therefore in this case it is 1.9 inch. The mean loss through regression is of that amount, or a little more than 0.6 inch. That is to say, taking one child with another, the mean amount by which they fall short of their mid-parental peculiarity of stature is rather more than six-tenths of an inch.

The stability of a Type, which I should define as "an ideal form towards which the children of those who deviate from it tend to regress," would, I presume, be measured by the strength of its tendency to regress; thus a mean regression from 1 in the midparents to in the offspring would indicate only half as much stability as if it had been to.

The limits of deviation beyond which there is no regression, but a new condition of equilibrium is entered into, and a new type comes into existence, have still to be explored.

With respect to my numerical estimates, I wish emphatically to say that I offer them only as being serviceably approximate, though they are mutually consistent, and with the desire that they may be reinvestigated by the help of more abundant and much more accurate measurements than those I have had at command. There are many simple and interesting relations to which I am still unable to assign numerical values for lack of adequate material, such as that to which I referred some time back, of the relative influence of the father and the mother on the stature of their sons and daughters.

I do not now pursue the numerous branches that spring from the data I have given, as from a root. I do not speak of the continued domination of one type over others, nor of the persistency of unimportant characteristics, nor of the inheritance of disease, which is complicated in many cases by the requisite concurrence of two separate heritages, the one of a susceptible constitution, the other of the germs of the disease. Still less do I enter upon the subject of fraternal deviation and collateral descent, which I have also worked out.

APPENDIX.

I.-Experiments on Seeds bearing on the Law of Regression.

I sent a set of carefully selected sweet pea seeds to each of several country friends, who kindly undertook to help me. The advantage of sweet peas over other seeds is that they do not cross fertilise, that they are spherical, and that all the seeds in the same pod are of much the same size. They are also hardy and prolific. I selected them as the subject of experiment after consulting eminent botanists. Each set contained seven packets, numbered K, L, M, N, O, P, and Q. Each packet contained ten seeds of exactly the

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