September 22, 1898] 



NA TURE 



501 



number of sixes did actually occur was as near to the icMilt 

 most probably with perfect dice as the asymmetry of the actual 

 dice allows one to expect. ^ 



These results will be enough to show you how absurd is the 

 attitude which so many writers have taken up towards chance 

 when discussing animal variation. The assertion that organic 

 variation occurs liy chance is simply the assertion that it obeys 

 a law of the same kind as that which expresses the orderly series 

 of results we have just looked at.- 



That is a matter which can be settled by direct observation. 

 But in order to express the law of chance in such a way that 

 we can apply it to animal variation, we must make use of a 

 trick which mathematicians have invented for that purpose. 



It is a well-known proposition in prol)abiIity that the fre- 

 quency with which one throws a given number of sixes in a 

 series of trials with twelve dice is proportional to the proper 

 term in the 'expansion of {\ + %y^. The values in this table 

 were calculated by expanding this expression. But if I 

 had wanted to show you the most probable result of ex- 

 periments with 100 dice, I should not 

 willingly have expanded (Tr + ^)'""- 

 The labour would be too enormous. 

 Then again, suppose we are given 

 a number of results, and are not told 

 how many dice were used, how are 

 we to find out the power to which we 

 must raise {J -f^), since this depends 

 on the number of dice ? 



Before applying the law of chance to 

 variations in which we cannot directly 

 measure the number of contributory 

 causes (the analogue of the number of 

 dice), we must find some way out of 

 this difficulty. 



The way is shown by the diagram 

 (Fig- I). 



The rectangles in this diagram arc- 

 proportional to the various terms of 

 \\ + 4)'^ ; and they represent the most 

 probable result of counting the number 

 of dice with more than three points in 

 a series of trials with twelve dice. The 

 heights of these rectangles were deter- 

 mined by expanding (^ -f ^i^'^ ; but you 

 notice thef dotted curve which is drawn 

 through the tops of them. The general 

 slope of this curve is, you see,, the same 

 as the general slope of the series of 

 rectangles ; and the area of any strip of 

 the curve which is bounded by the sides 

 of a rectangle is very nearly indeed the 

 same as that of the rectangle itself. 



The constants upon which the shape 

 of this curve depends are easily and 

 quickly obtained from any series of 

 observations ; so that you can easily 

 and quickly see whether a set of ob- 

 served phenomena obeys the symmetri- 

 cal law of chance or not. 



A good many characters of animals do vary in this symmetrical 

 way ; and I show you one, which will always be historically 

 interesting, because it was one of the principal characters used 

 to illustrate Mr. Gallon's invaluable applications of the law of 

 chance to biological problems. That is the case of human 

 stature. The diagram (Fig. 2) shows the stature of 25,878 

 American recruits ; and you see that the frequency with which 

 each stature occurs is very close indeed to that indicated by the 

 curve. So that variations in human stature do occur by chance, 

 and they occur in such a way that variation in either direction 

 is equally probable. 



In cases where a variation in either direction is equally likely 

 to occur, this symmetrical curve can be used to express the law 



1 It is unfortunate that I chose dice as instruments in these experiments. 

 Dice are not only ?x:Vf!Xh\y asymmetrical, but any ordinary dice are sensibly 

 dissimilar ; so that the result most probable with any actual dice is not 

 given by a simple binomial expansion. The result theoretically most probable 

 •or the actual dice used could not be determined without very careful measure- 

 ment of the dice themselves ; and I was unable to attempt measures of the 

 requisite accuracy. All that the records show, as they stand, is the amount 

 of agreement between four successive observations of a fortuitous event, 



- The law is not, however, identical in the two cases ; see infra. 



of distribution of variations. And the great difficulty in apply- 

 ing ihe law of chance to the treatment of other cases was, until 

 quite lately, that the way of expressing asymmetrical distribu- 

 tions by a similar curve was unknown ; so that there was no 

 obvious way of determining whether these asymmetrical dis- 

 tributions obeyed the law of chance or not. 



The form of the curve, related to an asymmetrical distribution 

 of chances, as the curve before you is related to symmetrical 

 distributions, was first investigated by my friend and colleague 

 Prof. Karl Pearson. In 1895 Prof. Pearson published an 

 account of asymmetrical curves of this kind, and he showed 

 the way in which these curves might be applied to practical 

 statistics. He illustrated his remarkable memoir by showing 

 that several cases of organic variation could be easily formulated 

 by the method he described : and in this way he made it possible 

 to apply the theory of chance to an enormous mass of material, 

 which no one had previously been able to reduce to an orderly 

 and intelligible form. - 



In this same memoir Prof. Pearson dealt with another 



NO. 1508, VOL. 58] 



problem in the theory of chance, which has special importance 

 in relation to biological statistics. It has doubtless occurred to- 

 many of you that the analogy between the complexity of the 

 results obtained by tossing dice, and the complexity of eveftts 

 which determine the character of an animal body, is false in an 

 important respect. For the events which determine the result » 

 when we throw a dozen dice on the table, affect each of the dice 

 separately ; so that if we know that one of the dice shows sis 

 points, we have no more reason to suppose that another will 

 show six points than we had before looking at the first.* But 

 the events which determine the size or shape of an organ in -m 

 animal are probably not independent in this way. Probably 

 when one event has happened, tending to increase the size of an 

 arm or a leg in an embryo, it is more likely than it was before 

 that other events will happen leading to increased size of this, 

 arm or leg. So that the chances of variation in the size of a 

 limb would be represented by a law similar to that which ex- 

 presses the result of throwing dice, but different from it. They 

 would more nearly resemble the result of drawing cards out of a 

 pack. Suppose you draw a card out of a pack. It is an even 

 1 That is to say, if we know beforehand that the dice are symn\eirical. 



