892 REPORT— 1898. 



A g;ood many characters of animals do vary in this symmetrical way ; and I 

 show you one. which will always be historically mteresting, because it was one of 

 the principal characters used to illustrate Mr. Galton'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 indi- 

 cated 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 prohable. 



In cases where a variation in either direction is equally likely to occur, this 

 symmetrical curve can be used to express the law of distribution of variations. 

 And the great difficulty in applying the law of chance to the treatment of other 

 cases was, imtil 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 distributions 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 Professor Karl Pearson. In 1895 Professor 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 Professor Pearson dealt with another 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 com- 

 plexity of the results obtained by tossing dice, and the complexity of events 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 

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

 than we had hefore looking at the first.' But the events which determine the 

 size or shape of an organ in an 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 varia- 

 tion in the size of a limb would be represented by a law similar to that which 

 expresses 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 chance whether you draw a red card or a black 

 one. Suppose you draw a red card, and keep it. The chance that your second 

 card will be red is not so great as the chance that it will be black ; because there 

 are only twenty-five red cards and twenty-six black cards left in the pack. 



Now Professor Pearson has shown how to deal with cases of this kind also ; 

 and how to determine, from the results of statistical observation, whether one is 

 dealing with such cases or not. 



I am no mathematician, and I do not dare even to praise the mathematical 

 process by which this result was achieved. I will only say that it is experimen- 

 tally justified by the fact that most statistics relating to organic variation are most 

 accurately represented by the curve of frequency which Professor Pearson deduces 

 for the case where the contributory causes are mutually inter-depeudent.'- 



' That is to say, if we know beforehand that the dice are symmetrical. 



'' Even the distribution of human stature, which has been so successfully treated 

 by the older, so-called ' normal ' curve, is more accurately represented by a curve of 

 Professor Pearson's type ; but in this case the difference between the two is so slight 

 as to be inappreciable for all practical purposes ; so that Mr. Galton's practice and 

 Professor Pearson's theor}' are alike justified. 



