438 



NATURE 



[September 8, 1892 



serius ocius Sons exitura" — is constituted as simply as the 

 urn in which we have supposed black and white balls to 

 be shaken up. This is a question in Applied Prababili- 

 ties to which we are just coming. 



Prof. Westergaard's applications of the calculus to 

 statistics are even more striking than his developments 

 of the pure theory of probabilities. The law of error may 

 be applied to concrete phenomena in two cases : where 

 the fluctuation of averages follows the analogy of the 

 simpler games of chance — as we just now assumed with 

 regard to deaths — and where this condition is not fulfilled. 

 Prof. Westergaard's contributions belong chiefly to the 

 first class. He has considerably added to the instances 

 discovered by Prof. Lexis, in which a set of 

 ratios — such as the proportion between the mor- 

 tality of male and-female infants in different years — are 

 grouped according to the same law of dispersion as the 

 percentages of white balls in a set of batches drawn at 

 random from an urn containing black and white balls 

 mixed up in a certain proportion. The uses of this dis- 

 covery are twofold — negative and positive. In the first 

 place, we may be deterred from a search after causes 

 which is hopeless. In the typical instance of the urn 

 and balls it would be vain to trace the reason why any 

 particular ball, or set of balls, extracted should be white 

 (or black). We cannot hope to analyse the " fleeting 

 mass of causes " — as Mill calls Chance — upon which the 

 event depends. We may have been able to break up 

 our batches of balls into two classes, say rough and 

 smooth, such that the rough balls are extracted from an 

 urn containing mostly white, while the smooth balls 

 are more frequently black. But when this process of 

 " depouillement " has been carried as far as possible, 

 when we have reached the ideal type of a single urn and 

 constant proportions, then the investigation of causes 

 halts. Then it is only crazy gamblers who hope to dis- 

 cover a principle underlying the " runs " of black and 

 white balls. We have reached the bounds of the territory 

 of science ; beyond there is only the sea of chance. 

 Prof. Westergaard has not only indicated this limit, but 

 also pushed many of his investigations up to it. 



These considerations do not preclude us from applying 

 the theory of error to detect delicate distinctions such 

 as the difference between a loaded and a perfect die, 

 which make themselves felt in the averages of great 

 numbers of observations. In fact, it is by the mathe- 

 matical method that we can best determine whether a 

 difference between two averages is' significant of a real 

 constant difference, or only-apparent and accidental. 

 Prof. Lexis, and Dr. Duesing after him had applied this 

 method to the investigation of the conditions under which 

 the excess "of male over female births is greater or smaller 

 than usual. Prof. Westergaard shows that the method is 

 applicable to many other subjects, among which the 

 mortality at different age-periods promises to be most 

 useful. We could wish, indeed, for more copious evidence 

 in favour of the premise that the mortality of a popula- 

 tion at a certain age-period (say of clergymen or inn- 

 keepers between the ages 35-44. See Grundzuge, 

 p. 82, with context, and cf. p. 52) fluctuates according 

 to the analogy of games of chance. 



Here the question arises : must the phenomenon under 

 consideration be known to vary after the manner of balls 

 NO. I 193, VOL. 46] 



extracted from an urn, in order that the mathematical 

 method may be applicable ? Certainly the apparatus of 

 the law of error — probable and improbable deviation — 

 may be employed to ascertain whether the difference be- 

 tween the average heights of two groups of two men is 

 significant or accidental ; though in this case the modulus 

 (or mean error) does not follow the analogy of the simpler 

 games of chance. Might we not similarly compare the 

 general mortality of two sections of population, although 

 the dispersion of such death-rates about their average is 

 much greater than it should be on the hypothesis of pure 

 sortition. The advantage, indeed, which we have above 

 distinguished as negative, would no longer exist in this 

 case. But might not the positive advantage still be 

 enjoyed in some degree ? Prof. Westergaard, so far as 

 we have observed, is silent on this topic. 



We have left ourselves too little space for noticing Prof. 

 Westergaard's other contributions to applied Probabilities. 

 His treatment of Insurance, together with the adjacent 

 theory of Life-Tables, involving the arts of Interpola- 

 tion, may dispute with Cournot's classical chapters the 

 honour of forming the best introduction to the subject for 

 the general reader — the reader prepared by a general 

 mathematical, as distinguished from a special actuarial 

 training. Nor must we pass over the chapter in which 

 the author surveys " economic " (exclusive of Vital) 

 statistics. He has here occasion to employ largely 

 the important principle of inference from samples. 

 For instance, in order to discover the amount of 

 wood in a country, we should first select one or more 

 sample surfaces {Probe-fldchen) , and a sufficient num- 

 ber of sample trees thereat, and then measure the 

 quantity of wood on those trees. " From the figures 

 so found conclusions can be drawn as to the whole 

 sample-surface, and from those to the total quantity 

 of wood in the country." So in order to determine the 

 quantity of milk, we must proceed by way of Probe- 

 kiihe. The method of samples is no doubt a potent 

 instrument when wielded by a trained hand like Prof. 

 Westergaard's. We may perhaps extend to economics 

 generally what he suggests with reference to its statistical 

 side : that a given effort and expense may be better laid 

 out in obtaining a detailed knowledge of a few parts with ■\ 

 a general view of the whole, rather than a more uniformly 

 distributed information. 



The last part of the work is devoted to a history of 

 statistics ; not a chronicle, but such a history as a great 

 tactician would write pf past wars. The criticism of 

 Quetelet's methods is particularly instructive. In con- 

 nection with Ouetelet we may note — without assenting to 

 —one of the Professor's objections to the principle of the 

 "Mean Man." It is in effect the same objection as 

 Cournot raised : that the average of one limb derived 

 from measuring several specimens might not fit the 

 average similarly found as the type of another limb. The 

 model man constructed by putting together these averages 

 of parts might. prove to be a monster. 



In conclusion we venture to express the hope. that this 

 important treatise may be translated into English ; in 

 order that the insular student may not have to encounter , 

 the difficulties of German and Probabilities at once. We 

 might advise the translator to follow the excellent English 

 custom of prefixing descriptive headings to all the 



