ON PRECIOUS METALS IN USE AS MONET. 229 



many averages, that this proportion must be very near the truth.' ' Still 

 stronger evidence is supplied by the consilience between the results of the 

 French enquetes made in 1878 and 1885. The two curves which repre- 

 sent the proportions in which the circulation in the years 1878 and 1885 

 respectively comprised coins of each date earlier than 1878 correspond, not 

 only in their general character, but even as to their minuter traits.^ 



If farther accuracy is required, it would be proper to apply the 

 Mathematical Method of Statistics, the doctrine of Errors, to the data. 

 The case is as if we had observed the proportion of males in a great number, 

 say 100,000, births taken at random, so to speak, from the general popu- 

 lation. We might suppose a number of subdistricts all over the country 

 selected on some random principle (such as the alphabetical order of 

 their initial letter) and the birth-rate observed for the heterogeneous 

 aggregate thus formed. It is required to determine with what accuracy 

 we can infer, from this aggregate of samples, the proportions prevailing 

 in the total number of births, say 1,000,000, appertaining to the entire 

 population. Suppose that for the aggregate of 100,000 the observed 

 ratio was "51. What extent of deviation from that ratio are the returns 

 for the entire population likely to present ? 



The proper course would be to break up the aggregate of 100,000 

 samples into a good number, say 50 (or 25), of smaller parcels consisting 

 each of about 2,000 (or 4,000) units. Let the proportion of male births in 

 each of these small groups be observed. Call these partial ratios r , r, 

 &c. . . . r^; the ratio for the entire aggregate being "51. Form the 

 mean-square-of-error 



(•51-r,)^+(-51-r,)^-h&c.-K-51-r,,)=' 

 50 ' 



and put the square-root of double the mean-square-of-error, say fi, for 

 the modulus, that constant upon w'hich all the higher operations of the 

 Calculus of Probabilities turn. The modulus for the proportion of male 

 births in a group of 2,000 being /x. say 016,^ the modulus for the same 



ratio in an aggregate of 100,000 births should be — ^= ; or some rather 



\/50 

 larger figure which good sense tempering theory may prescribe, say "OOS. 

 Then we may affirm with confidence that the ratio presented by 100,000 

 bii'ths will not difier I'rom that which would be presented by a much 

 larger number by more than 3 X "003 ; and, therefore, that the proportion 

 of males in 1,000,000 births does not differ from '51 (the observed pro- 

 portion in 100.000) by more than the calculated limit of eri'or, say •01. 



These are not mere anticipations of theory, but practical rules which 

 liave been verified by copious experience. 



For illustrations of the Theory of Errors applied to statistical reasoning see the 

 writer's ' Methods of Statistics,' Journal of the Statistical Society, jubilee vol. 1885, 

 and ' Methods of Ascertaining Rates,' ibiil. Dec. 1885. It may be observed that in 

 the illustration just given we are not, as in most statistical inductions, inferring 

 from what has been observed of one time or place, to what is true of another time 

 or place ; but, from the constitution of samples selected at random to the constitu- 

 tion of the aggregate from which they were taken. The validity of the inference 

 ■depends upon the impartiahty of the selection. 



' Currency and Finance, p. 265. 



2 See De Foville, Jmirnal de Statistique, 1886, p. \2,a.ndi Bulletin de France, 1885. 



^ As may be gathered from the researches of Professor Lexis. 



