634 TRANSACTIONS OF SECTION F. 



as to wlietlier two series or groups are connected, or whetlier the observecl 

 variations are independent, or as to wliether a difference between two measure- 

 ments is significant or the accident of observation, or as to whether a change 

 observed in average or grouping can be accounted for without assuming a change 

 in the plexus of causes governing the phenomena. 



It may be well to give commonplace instances of each of these methods. The 

 ages of persons, as tabulated from the Census forms, are systematically smoothed 

 for the Table of Survivals. Records of prices are averaged to give index numbers 

 independent of individual variations. Records of unemployment are averaged to 

 give the seasonal variation apart from the general trend. Records of wages may be 

 rejected or doubted if they show too close a grouping at round numbers, if the 

 grouping found in two similar establishments is markedly different, or if the 

 relation of the various grades is not that obtained in properly chosen samples. 

 The relation between infantile mortality and the employment of married women 

 is a problem in correlation very difficult from the dearth of data. The observed 

 difference in death-rates in two occupations requires a delicate mathematical test 

 before it can be establislied as a real phenomenon. Tlie proof that a known 

 change in tariff has or has not affected prices or trade requires an adequate 

 measurement of the variations when there has been no such change. The question 

 whether the national income has in recent years become more or less uniformly 

 distributed supplies a mathematical problem of considerable difficulty. Most of 

 these illustrative problems can be treated arithmetically ; the essential thing to 

 observe is that the choice of the right method of treatment requires mathematical 

 analysis. 



The time is not ripe, nor have I the knowledge, for writing the history of 

 mathematical statistics ; but a slight sketch may be offered of some of the main 

 developments, from Gauss and Laplace to Quetelet, and to Professors Edgeworth 

 and Karl Pearson. I leave on one side the mathematics of graduation and inter- 

 polation, Newton's interpolation formula, Farr's life table, Hain's application to the 

 smoothing of statistical series, and Mr. Sheppard's central differences ; and I omit 

 references to the method of least squares, used by Gauss in 1795 and Legendre in 

 1806, since this has been developed on non-statistical lines and its statistical use 

 is merged in the development of the law of error. 



The fundamental formula of the normal curve was known at least as early as 

 1809. Hagen (of Berlin) used it in 18;j7, deducing it from the binomial form. In 

 1837 Poisson defined ' the law of great numbers,' a phrase whose meaning has been 

 enlarged by Edgeworth. In 1846 Quetelet showed its very extensive applica- 

 tion to anthropometry, and enriched his letters with illustrations culled from a 

 very wide field. In the same year Bravais discussed the surface of error for two 

 variables. In 1852 Hain (of Vienna) applied Quetelet's method to the observa- 

 tion of the constancy of many important statistical totals and to the measurement 

 of their variation. He measured the deviation by the method of mean square 



( A / — ), whereas Quetelet had dealt mainly with the binomial measure 



{^y2pgn). In 1877 Lexis showed the importance of the difference between the 

 two methods of measuring deviation just stated, and applied the law of great 

 numbers to the ratio of male to female births and to the normal span of man's 

 life (a subject continued in Professor Karl Pearson's ' Chances of Death '). In 

 1885 Professor Edgeworth brought into prominence the means of testing the 

 significance of observed differences between the averages of kindred groups, and 

 showed the very large practical field of possible applications of this and other 

 methods based on the law of error. More recently, and especially in the kst ten 

 years, the theoretical foundation and the extensions and modifications of the 

 normal law have been examined and the formulae developed. The skewness of 

 the binomial form is shown in Laplace's formula (1814), and was commented on 

 by Quetelet (in 1846) ; Poisson gave the second approximation to the normal 

 curve ; this has been developed, varieties of treatment .suggested, and further 

 approximations givep, hj Pechner, Lipps, Brvws, Werjier, J^ud-\vig) Oharlier, 



