PRESIDENTIAL ADDRESS. 635 



Kapteyn, and especially by Edgeworth, who has arrived at a temporary conclusion 

 of his labours this year in his analysis of the complete form of the law of great 

 numbers and of the conditions under which it may be expected to hold. On the 

 other hand, luatheraatical formulae on a double basis of rt/)?w/ hypothesis and 

 of empirical justi6cation have been elaborated (primarily for biological purposes) 

 by Professor Karl Pearson, whose method of fitting by moments has proved 

 fundamental for work of this kind. 



The measurement of correlation, implied by Bravais in 184G,' received a great 

 impetus from Galton in 1888, and, after its analysis in successive papers by 

 Edgeworth, Pearson, and Yule, is now in general use. The more elementary pro- 

 cesses of measurement by averages have been examined and extended by Galton, 

 Venn, and Fechnev, till the ideas of median and quartiles (used implicitly in 

 Quetelet's method of fitting), mode, arithmet-.c average, dispersion, and mean and 

 standard deviation, are common property with even the least advanced statisticians, 

 and are coming into use in official statistics here and in the U.S.A. The most 

 important inroad into official statistics has been made by index numbers. This 

 method has grown very gradually, and has received impetus from many economists 

 and statisticians ; the most complete analysis of its mathematical basis is in the 

 report of this section's Committee in 1887-1889. 



In recent years progress in the development of theory has, indeed, been rapid, 

 and a great number of important and thoroughly criticised methods are ready for 

 use, and are, in fact, in constant use by biologists and botanists ; but there has 

 been remarkably little application to practical statistical problems. In the thirty 

 years following the publication of Quetelet's ' Lettros,' attention was mainly given 

 to establishing the constancy of great numbers and averages based thereon, an 

 important but limited work, while the relation of the frequency of deviations to the 

 law of error was regarded rather as a statistical curiosity. Professor Edgeworth's 

 illustrations in 1885 of the importance of mathematical methods in testing the 

 truth of practical deductions has as yet borne singularly little fruit. The atten- 

 tion of mathematical statisticians has been mainly directed to theory, and to actual 

 measurement of anthropometrical and biological correlations ; it is time that it was 

 brought to bear on the criticism and analysis of existing industrial statistics. 

 Something has been done by Yule and Hooker "in England, by Norton in the U.S.A., 

 and others, to test correlation and periodicity, and in other practical problems, 

 but most of our statistics remain untested and their significance not analysed. 

 The simple method of samples, illustrated below, for which all the materials have 

 existed for at least twenty years, has (so far as I know) been completely 

 ignored. 



The region to which I am devoting particular attention is that where the 

 theory of probability is invoked, not because there are not many other directions 

 in which mathematical methods are useful, but because this is of the greatest 

 importance and the least generally understood. All depends on a complete grasp 

 of the nature of the measurement when we say, for example, that from certain 

 data the most probable estimate of average wages is 2-is. ; it is as likely as 

 not, however, to be as much as 4d. from this value : the standard deviation 

 is 6d. ; the chances are 10 to 1 against the average being over 24s. 8f/., 100,000 to 1 

 against it being over 26s. This is the kind of .statement to which calculations 

 lead. The result may be briefly indicated as 24s. ± Gd., when the 'standard 

 deviation ' is adopted as the measure of accuracy. In a normal curve of frequency 

 about two-thirds of the area is within the standard deviation ; the chance that a 

 given observation should be within this distance of the true average is 2 : 1. The 

 unit of measurement thus devised is most subtle and most complex. When it is 

 applicable it gives the only complete measure of precision. When the initial 



■ ' Bravais' formula relates to the position of a point given by" two co-ordinates, 

 the sources of error of which are not independent. The term arising from this 

 interdependence proved to be essentially the same .%s that re.ichcd hy the later 

 writers working from cjuite different standpoints, 



