382 Mr. F. Y. Edgeworth's Problems in Probabilities. 



degree of probability just mentioned it would be necessary to 

 make the limit 2M. These and similar results may be used 

 to find in a rough and ready fashion a superior (or inferior) 

 limit to the fluctuations of any phenomenon ; the Mean and 

 the Maximum of a certain number of observations being 

 given. Thus, consider the Registrar-General's returns of the 

 proportions of male to female children for the different 

 counties and for several consecutive years, e.g. Report 46, 

 table 16. Omitting Lancashire, West Riding, Huntingdon- 

 shire, and Rutlandshire, we have 40 observations of pretty 

 much the same weight in each of the first ten columns in the 

 table referred to. Take any one of these, e.g. the fourth, 

 and observe the difference between the maximum return in 

 that column and either its own or the general Mean. The 

 Mean + twice that difference constitutes a fairly safe superior 

 limit. In the case selected, the maximum return is 1089. And 

 the Mean (both of this column and all the columns) is 1038. 

 Hence for the superior limit we have 1140, which is not I 

 think reached by any of the four hundred returns in the table. 

 I have tested the proposed canons in the Art of Conjecturing 

 by similarly applying them to various phenomena more or 

 less obeying the law of error ; such as* a series of figures, 

 each of which is formed by adding together a certain number 

 of digits taken at random from mathematical tables; recordsf 

 of temperature ; statistics % of the dactyls in Virgilian lines, 

 &c. 



IV. My fourth problem is : Given the Modulus of the law 

 of error for a certain species of demand made upon a bank, 

 to find what the Modulus becomes when the liabilities are 

 increased in any proportion cceteris paribus. The general 

 answer is that the Modulus increases, not in the ratio of the 

 volume of business, but in the square root of that ratio. The 

 general rule and the exceptions may be illustrated by ex- 

 amples taken from Vital Statistics. Consider § a Table of 

 Proportions of Male to Female children born in Registration 

 Counties. Form the sums of the columns. Compare the fluc- 

 tuation of the fringe-row thus formed with the fluctuation of 

 the figures in the ordinary rows. It will be found that the 

 Modulus for the former is not n times, but about \/n times, 



* ^ See ray paper on " Methods of Statistics," Journal of the Statistical 

 Society, Jubilee volume. 



t Such as Mr. Glaisher's in ' Philosophical Transactions,' and in the 

 ' British Meteorological Journal 'j where the means and the maxima are 

 given ready to hand. 



X " On Rates," Journal of the Statistical Society, Dec. 1885. 



§ See ' Methods of Statistics,' p. 198. 



