Mr. F. Y. Edgeworth's Problems in Probabilities. 373 



95, 99, 99, 92, 95, 94, 94, 99, 101, 100, 108, 108, 106, 



105, 103, 103, 107, 112, 105, 105, 101, . . . ; 



where the first term and the last term are the only indepen- 

 dent observations ; and where there occur twenty observations 

 running above the Mean of the series (supposed indefinitely 

 prolonged). In case of such entangled series, I submit that 

 the advantage of accuracy generally attaching to the Arith- 

 metic Mean and the determination of Modulus by way of mean 

 square of error disappears. There is no set-off against their 

 disadvantage of inconvenience. Accordingly the proper mode 

 of reducing such observations is what may be called the 

 Galton-Quetelet method, by way of Median and " Quartile." 



I have applied this method to find what was the probability 

 in 1844 that the Notes in the hands of the public would not 

 in the proximate future fall below the limit fixed in that year 

 for uncovered Notes. The returns upon which the calculation 

 is based* are the monthly returns from 1833-1884, and cer- 

 tain biennial returns from 1826-1844. I find that these 

 returns conform pretty accurately to a probability-curve, 

 whose Median is £18,400,000, and whose probable-error is 

 £1,000,000. The Modulus then is about £2,000,000. And 

 the proposed limit is £15,500,000f ; that is, at a distance from 

 the Median of about 1 J the Modulus. The probability, then, of 

 the Notes not falling below the regulation-limit in the proxi- 

 mate future is about -98|. 



II. The problem becomes complicated if we suppose the 

 number of (independent) observations to be not large, and 

 seek to determine, by way of Inverse Probability §, the error 

 due to that limitation. 



The error incurred by the Galton-Quetelet method depends 

 upon the errors attaching to the assumptions that the point 

 dividing the given set of observations into two equal groups 

 is the real Median, and that the points dividing the given set 

 of observations in the ratio of three to one are the real " Quar- 

 tiles." The errors of these assumptions may be investigated 

 by the following method, which was suggested by Laplace's 

 determination of the error incurred by his " Method of Situa- 

 tion " (Theorie Analytique, Supplement 2, Sect. 2). 



* See f Mathematical Theory of Banking.' 



t £14,000,000+1,500,000 (Bills and Lost Notes), ibid. 



X More exactly -98 is the probability of the next observation falling 

 below the limit. Within what time the next observation must take 

 place depends upon the number of (independent) observations assigned to 

 the series. It would be safe, I think, to say — within six months. 



§ Problem I. belongs to _the class termed d in my ' Observations and 

 Statistics ; ' Problem II. to d. 



