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



Let n be the number of observations supposed to range 

 under one and the same curve y = $(x) ; whereof P is the 

 (real) ordinate at the Median point ; which point may be 

 taken as the origin. The probability of an error z being 

 committed by the apparent Median is equal to the proba- 

 bility of half of the n observations falling on one side and 

 half on the other side of z. Now the probability of a single 

 observation falling outside z is 



And the probability of a single observation falling inside z is 



Hence the probability of z being the apparent Median is 



n n 



ac(i-fV(*)&y (i+j/o)^) 2 a 



if z is small 



n n 



The probability of the error z is therefore proportioned to 



2P 3 

 1 z 2 i or (by a step* frequent in this region of mathe- 

 matics, and which success and the authority of Laplace 



-*¥* 



sanction) e « . Now if 



it "* 



<£(#) = — -=-e c% P = 



And the law of facility for the error of the Median is 



7j = e * c2 , multiplied by a proper coefficient. 



The error incurred by assuming that the point which di- 

 vides the given observations in two groups numbering respec- 

 tively \n and \n is (the extremity of) the real Quartile, may 

 be similarly calculated. Let Q be the real Quartile, and let 

 us consider the probability of the point (Q + s) dividing the 

 observations in the ratio of j to J. The probability of an 

 observation falling outside (Q + s) is 



[i-(«*(Q) + 5-^(0))} 



* I desiderate a more rigid proof of this step ; such as I have attempted 

 to give for the general case of the law of error j "Observations and 

 Statistics/' Camb. Phil. Trans. 1885, p. 142. 



