Mr. F. Y. Edgeworth's Problems in Probabilities. 379 



The fifth column is obtained by dividing each entry in the 

 fourth column by the sum of them all. The seventh column 

 gives the probability that, if any hypothesis is true, the effect 

 under consideration would follow from it. The eighth column 

 is obtained by multiplying each entry in the seventh column 

 by the corresponding entry in the fifth column. The sum of 

 the entries in the eighth column gives, very roughly of course, 

 the probability that the next observation above the Mean will 

 exceed 4Q f . The probability that the next observation above 

 the Mean will fall within that limit is *987, corresponding to 

 '985 *, as found by the approximative method. 



So far we have been supposing the Mean given ; but we 

 must now investigate the error incurred in assigning the 

 probability that a subsequent observation will not exceed a 

 certain multiple of the probable-error measured not from 

 the real, but the apparent, Mean. Let be the real, 0' the 



o"_ Q Q' 



apparent Median. Let Q be the real Quartile point, Q ; the 

 apparent one, namely, that which separates the given obser- 

 vations into two groups numbering respectively j n and £ w. 

 Let 0'P = aO'Q'. Let us find the probability that the 

 (w + l)th observation f will fall outside P. 



Let 2 = 00', ? = QQ'. And let k be the Modulus for the 

 error of the Median, k for that of the Quartile point. It is 

 shown above that 



V 2n V 1-7 n 



For any particular values of z and ? the probability that a 

 subsequent observation will exceed P is proportioned to 



'0P> 



where, as before, 



* [i->m. 





e-< 2 dt. 



Now 0P=00' + 0'P=2+*0'Q'. 



* The method of quasi-integration by means of a table appears theo- 

 retically safer, however practically rough. 



t It is a little awkward here to introduce the condition u being above 

 the mean," as in the preceding problems, on account of our uncertainty 

 where the real mean is. 



