►per 



366 Mr. F. Y. Edgeworth 



The hypothesis entitles us to assert that 23 is an error-prop 

 — an accidental deviation from 45 ; though the odds against 

 such an event before its occurrence are considerable, about 100 

 to 1. On the other hand, we may know for certain that 136 

 is a mistake. 



(/3) According to the second hypothesis, the type of error 

 is still the probability-curve with unvarying constant. But 

 the range of its applicability is not so accurately known before- 

 hand. We cannot at sight distinguish errors proper from mis- 

 takes. We only know that mistakes may be very large, and 

 that the large mistakes are so infrequent as not to be likely to 

 compensate each other in a not unusually numerous group of 

 observations. This hypothesis may thus be exemplified : — 

 As before, we have a series of numbers, each purporting to be 

 the sum of ten random digits. But occasionally, by mistake, 

 the sum (or difference) of two such numbers is recorded. The 

 mistake might be large, but it would not always exceed the 

 limits of accidental deviation (100 and 0); which need not be 

 supposed known beforehand. Here is a sequence of seven 

 such numbers, which was actually obtained by me (in the 

 course of 280 decades) — 



50, 54, 41, 73, 46, 38, 49. 

 The hypothesis leaves it doubtful whether 73 may not be a 

 mistake ; the odds against it being an ordinary accidental 

 deviation being, before the event, about 250 to 1. 



(y) According to the third hypothesis all errors are of the 



type y = — -=e~ h2x2 . But the h is not the same for different 



observations. Mistakes may be regarded as emanating from 

 a source of error whose h is very small. This hypothesis may 

 be thus illustrated. Take at random any number n between 

 certain limits, say 1 and 100. Then take at random (from 

 Mathematical Tables) n digits, add them together and form 

 their Mean (the sum -r- n) , and multiply this Mean by ten. 

 The series of Means so formed may be regarded as measure- 

 ments of varying precision ; the real value of the object mea- 

 sured being 45. The iveic/ht, the A 2 , being proportionate to n, 

 one weight is a priori as likely as another. In order that the 

 different degrees of precision, the equicrescent values of h, 

 should be a priori equiprobable, it would be proper, having 

 formed our n as above, to take the mean of (and then mul- 

 tiply by 10), not n, 

 in this latter fashion : — 



lOxMeanofw 2 

 random digits 



but n 2 digits. 



Here is 



a series foi 



5 7 6 



25 49 36 



1 10 

 1 100 



8 1 

 64 1 



31 45 43 



100 43 



47-5 100 



