496 Theory of the Average. [CHAP. xix. 



was absolutely essential that the work should be true to the 

 tenth of an inch for it to be of any use. But conceive also 

 that two specifications had been sent, resting on different 

 measurements, in one of which the length of the requisite 

 piece was described as sixty and in the other sixty-one 

 inches. On the assumption of any ordinary law of error, 

 whether of the binomial type or not, there can be no doubt 

 that the firm would make the best of a very bad job by con 

 structing a piece of 60 inches and a half: i.e. they would 

 have a better chance of being within the requisite tenth of 

 an inch by so doing, than by taking either of the two specifi 

 cations at random and constructing it accurately to this. 

 But if the law were of the kind indicated in our diagram 1 , 

 then it seems equally certain that they would be less likely 

 to be within the requisite narrow margin by so doing. As a 

 mere question of probability, that is, if such estimates were 

 acted upon again and again, there would be fewer failures 

 encountered by simply choosing one of the conflicting 

 measurements at random and working exactly to this, than 

 by trusting to the average of the two. 



This suggests some further reflections as to the taking of 

 averages. We will turn now to another exceptional case, 

 but one involving somewhat different considerations than 

 those which have been just discussed. As before, it may be 

 most conveniently introduced by commencing with an ex 

 ample. 



1 There is no difficulty in conceiv- in the estimate at random (within 

 ing circumstances under which a certain limits), the firm having a 

 law very closely resembling this knowledge of this fact but being of 

 would prevail. Suppose, e.g., that course unable to assign the two to 

 one of the two measurements had their authors, we should get very 

 been made by a careful and skilled much such a Law of Error as is sup- 

 mechanic, and the other by a man posed above, 

 who to save himself trouble had put 



