174 On Deviations from the Probable. 



Illustration YIII. 

 The reader may ask : Is it not possible to find material 

 which obeys within probable limits the normal law ? I reply, 

 yes ; but this law is not a universal law of nature. We must 

 hunt for cases. Out of three series of personal equations, I 

 could only find one which approximated to the normal law. 

 I took 500 lengths and bisected them with my pencil at sight. 

 Without entering at length into experiments, destined for pub- 

 lication on another occasion, I merely give the observed and 

 normal distribution of my own errors in 20 groups. 



Group. 



Observation. 



Theory. 



Group. 



Observation. 



Theory. 



57-0 

 47-1 

 34-0 

 22-7 

 13-5 



70 



35 



1-6 

 ■6 

 ■3 



1 



1 



3 



11 



14-5 



21-5 



30 



47 



515 



72 



65-5 



23 

 3-4 

 6-9 

 13-1 

 22 2 

 336 

 47-5 

 57-8 

 63-2 

 62-7 



11 



53 



50-5 



28-5 



27 



135 



7-5 







1 







2 



2 



12 



3 



13 



4 



14 



5 



6 



15 



16 



7 ....... 



17 



8 



18 



9 



10 



19 



20 



■ 





Calculating ^ in the manner already sufficiently indicated 

 in this paper, w T e find 



% 2 = 22-0422. 

 We must now use the more complex integral formula for P, 

 and we find 



P = -2817. 



Or, iu every three to four random selections, we should expect 

 one with a system of deviations from the normal curve greater 

 than that actually observed. 



I think, then, we may conclude that my errors of judg- 

 ment in bisecting straight lines may be fairly represented by 

 a normal distribution. It is noteworthy, however, that I found 

 other observers' errors in judgment of the same series of lines 

 were distinctly skew. 



(8) We can only conclude from the investigations here 

 considered that the normal curve possesses no special fitness 

 for describing errors or deviations such as arise either in 

 observing practice or in nature. We want a more general 

 theoretical frequency, and the fitness of any such to describ 

 a given series can be investigated by aid of the criterion dis- 

 cussed in this paper. For the general appreciation of the 

 probability of the occurrence of a system of deviations defined 

 by x 2 ( or anv greater value), the accompanying table has 

 been calculated, which will serve to give that probability 

 closely enough for many practical judgments, without the 

 calculations required by using the formulae of art. 4. 



