172 Prof. Karl Pearson on Deviations from the 



and theoretical. In order to avoid so many different groups, 

 I have tabulated his groups in *10" units, and so reduced them 

 to 21. From these 21 groups I have found ^ 2 by the method 

 of this paper. By this reduction of groups I have given 

 Sir George Airy's curve even a better chance than it has, as 

 it stands. Yet what do we find ? Why, 



X 2 = 38'2872. 



Or, using the approximate equation, 



P = -01423. 



That is to say, only in one occasion out of 71 repetitions of 

 such a set of observations on Polaris could we have expected 

 to find a system of errors deviating as widely as this set (or 

 more widely than this set) from the normal distribution. Yet 

 Sir George Airy takes a set of observations, the odds against 

 which being a random variation from the normal distribution 

 are 70 to 1 , to prove to us that the normal distribution applies to 

 errors of observation. Nay, further, he cites this very impro- 

 bable result as an experimental confirmation of the whole 

 theory ! " It is evident," he writes, " that the formula repre- 

 sents with all practicable accuracy the observed Frequency 

 of Errors, upon which all the applications of the Theory of 

 Probabilities are founded : and the validity of every investi- 

 gation in this Treatise is thereby established." 



Such a passage demonstrates how healthy is the spirit of 

 scepticism in all inquiries concerning the accordance of theory 

 and nature. 



Illustration VII. 

 It is desirable to illustrate such results a second time. 

 Professor Merriman in his treatise on Least Squares * starts 

 in the right manner, not with theory, but with actual expe- 

 rience, and then from his data deduces three axioms. From 

 these axioms he obtains by analysis the normal curve as the 

 theoretical result. But if these axioms be true, his data can 

 only differ from the normal law of frequency by a system of 

 deviations such as would reasonably arise if a random 

 selection were made from material actually obeying the 

 normal law. Now Professor Merriman puts in the place of 

 honour 1000 shots fired at a line on a target in practice for 

 the U.S. Government, the deviations being grouped according 

 to the belts struck, the belts were drawn on the target of 

 equal breadth and parallel to the line. The following table 

 gives the distribution of hits and the theoretical frequency- 



* 'A Textbook on the Method of Least Squares,' 1891, p. 14. 



