Probable in a Correlated System of Variables. 171 



calculating the areas of the curve instead of using its ordinate", 

 I think we may consider it not very improbable that the 

 observed frequencies are compatible with a random sampling 

 from a population described by the skew-curve of Type I. 



Illustration VI. 



In the current text-books of the theory i of errors it is cus- 

 tomary to give various series of actual errors of observation, 

 to compare them with theory by means of a table of distri- 

 bution based on the normal curve, or graphically by means of 

 a plotted frequency diagram, and on the basis of these com- 

 parisons to assert that an experimental foundation has been 

 established for the normal law of errors. Now this procedure 

 is of peculiar interest. The works referred to generally give 

 elaborate analytical proofs that the normal law of errors is the 

 law of nature — proofs in which there is often a difficulty (owing 

 to the complexity of the analysis and the nature of the approxi- 

 mations made) in seeing exactly what assumptions have been 

 really made. The authors usually feel uneasy about this process 

 of deducing a law of nature from Taylor's Theorem and a 

 few more or less ill- defined assumptions; and having deduced 

 the normal curve of errors, they give as a rule some meagre 

 data of how it fits actual observation. But the comparison of 

 observation and theory in general amounts to a remark — based 

 on no quantitative criterion — of how well theory and practice 

 really do fit ! Perhaps the greatest defaulter in this respect 

 is the late Sir George Biddell Airy in his text-book on the 

 ' Theory of Errors of Observation/ In an Appendix he gives 

 what he terms a " Practical Verification of the Theoretical 

 Law for the Frequency of Errors/' 



Now that Appendix really tells us absolutely nothing as to 

 the goodness of fit of his 666 observations of the N.P.D. of 

 Polaris to a normal curve. For, if we first take on faith what he 

 says, namely, that positive and negative errors may be clubbed 

 together, we still find that he has thrice smoothed his obser- 

 vation frequency distribution before he allows us to examine it. 

 It is accordingly impossible to say whether it really does or 

 does not represent a random set of deviations from a normal 

 frequency curve. All we can deal with is the table he gives of 

 observed and theoretical errors and his diagram of the two 

 curves. These, of course, are not his proper data at all : it is 

 impossible to estimate how far his three smoothings counter- 

 balance or not his multiplication of errors by eight. But as I 

 understand Sir George Airy, he would have considered such a 

 system of errors as he gives on his p. 117 or in his diagram 

 on p. 118 to be sufficiently represented by a normal curve. 

 Now I have investigated his 37 groups of errors, observational 



