MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 307 



Now it will be seen at once that the probable errors in the algebraic constants are 

 large, but that the probable errors in the position of the mean, of the mode, and in 

 the magnitudes of the mean frequency and standard deviation are small. The position 

 of the mean is sensibly more correct than that of the mode. On the other hand, the 

 distance of the mean from the mode and the skewness have large probable errors, not, 

 however, so large but what these quantities are probably significant. The frequency 

 distribution probably differs significantly from the normal distribution, but the 

 difference is small and would require a very large number of observations to deter- 

 mine it with extreme accuracy. That there is a significant divergence from normality 

 is also indicated by the sensible difference between the percentage errors in y., and cr, 

 which would be equal for a normal distribution. Had we taken a normal distribution, 

 the pi-obable error of the mean would have been '0400, and of the standard deviation, 

 02831. In fact, the standard deviation of the standard deviation, if calculated for 

 the normal curve = '04197, if calculated by our present method = '044246, and if 

 calculated by a modified form of the fourth moment formula given by CZUBER* 

 = '044240. This shows that the arithmetic of our process has been substantially 

 correct. 



We now place together the umbral equations for the correlations of the errors in 

 the " physical " constants. They are 



Xm Antl. l'492,5897x/, + And. r082,4588x + Antl. 

 X, = Antl. l-272,7484x - Antl. 1-282,1306 X , + Antl. r913,0028x* 

 x , 2 = Antl. T-816,696Gx^ Antl. l']G7,3480x, + Antl. 1-155,7688 X> . 

 X,i = Antl. -266,3616x, + Antl. T'740,1625x,, Antl. -323,825Gx, 

 X*.= Antl. 1-865,6420x4, - Antl. 'Ol7,0466x i . 



From these results any correlation between pairs of errors, " physical " or algebraic, 

 can be found at once. The following table gives the chief results : 



CORRELATION Coefficients between Errors in Constants. 



* ' Theorie der Beobachtnngsfehler,' p. 133. 

 2 R 2 



