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VII. Mathematical Contributions to tlie Theory of Evolution. JV. On the Probable 

 Errors of Frequency Constants and on the Influence of Random Selection on 

 Variation and Correlation. 



By KARL PEARSON, F.R.S., and L. N. G. FILON, B.A., University College, London. 



Received October 18, Read November 25, 1897. 



CONTENTS. 



PAGE 

 I. Introductory ......................... 229 



II. On the probable errors of a system of frequency constants. General Theorem . . 231 

 On the determination of the probable errors and the error correlations of the 



frequency constants ..................... 236 



III. On the probable errors and the coefficients of correlation between errors made in the 



determination of the constants of a normal frequency distribution ..... 237 



Probable error of a regression coefficient for two organs ......... 244 



Probable errors of variation and. correlation for three organs ........ 245 



Case of four or more organs ................... 250 



IV. On the probable errors and the coefficients of correlation between errors made in the 



determination of the constants in skew variation ........... 2G5 



In the case of the curve 



266 



Numerical illustration : Incidence of enteric fever ........... 279 



In the case of the curve 



y = y l (l + */a,) 7 "' (1 aj/aj)'"* ........ 282 



Numerical illustration : Glands of the forelegs of swine ......... 289 



In the case of the curve 



___ - _ ,,-tan-i(*M) OQ7 



y- y >{l + (a/a*)}"' 



Numerical illustration : Stature of children .............. 303 



Conclusion ...... ............ ....... 309 



Note on the probable error of the criterion . . .............. 310 



I. INTRODUCTORY. GENERAL THEOREM. 



(1) In earlier memoirs by one of the present authors, methods have been discussed 

 for the calculation of the constants (a) of variation, normal. or skew,* (6) of correla- 



* "Mathematical Contributions to the Theory of Evolution. II. On Skew Variation," 'Phil. Trans.,' 

 A, vol. 186, pp. 343-414. 



12.7.98. 



