Mathematical Contributions to the Theory of Evolution. 173 



Theory of Error. 



Let a community be divided into a number of classes, the propor- 

 tions in the different classes being z x , a%, z 3 . . . . , so that z x + Zz + *3 + 

 .... = 1. Suppose a random selection of n individuals to be made, 

 the numbers drawn from the different classes being n{z x + gi), 

 n{z-i + e 2 ), n(zs + 63). . . . It is proved geometrically, with the aid 

 of the binomial theorem, that the values of the errors e 1} e 2 , e 3 . . . . are 

 normally distributed, and that the distributions are normally cor- 

 related. It follows that the values of any expression of the form 

 2Ae = 4- A 2 e 2 A363 +. . . . are normally distributed. The mean 

 square of 2Ae is shown to be {SA 2 ? — (2A^) 2 }-f-?i, and the mean pro- 

 duct of 2Ae and 2Be to be {2ABs — 2As . 2Bz}-f-rc. The applica- 

 tions are of two kinds : — 



(1) The values of the probable errors in the determination of 

 certain quantities are obtained, and, in particular, the probable errors 

 in the mean, mean square of deviation, mean product of deviations, 

 and divergence. 



(2) Formulas are obtained for testing particular hypotheses ; e.g., 

 whether two distributions (of any kind) are independent ; whether a 

 distribution is normal ; and whether two normal distributions are 

 correlated. 



" Mathematical Contributions to the Theory of Evolution. IV. 

 On the probable Errors of Frequency Constants and on 

 the Influence of Random Selection on Variation and 

 Correlation." By Karl Pearson, M.A., F.R.S., and 

 L, N. G. Filon, B.A., University College, London. Re- 

 ceived October 18, — Read November 25, 1897. 



(Abstract.) 



1. This memoir stains with a general theorem, by which the 

 probable errors made in calculating the constants of any frequency 

 distribution may be determined. It is shown that these probable 

 errors form a correlated system approximately following the normal 

 law of frequency, whatever be the nature of the original frequency dis- 

 tribution, i.e., whether it be skew or normal. The importance of this 

 result for the theory of evolution is then drawn attention to. It is 

 shown that any selection, whether of size, variation, or correlation, 

 will in general involve a modification not only of the size, but the 

 variation and correlation of the whole complex of correlated organs. 

 The subject of directed selection, of which this random selection is 

 only a special case, is reserved for another memoir, nearly completed. 



