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THE AMERICAN NATURALIST [Vol. LI 



sample from a population from which a first sample has 

 produced a certain value is given, not by the ordinates of 

 a normal curve of errors, as is commonly assumed in writ- 

 ings on the theory of probability, but by the successive 

 terms of a simple hypergeometrical series. In an earlier 

 paper the same author^ had solved the problem of the 

 momental properties of the hypergeometrical series. 

 Combining the two results he was able to derive the neces- 

 sary equations for the complete solution of the problem 

 of probable errors in sampling. We may proceed at once 

 to the exposition of these results, referring the reader for 

 the proofs to the papers of Pearson cited. 



Let it be supposed that a first sample of n = p-}- q be 

 drawn from the population, p denoting the number of 

 times the event dealt with occurs in the n trials, and q 

 the number of times it fails. 



Write 



whence of course 



We then have for the chief constants of the error dis- 

 tribution for a second sample, of magnitude m, drawn 

 from the same population the following values : 



Mean^ = mp + ^"^ 9 (q -p)y (i) 



Mode = the integral portion of mp + p, (ii) 



These values are entirely general, and independent 

 the values of n, m, p and q. Under certain circumstances, 



8 Pearson, K., "On Certain Properties of the Hypergeometrical Series, 

 and on the Fitting of such Series to Observation Polygons on the Theory 

 of Chance," Fhil. Mag., Feb., 1899, pp. 236-246. 



i 



