368 Prof. Karl Pearson on the Influence of 



Now it is clear that the terms of this hypergeometrical 

 ' mg as 



in — r-1-1 p + r 



series increase as long as 



r y-f in — r + 1 

 or as long: as 



> 1. 



» 



r<K»' + l) ... (v) 



n v v 



The nearest whole integer to p(m+l)/w accordingly gives 

 ns the modal value, or maximum frequency to be expected in 

 second trials. This result is wholly independent of the values 

 of p, q, m, and n. 



Let us write p = p/n, q = q/n so that p + q = 1. Then 

 p and q are the chances calculated from the first trial, and 

 the modal or most probable value on the second trial is the 

 nearest whole integer to (m + iyp. 



The value of the mean of the hypergeometrical series is 

 given in Eqn. (8) of the paper cited. Measured from r = 

 we find : 



Mean = —^p^ = m P + ~^ 1 (*-?), • • 00 



whereas from the same origin 



Mode = greatest integer in mp-\-p. . . (vii.) 



Hence for mode and mean to agree — the first condition 

 for an approach to a Gaussian curve — it is necessary that p 

 should not be very small, and further m/(n + 2) should be 

 small. In other words, characters which occur in a very 

 small percentage of the population, or when a relative large 

 second sample is taken, cannot be expected to give on further 

 sampling a mean simply determined by the experience of the 

 first sample. 



As illustration, let us take the following : 2 p. c. of a 

 certain population, on the basis of a sample of 100, are found 

 to.be suffering from malaria. The mean value to be expected 

 on the basis of this sample in future samples of 100 is 2*94 

 or nearly 3 and not 2 p. c. If the first sample were 1000, 

 future samples of 100 might be expected to have an average 

 percentage of 2*10 p. c. For both cases of first sampling the 

 most probable frequency in second samples would be 2 p. c. 

 But it must be remembered that in dealing with the probable 

 error, ice generally take the probable error of the mean and 

 not of the mode. Hence in rare characteristics considerable 

 caution must clearly be used in tacitly replacing the modes 



