NOTE 

 In the course of the present memoir it is shewn that 



tP'hk oO'q o^"* 



It may then be asked why not measure the improbability of <^ exceeding ^cr^ by 

 the ordinary theory of the probability integral ? We know that (^^ is by its essence 

 positive and we should probably have to take ^ positive also. The distribution of ^^ 

 for samples from a population in which ^^ is zero, has not been studied ; we know the 

 mean value, ^, for such a population, but we do not know the frequency of ^ in terms 

 of jcr^ and we have no reason to suppose that it can be expressed by aid of a Gaussian 

 distribution in terms of the constant jO-^. Similar remarks apply to Qj,p'q, ^"^^ '"'hkU'^Kk^ 

 the latter will clearly be a limited range frequency, and a normal curve distribution 

 especially for large values of r^^j will certainly be inadmissible. As a matter of fact 

 the probability that a sample occurs with a ^ over a given value is 



which connotes a frequency distribution 



and this is not a normal distribution*. The above relations between x. ^a*. Q, ^ 

 and their standard deviations are interesting, but they do not provide us, by aid of 

 a Gaussian probabiHty table, with the requisite " equality in probability," which we 

 are seeking in this memoir. There is very grave danger when, having found in some 

 case the value of the standard deviation of a statistical quantity, we then assume that 

 for this case the Gaussian distribution must apply, and deduce thereby a measure of 

 the significance of the quantity, e.g. estimate the significance of Q from a knowledge 



of ^(Tq^. 



* It also is only a close approximation; we have actually assumed the Gaussian may be used to 

 describe the frequency binomials, but it is a far closer approximation than using a Gaussian to describe 

 the frequency of Xi or of C/oO-q. 



FEB 18 1946 



PEPT. f •!- 

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