clxxx Tables for Statisticians and Biometricians [XXXI- -XXXVI 



But no biometrician would admit the absolute ignorance requisite for deducing 

 in this case the " most likely value." The correlation of skull length and breadth 

 has been determined rather vaguely and not on adequate numbers*, and it pro- 

 bably varies from race to race, but it undoubtedly lies between and '6. This 

 a priori knowledge leads on precisely the same logical basis as Bayes' Theorem 

 to the value p = '3480, a result very much closer to previous experience of the 

 mean value, than to the observed result, '50. There are relatively few cases in 

 which some such, even if only vague a priori, experience does not exist, and 

 accordingly in practice an attempt to find p the most reasonable value should be 

 preferred to a determination of p, which must be reserved for cases in which we 

 are forced to distribute our ignorance " equally." 



TABLES XXXV XXXVI. 



To determine the Probability that a Small Sample has been drawn from a 

 Normal Population with a specified Mean and Standard Deviation. ( J. Neymah and 

 E. S. Pearson, Biometrika t Vol. XX A . pp. 235240.) 



Let us suppose there is reason to believe that a sample of n individuals 

 measured for a single character, x, with mean = x and standard deviation = s, has 

 been drawn from some normal population, and that we wish to test the hypothesis 

 that the mean and standard deviation in the population are respectively a and a. 

 It would be possible to consider separately the significance of the difference 

 between x and a and between s and cr, but the use of a single test based on the 

 conception of "likelihood" has been suggested by Neyman and Pearson in the 

 paper referred to above. According to this method of approach the likelihood of 

 the hypothesis tested decreases as the criterion 



c- -%(M* + S*-l) 

 A.O e * (i) 



decreases from unity towards zero, where 



M = (x a)/<r, S = s/o: 



The test, therefore, consists in finding the chance, P A , that a value of X as small as 

 or smaller than that calculated from the given sample would occur in random 

 sampling from the hypothetical population. 



Using the above notation the simultaneous frequency distribution of M and S 

 in samples of n is 



/( M, S) = constant xS n ~*e~ 2 {M '' } (ii), 



and PA is the integral of the frequency taken outside the curve on which X is 

 constant, i.e. upon which 



lOgio A, ^ / ^111^. 



* This was written in 1917, longer series worked out since suggest -37 to -39 as a more suitable 

 value than -30 for p. 



