bis 



clxxxvi Tables for Statisticians and Biometrwians [XXXVI 



A limiting case of this problem occurs when n 2 -*-vc but Wj remains finite 

 (or vice versa). The second sample then becomes indistinguishable from an infinite 

 population, and the problem is reduced to that discussed in the previous section, 

 namely that of testing the hypothesis that a sample of n (= n^) with mean x (= x-^) 

 and standard deviation s(=i) has been drawn from a normal population with 

 mean a (= ac 2 ) and standard deviation cr (= s 2 ). The values of P A found accurately 

 by quadrature in this earlier problem provided therefore a method of checking the 

 marginal columns (i.e. those for % or n 2 = ) of Tables 



Illustration. The following figures represent the values of the Cephalic Index 

 calculated for two series of 10 human skulls. 



Sample I. 74'1; 777; 744; 74"0; 73'8; 72"2; 75'2; 78'2; 771; 78-4; 

 whence ??x = 10; Xi = 75'51 ; i = 2'059. 



Sample II. 66'7; 69'4; 67'8; 73'2; 79'3; 80'7; 64'9; 82'2; 72-4; 781 ; 

 whence n 2 = 10; x 2 = 73'47; s 2 - 5'94>2. 



We may now test the hypothesis that the two samples have been randomly 

 drawn from the same population, only making the assumption that the distribution 

 of cephalic index does not differ so much from normality as to invalidate the test. 



On combining the two samples it is found that S Q = 4'562, and hence 



X = (siV*o 2 ) 10 = '00493. 



The table shows that for n 1 =n 2 = 10 this value corresponds closely to P A = '01; 

 that is to say only once in a hundred times should we expect the criterion, X, to 

 have as low or a lower value were the hypothesis tested true. We should therefore 

 conclude that it was very unlikely that the two series of skulls were random selec- 

 tions from the same population. 



TABLE XXX VII to . 



Further Tests of Normality. Table showing the 5 / and 1 / probability limits 

 for vfii and j3 2 in random samples drawn from a Normal Population. (E. S. Pearson, 

 Biometrika, Vol. xxn. p. 248.) 



The table shows for samples of various sizes the upper and lower limits of V/3i 

 and of /3 2 which will be exceeded in (a) 5 / , and (6) 1 / of random samples. The 

 distribution of v/Si is symmetrical about zero so that the corresponding upper and 

 lower limits are of the same magnitude but with opposite signs. The table may 

 also be entered with /3j; in this case positive and negative values of V/3 X will be 

 clubbed together and the limits will therefore be those exceeded in 10 % an d 2 / c 

 of samples. Since in samples from a normal population \//3i and /3 2 are completely 

 uncorrelated, we have two separate and independent tests. 



The table has been based upon the known values of the first four rnoment- 

 eoefficients of the sampling distributions of V/Sj and /3 2 , and it has then been as- 



