x x x v i x x x vii"-"] ii,tru,i,,.-t;, m 



index within a .single rare lay within tin- ran^e ~1 ."< I... !<), u, .-..uld |,, l \,. mi |, 

 upon the c-tcst used in (ii). We could say I hat the sample jx,iuL in iJia^rum 2, 

 |i. rl\\\iii, must. lie Mum-when- ,,;i Hi,. Im,- : = '2(>.S which join* the origin, J/=0tf, 

 to the oorreflponding division of the radiating r-scale given in the right-hand margin 

 <>l I he diagram. Tin; exact, position depends <>n the value given to a; but if we may 

 inter that <r < 4'0 we can say that it is very unlikely that the sample j*>int will 

 lie further to the left along this 2-line than the point for which 



S = 8 /<r = 5-942/4-0 = 1-485. 



To the right of this point the contours cutting the lino have k > -OS and there- 

 fore, as shown by the table, P A < -08. We should therefore argue that the hypo- 

 thesis as to the origin of the sample is less likely to be true than the value of 

 P z obtained in (ii) would suggest. The knowledge of the probable limits of a, drawn 

 from wider experience, has enabled us to apply a more searching test, although still 

 not so exact as that which could be applied in (i). 



TABLES XXXVII -*. 



The Case of Two Samples. (E. S. Pearson and J. Neyrnan, "On the Problem of 

 Two Samples," Bulletin de I' Academic Polonaise des Sciences et des Lettres, pp. 73 

 96, Cracovie, 1930.) 



A further problem is that of judging the probability that two samples in which 

 the individuals have been measured for a single character, x, have been drawn from 

 the same population. In the test to be described it is assumed that the character 

 in the population sampled is approximately normally distributed; the first sample is 

 of size HI with a mean x and standard deviation s\, the second of size n t with 

 the corresponding quantities a! 2 and s z . Then it would be possible to consider 

 separately the significance of the difference between xi and x t) and between i and 

 s a , but Pearson and Neyman, in the paper referred to above, have again suggested 

 the use of a single test based upon the principle of likelihood. 



In this case the appropriate criterion is 



where * is the standard deviation obtained by combining together the HI + n t vari- 

 ables of the two samples. The test consists in finding the chance, P A , that a value 

 of X smaller than that calculated from the observations would occur in drawing two 

 independent random samples from a common normal population. The distribution 

 of \ is independent of the mean and standard deviation of the population sampled. 

 Although its exact form has not been determined the moment-coefficients of this 

 distribution can be obtained; making use of these coefficients and certain other 

 considerations the writers provided two approximate tables giving for a variety of 

 combinations of Tij and w a the valu2s of X corresponding to P A = '05 and "01. It is 

 believed that the values of X tabled are not in error by more than two units in the 

 4th decimal place. 



i;. II. <"' 



