no. 2.] PSYCHOLOGICAL EXAMINING IN THE UNITED STATES ARMY. 



455 



this ratio for different tests in any one comparison of the two groups may be taken to indicate 

 roughly the relative values of the tests for differentiation between the two groups. This method 

 has certain limitations, however, which become very acute when studying tests which are so 

 difficult that a large proportion of those tested fail to make any score. In such a case calculation 

 of the mean and the standard deviation of the distribution mathematically implies a situation 

 that psychologically is far from the truth, because zero, mathematically, is one unit less than 

 one, while a score of zero for a test may mean one arbitrary unit of ability less than that repre- 

 sented by a score of 1, or it may mean any number of such units. The comparison of two tests 

 by this method where one test covers a much greater range of ability than any other is at best 

 very unsafe. 



A second method which avoids this difficulty to a considerable extent is an application of 

 Pearson's criterion 1 for significance of difference between two distributions. This method 

 makes no assumption regarding the type of distribution under consideration. It takes into 

 account the undistributed zero cases in a much more adequate manner than the first method 

 described above. The only assumption made by this process of comparing distributions is that 

 the deviations of the proportional frequency in any class interval, due to random sampling, 

 from the true proportional frequency in the sampled population, form a normal or Guassian 

 distribution. 



In applying this method we calculate a quantity, x 2 , from the formula 



X 2 



= mn.s\\m~n) 



rn + n 



c, 



h 



where M and N are the total numbers of cases in the distributions and m and n are frequencies 

 in corresponding class intervals. 



The quantity x 2 thus obtained may be used in two ways: (1) By means of the formula — 



_f "I* 



M + N+x 2 J 



a coefficient of mean square contingency might be obtained. This would be rather unsatis- 

 factory for the reason that the proper class-index corrections could not be made even approxi- 

 mately. (2) A measure of the probability that the observed divergence would be equaled or 

 excelled hi two samples of the same size drawn from a single population, may be obtained from 

 Elderton's tables. 2 This is, in principle, the procedure followed. 



Before applying the x 2 test for divergence all score distributions for all groups ha/e been 

 reclassified as indicated by the following table: 



ALPHA TESTS. 



BETA TESTS. 



'Biometrika, vol. 8, p. 2502.; vol. 10, p. 85fl. 



» W. P. Elderton, Biometrika, vol. 1, pp. 155-163. 



