The fundamental difference between 

 these two methods of computation is one of com- 

 paring samples or comparing estimates of 

 population parameters. In Ginsburg's method 

 a simple comparison of samples is made and 

 no precise mathematical inferences about the 

 populations are possible. In the other method 

 it is assumed that the samples have been ob- 

 tained from normal distributions and precise 

 mathematical inferences about the populations 

 may be made. In addition it becomes possible 

 to use the immense background of mathematical 

 and statistical experience which has been con- 

 cerned with normal populations . 



We prefer to change slightly formula (4) 

 to a form which will be a starting point for the 

 generalization which follows. It becomes 



x l" x 2 



= X! + x 2 



will have equal 



D- KH 



(5) 



W 



in which ~ is the pooled within- sample standard 

 deviation computed from the pooled variance of 

 the two samples and D is the distance between 

 the means in the standard measure of statistics, 

 i.e., in units of the standard deviation. It will 

 be obvious that n 



CD.^^r-. (6) 



The difference arises because in (5) the average 

 standard deviation is computed from the pooled 

 variance, which is the usual statistical way of 

 estimating the standard deviation in the popula- 

 tion. 



A graphic presentation of the normal 

 frequency distribution will illustrate this concept. 

 If we simplify the illustration by assuming large 

 sample size and equal variance—' with D = 2.5, 

 then the plotted normal frequency distributions 

 are as in figure 1 . The area of overlap is shaded 

 It may be seen that an individual from one of the 



two samples with a character of size 



2/ It should be noted that it is not necessary to 

 have equal sample size for the method to be 

 valid, because the means and variances are 

 practically independent of sample size . It 

 will be shown later that moderate departures 

 from equal variance are also permissible. 



probability of being correctly classified on the 

 basis of the character . As X becomes greater 

 the probability increases that the individual 

 "belongs" to population 2 and as X becomes 

 smaller the probability increases that the in- 

 dividual "belongs" to group 1. Since, in any 

 normal distribution, X may be infinitely large 

 or small the probability of correct classification 

 never reaches 1 but soon approaches close 

 enough for practical purposes. 



Instead of considering an individual hav- 

 ing a particular character X_ we may consider 

 all individuals in which X^ x l + *2 



3/ 2 

 or those in areas 2A and IB- in figure 1 . It 



will be obvious that the area of IB is equal to 

 2B with our simplifying assumptions, and hence 

 we may use the relative areas 2 A and 2B to de- 

 termine a probability. The area corresponding 

 to 2A may be obtained from a table of the normal 

 probability integral, such as Pearson's (1948, 

 table 2), if the samples are large, i.e., 

 n l + n 2 ^ ^0 ■ In tms table is given the area of 

 half of the normal curve plus the space from the 

 mean to the argument, x = _D . We shall call 



2 

 this relative area the probability 1 - p. 



If the samples are small and greater ac- 

 curacy is desired, it is necessary to use a table 

 of t, such as table 3 in Fisher and Yates (1948). 

 When this table is entered with the arguments 



t = D_ and nj + n 2 = n, a probability P t 

 is found, which refers to both tails of the dis- 

 tribution. Our p defined above may be found by 



P = P t 



(7) 



Despite its increasing use, considerable 

 confusion exists regarding the name of the 

 phenomenon and the exact meaning of the prob- 

 ability figure. Mayr et al. (1953:146) call 1 - p 

 3/ Area 2A includes all of the area of distribu- 

 tion 2 greater than x i + x 2 , area IB all of 



2~ 

 the area of distribution 1 greater than X I + x 2 . 



2 



Hence, 2A and IB are overlapping. 



10 



