(for example area 2 A in fig. 1) the joint non- 

 overlapping; Ginsburg (1938:255) calls it a 

 measure of divergence; Mather and Dobzhansky 

 (1939:15) use p and consider it the frequency 

 of misclassification. To call it the amount of 

 overlap is troublesome because as 



and 



1 - p £ 0.5. 



A condition of complete overlap is indicated by 



p = 0.5. 



It is probably simplest to consider p as the 

 probability of misclassifying an individual from 

 one of the two samples by use of the character 

 in question. The value of 1 - p is the probabil- 

 ity of correct classification and varies from .5 

 or 50 percent with complete overlap to nearly 

 1 or 100 percent with no overlap . 



(Edwards 1954) . Ginsburg (1938), whose concept 

 of overlap p* was similar to our p, used the 

 overlap of frequency distributions to show a con- 

 tinuous gradation from p = . 5 in two samples of 

 the same species to no overlap in two separate 

 but closely related species of the same genus. 

 He suggested that a p = . 1 would commonly be 

 found between species, .3 between sub-species, 

 .4 between races, and up to .5 between varieties. 

 Mayr et al. (1953) suggest that the conventional 

 level of sub-specific difference is p = . 1, 

 U L = 20 percent; that is, a difference between 

 the means of 2.56 times the average standard 

 deviation (D = 2.56). Hubbs and Hubbs (1953:56) 

 state that the more usually accepted sub -specific 

 difference amongst fishes is equivalent to p = .25, 

 c/i = 50 percent. It is evident that the amount 

 of overlap is a continuous positive distribution 

 and the level used for a decision must be chosen 

 arbitrarily or in association with other factors. 



There is, however, a need for another 

 concept of overlap. This we define as the pro- 

 portion of one sample which might "belong" to 

 another specified sample . We consider it to be 

 the entire area under one curve which is also 

 included in the other. We assign to this concept 

 the Greek letter omega <Vland express it as a 

 percentage. It will be obvious that under the re- 

 strictions of equal sample size and equal variance 



( fi g- 2 > Jl = 200p 



This concept has an obvious advantage 

 over p in some applications because a condition 

 of complete overlap is indicated by 100 percent; 

 in terms of our definition all of one sample might 

 belong to the other. It also is a complete de- 

 parture from the concept of the probability of 

 misclassifying an individual (which with no in- 

 formation at all would be misclassified only half 

 of the time) to the concept of the proportion of a 

 sample which might belong to another group. 

 From the samples, if they are representative, 

 we may then make inferences about the populations 



USE OF OVERLAP 



A concept of overlap has been used along 

 with other data on geography, ecology, or qualita- 

 tive characters in the definition of species and 

 sub-species, but there is no general agreement 

 on the meaning of different amounts of overlap 



Comparisons of overlap computed frou. 

 the estimates of the population parameters 

 (formula 5) with overlap computed directly from 

 the samples (formula 3) indicate that when large 

 normally distributed samples are considered the 

 two methods yield nearly identical results. To 

 show this we shall compare p* and p using 

 data selected from published material to include 

 a range of values of D and to include some 

 material which we shall refer to later in this 

 paper . 



An example of overlap near the borderline 

 of subspecific differentiations is given by Gins- 

 burg (1938:269) as the frequency distribution 

 (table 1) of the articulate dorsal rays in the weak- 

 fish Cynoscion regalis regalis from the Atlantic 

 coast of the U.S. and Cynoscion regalis arenarius 

 from the Gulf of Mexico. A summation of the 

 smaller percentages (formula 3) provides an 

 estimate of overlapping of 



p* = 2.63 + 13.68+ 18.49+ .84 = 17.8%. 

 2 



Substituting the values of the means and pooled 

 standard deviation in formula 5 we have 



D 



27.258 - 25.874 = 1.683 



D 



2 



.8221 

 = .841 



P = 



.20 . 



12 



