278 



For the computation of these ratios, I have used, with two 

 exceptions to be presently stated, the thousand Illinois collec- 

 tions most available for these studies, excluding five hundred 

 and forty-four additional collections, which, because of imper- 

 fect data and for various other reasons, are undesirable mate- 

 rial. I find that the species Hadropterus aspro has been taken 

 in 159 of these thousand collections, which ratio of average 

 frequency may be expressed by the fraction .159 ; and that the 

 species Hadropterus phoxocephalus has been taken 85 times, 

 which gives a frequency ratio of .085. That is, in any thousand 

 similar miscellaneous collections distributed over the area in- 

 habited by these species we may, according to these data, ex- 

 pect to get the first species 159 times and the second species 85 

 times; and the chance that any single collection will contain 

 the first species is .159, and that it will contain the second species 

 is .085. From this it follows that the chance that the two species 

 will occur together in any single collection of the thousand, 

 provided that the distribution of each is arbitrary and acci- 

 dental with reference to that of the other, is the product of 

 these fractions; and the probable number of chance joint occur- 

 rences of the two species in the thousand collections is, of 

 course, a thousand times that product, or 13.515. As a matter 

 of fact, however, these two species were found together in my 

 collections 40 times instead of approximately 13.5 times, or three 

 times as frequently as there was reason to expect provided 

 that there had been no associative bond between the spe- 

 cies. This number 3, indicative of the frequency of actual asso- 

 ciation as compared with the chance or accidental, is the coeffi- 

 cient of association for these two species. If the numbers of pre- 

 sumable and actual joint occurrences were equal, this coefficient 

 would evidently be 1, in which case no associative bond would 

 be indicated; and if it were notably less than 1, we should have 

 some reason to suppose that the two species belonged to dif- 

 ferent ecological groups — that their ecological affinities and 

 relationships tended to separate them instead of to bring them 

 together. 



