in order to eliminate any variation associated 

 with age. Samples from the four Yellowstone 

 Lake tributaries were combined in the compar- 

 isons because differences among spawning runs 

 in spotting and hyoid teeth were not statistically 

 significant. The stocks in Yellowstone Lake 

 tributaries were thoroughly mixed annually by 

 hatching operations which continued until 1953. 

 (Benson and Bulkley, 1962). Eggs were collected 

 from fish in the spawning runs entering these 

 and other tributaries of Yellowstone Lake, and 

 eggs and fry were planted back into the streams. 

 The eggs were not segregated by stream within 

 the hatchery. This procedure caused consider- 

 able mixing of stocks among the various spawning 

 runs in the latce. Cope (1957) found sufficient 

 differences in size of fish and eggs, time of 

 migration, etc., to describe them as distinct 

 races, but significant differences in spotting 

 and hyoid teeth have apparently not had time to 

 develop . 



Hubbs and Hubbs (1953) describe a simple 

 method for determining statistically significant 

 differences in samples when illustratedas in 

 figure 2. This method of working directly from 

 the graphs could not be used here. A careful 

 scrutiny of data in figure 2 will reveal that the 

 variances (S^) for the samples of mature fish 

 from different areas appear to be related to the 

 size of the mean (x) and hence cannot be pooled 

 for testing differences. The fact that the vari- 

 ances are distinctly different indicates that the 

 populations are quite different. The frequency 

 distribution in certain samples (e.g. , Pelican 

 Creek hyoid teeth) is also skewed and deviates 

 from a normal distribution. All meristic counts 

 were converted to base -10 logarithms to fulfill 

 the requirements of equal variance and normal 

 distribution so that "t" tests could be made of 

 differences between samples . 



Although a statistically significant differ- 

 ence among populations is of interest, the amount 

 of change (overlap) that has occurred since isola- 

 tion is the factor of primary importance. The 

 amount of overlap (p) as defined by Royce(1957) 

 is the probability of misclassifying an individual 

 from one of two samples by use of the character 

 in question. A p value of 0.5 indicates complete 

 overlap of two samples, whereas a value of 0.0 

 indicates complete separation. Calculation of 

 p is illustrated in appendix A. 



Another useful measurement is the per- 

 centage of one sample which might belong to 

 another sample; this percentage has been assigned 

 tne Greek letter omega (Q) by Royce. A. condition 

 of complete overlap of two populations is indicated 

 by ano value of 100 percent. With equal sample 

 size and equal variance, Q = 200 p. Our samples 

 are not of equal size, but value of p and approx- 

 imate values of il for populations compared in 

 this study are presented in table 1. Some sam- 

 ples are obviously very divergent; for example, 

 the percentage of the Sedge Creek sample that 

 could theoretically belong to the Creston hatchery 

 stock is 1 .20 percent on the basis of spotting 

 counts and 2 .50 percent on the basis of hyoid 

 teeth counts. In contrast, Creston and Yellow- 

 stone Lake samples were quite similar, with an 

 overlap in spotting of 81.6 percent and in hyoid 

 teeth counts of 35.0 percent. 



Spotting 



Spotting differences among samples were 

 apparent not only in number of spots below the 

 lateral line, which is indicative cf the amount of 

 spotting on the entire body, but also in distribu- 

 tion of spots . The relative number of spots on 

 the anterior part of the body varied among in- 

 dividual fish. To illustrate this difference 

 among populations , the number of spots anterior 

 to a line drawn perpendicularly from the in- 

 sertion of the ventral fin to the lateral line was 

 compared with the total number of spots on the 

 side of the body below the lateral line (table 2). 

 In the Sedge Creek collection an average of less 

 than 4 percent of the total spots below the lateral 

 line were on the anterior part of the body. Spot- 

 ting was more evenly distributed on fish from 

 Bear Creek, Yellowstone Lake, and Creston. 

 The Creston fish had the largest percentage of 

 spots anterior (21.24), indicating a relatively 

 even distribution of spots on the average fish in 

 this population. 



The regression of the number of spots 

 anterior to the ventral fins to the total number 

 of spots was determined to see if another 

 measure of the amount of overlap between samples 

 could be obtained. Assumptions necessary to de- 

 termine statistical differences and to calculate 

 the degree of overlap in spotting distribution 

 were difficult to fulfill. Logarithmic transforma- 



