M 



b2 



be the mean in Region B of the character 

 in the second year. 



Let there be 



N, fish in Region B in the second year that 

 were in Region A in the first year 



N fish in Region B in the second year that 

 2 were in Region B in the first year . 



The percentage of fish in Region B in the 

 second year that came from Region A since the 

 first year, then, is P in 



Nl 



W P ■ N 1+ N 2 



(100) 



The sum over all fish of the value for the char- 

 acter in question in Region B in the second year 

 is 



N l + N. 



b2i 



(N 



+ N 2 )M b2 



and is also equivalent to 

 (2) (N 1 + N 2 )M b2 



NjM al + N 2 M bl 



Rearranging the expression (2) above, 

 one finds that 



P = ^2 - M bl (100) 



(3) 



M 



al - 



Ttf 



bl 



So if the three means are Known (the mean 

 of fish in Region B in the second year, of fish in 

 Region B in the first year, and of fish in Region 

 A in the first year), the percentage of fish in the 

 second year in Region B that came from Region 

 A since the first year is uniquely determined. 



Two conditions are necessary to the 

 above conclusions: 



1 . The means 

 change with time and, 



M and M, do not 



2. The fish that move, N L in number, 

 must have the same mean as those remaining 

 in Region A, namely M ■ 



These two conditions are probably satis- 

 fied in the case of meristic characters of adult 

 fish, but probably are not for many morpho- 



metric characters. One morphometric char- 

 acter that normally would satisfy them, however, 

 is the "calculated" length of a fish at some given 

 age . An example given later will be based on 

 such data . 



GENERAL APPLICATION; SAMPLING 

 AND CONFIDENCE LIMITS 



Rarely will a situation as simple as the 

 example given above be found in practice. Usu- 

 ally the biggest problem will be to describe the 

 means of populations in all adjacent areas in- 

 habited by the fish in question. The difficulty 

 then will be one of deciding from which region 

 the movement originated. A change in the mean 

 between time periods at a given region proves 

 that some fish have entered that region. To find 

 out what proportion are immigrants, one must 

 know the mean value of those that entered. 



If adjacent regions contained fish pre- 

 viously that could, by immigrating, also have 

 altered the observed mean, then the amounts of 

 each, taken one at a time, necessary to have 

 accomplished the change, can be computed. It 

 remains a matter of judgment from other evidence 

 as to which region or regions affected the local 

 mean 



In practice, the true mean value of a 

 character is never known, since only a part of the 

 population is observed. The mean of the samples 

 will approach the true mean to the degree that 

 the samples are representative of the population. 

 If the definition of populations is restricted to 

 only those regions that are uniformly or repre- 

 sentatively sampled, apprehension and bias will 

 be avoided. Within representatively sampled 

 areas, larger samples will be more accurate in 

 describing the population. If some areas known 

 to contain fish are not sampled representatively, 

 it is better to restrict conclusions to only those 

 that are sampled representatively, rather than 

 to try to extend them to all areas, sampled or 

 not. While the former results will be limited, 

 the reason will be incomplete coverage and not 

 improper sampling. The areas not sampled will 

 then stand out as those in which future work 

 should be done . 



The distribution of most characters will 

 not be normal, but will be skewed with a definite 



30 





