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FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



in their formula (8), they arrive at an estimate of 

 range of the error due to sampling. They conclude 

 that — 



If we use 2(T„ as our range on either side of the figure 

 obtained ... we are almost certain to include the correct 

 figure for p, since twice the standard deviation on either 

 side of the mean includes 95 percent of a normal distribu- 

 tion curve. 



"While this estimate of the reliability of the 

 population estimate is better than none and, 

 indeed, will give an idea of limits mthin which 

 the population may be expected to fall, it suffers 

 from a lack of precision. The method of compu- 

 tation may be illustrated by the simple numerical 

 example we have employed before. Here P=450, 

 A^=500, and p = 0.90. Formula 9 then yields 



ffp=y- 



9X.1 



500 



.01342 



and 2o-pA^= 13.42. The corresponding values of 

 463.4 and 436.6 may be employed in the second 

 quotient of formula 8 in place of 450 for P to 

 obtain estimates of 927 and 873 for limits of the 

 estimate of "other hares present in precensus 

 period." Corresponding values of total population 

 arc 1,027 and 973. 



Formula 9 gives the standard deviation of 2' in 

 repeated samples of size N from a population of 

 infinite size. Since in the present case the popula- 

 tion is finite, and A'^is large with respect to it, the 

 formula for the standard deviation of p should be 



,_P-Np.q 



"""F-l N 



where P=the number in the population (Cramer 

 1946, p. 516; Kendell 1944, p. 203). Thus Green 

 and Evans' limits for p would tend to be too broad. 

 For the same simple example used above, this 

 formula gives us 



(1000-500) 0.9X0.1 



= .0000901 



" (1000-1) 500 



ap = . 00949 



Green and Evans' estimate also has, however, 

 the same objection that Pearson raised to Laplace's 

 solution, rather important in this instance, that 

 this treats the problem of a further sample from a 

 population in which the value of p is known. 



which is not the same thing as determining the 

 error of the estimate of the population from the 

 single sample available. 



Dice (1941) refers to the paper by Green and 

 Evans and considers a number of practical factors 

 to be taken into account in carrying out the 

 sampling. 



Knut Dahl (1943, pp. 139-143), has applied the 

 method of marked members to enumeration of 

 trout in a lake. In a small lake on the west coast of 

 Norway, of 250,000 square meters, trout were 

 captured by beach seine and marked. During a 

 second fishing 8 to 14 days later he determined the 

 number of marked and unmarked fish captured. 

 From the number of marked fish liberated, divided 

 by the number of marked fish recaptured, he com- 

 puted a "Gjenfangstkvotient" by which the total 

 fish taken in the second fishing was multiplied to 

 obtain the total population. This is, of course, 



T . 

 the same a» formula 1, where y is the "Gjenfangst- 

 kvotient." 



Rieker (1942) mentions the simple case here con- 

 sidered, although he uses a method of repeated 

 tagging and sampling on the stationary pojmla- 

 tions of pond fishes dealt with in his paper. This 

 method will be reviewed subsequently. 



In a later paper, Rieker (1945a) employs for- 

 mula 1, which he calls "the Peterson method," 

 after the Danish investigator who is said to have 

 used it on plaice. Ricker's field procedure is 

 siniilar to that of Green and Evans on hares in 

 that he used the number of fish marked during a 

 precensus period and the mark ratio of a later 

 period. He also writes in regard to the sampling 

 consideration we have discussed earlier in relation 

 to Jackson (1936) that: 



The principle involved here is that if either the marking 

 or the search for recaptured fish is made on only a part of a 

 homogeneous population, the Peterson estimate will still 

 apply to the whole population. If both marking and 

 search are made in only a fraction of the population, the 

 estimate applies to whichever fraction is larger. 



Cagle (1946) employed marked lizards to esti- 

 mate their population on a section of Tinian 

 Island by the employment of the method formu- 

 lated in formula 1. He marked 127 individuals by 

 cUpping their toes and in a sample of 52 found 

 12 marked, jdelding an estimated population of 

 roughly 500 individuals. He did not consider the 

 problem of sampling error. 



