198 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



(1943) has done the same in estimating the white- 

 bass population of an Ohio lake. 



Schumacher and Eschmoyer (1943) have de- 

 vised an estimate of A'^ from repeated samplings 

 which is different from that of Schnabel. They 

 assume that the weight, or value, of each sample 

 is proportional to the nimiber of fish in the sample. 

 Under this assumption, an estimate of A^ is arrived 

 at by minimizing the sum of the squares of the 



T 

 weighted discrepancies of the ^' from their esti- 



mates 



This leads to the formula 



A^^ 



1=1 



i=l 



(20) 



which is applied by these authors to the estimation 

 of fish populations of a pond in Tennessee. 



These authors have also derived an expression 

 for the sampling error of A'^. They take as the 

 standard error of A'^ the square root of 



JVV_ 



(21) 



where 



-^-iLSi^ivS^'^'J 



In the last formula I have corrected a typo- 

 graphical error which appears in the original paper 

 (formula 3, page 234) and which Professor Schu- 

 macher has kindly pointed out in a private 

 communication. 



In table 2 is recapitulated a numerical example 

 from Schumacher and Eschmeyer (1943), pertain- 

 ing to the estimation of a population of bullheads 

 in Yellow Creek Pond, Tennessee. Substituting 

 the appropriate sums from this table in formulae 

 [20] and [21], we obtain the following estimates 

 for A'^ and its standard error (o-at): 



A^= 



49,398,907 



35,121 



1,412 



14V 



27.992263- 



35,121' 

 1412 , 



= 0.222S 



afi 



^^ 



412)3(0.2228)^ 



35,121 



:134 



Table 2. — Schumacher and Eschmeyer' s method oj com- 

 puting a fish ■population by repeated sampling and marking 



[Data from table 2 of Schumacher and Eschmeyer (1943)] 



Kicker (1945b) has investigated the relative 

 efficiency of Schumacher's estimate (20) and 

 Schnabel's formula (18). He states: 



From an exchange of letters with Dr. Schumacher it 



appears that the efficiency of this expression is at a maxi- 



T 

 mum when -^ is equal to 0.5, whereas Schnabel's second, or 



approximate formula becomes most efficient as (T/N) —►0, 

 and the two formulae are of equal efficiency when 

 T/N=0.2o. Consequently Schnabel's form will ordinarily 

 be best, since the value of T/N rises gradually from a very 

 small initial magnitude, and, except on quite small bodies 

 of water, will not often exceed 0.25 even when the experi- 

 ment comes to an end. Of course Schnabel's long formula, 

 carried to several terms, can always be used if the best 

 possible estimate is desired; but the labor of computation 

 will rarely be warranted, considering the magnitude of the 

 sampling and probably systematic errors in such experi- 

 ments. 



Krumholz (1944) has made a practical check of 

 the accuracy of estimation of a fish population by 

 repeated sampling, marking by clipped fins, and 

 the application of Schnabel's formula (18). He 

 computed the population of fish over 45 milli- 

 meters in length in the north basin of Twin Lake, 

 Mich., in this manner and then poisoned the area 

 with rotenone and counted the fish poulation 

 directly. He concluded: 



The estimate from netting operations was very close to 

 that obtained by poisoning in this first check on the fin- 

 clipping method for estimating fish populations. Further 

 studies of this type are needed to prove definitely the 

 accuracy of the method. . . . Other checks of this method 

 will be made when conditions permit. 



