ESTIMATION OF SIZE OF ANIMAL POPULATIONS 

 BY MARKING EXPERIMENTS 



By MILNER B. SCHAEFER, Fishery Research Biologist 



Determination of population numbers is basic 

 to studies of changes in populations of animals 

 and of the causes of the changes, such as the 

 effects of fishing on a population of fishes. For 

 many animals this cannot be accomplished by 

 direct enumeration, and recourse must be had to 

 indirect methods. One technique which has 

 been employed in the study of fishes, and other 

 organisms as well, and wliich is still in course of 

 development, is the use of marked members to 

 estimate population numbers. 



SIMPLE CASE 

 THE PROBLEM 



The simplest case vnth which we have to deal, 

 and which can be applied to many fish populations, 

 is where we have a population containing A'' 

 members (unknown) which is known to contain T 

 marked members and U=N—T unmarked, and 

 where we have drawTi a single representative 

 sample of n members containing t marked and 

 u = n—t umnarked. The term "representative" is 

 used here to mean that the character estimated 

 from the saniple will have a mean value in repeated 

 samples equal to the population value. This cor- 

 responds with the commonly accepted sense of the 

 term, and also with its usage by Neyman (19.34). 

 A simple random sample of the population is repre- 

 sentative, but so also may be various others. 



The problem of estimating N consists in making 

 such an estimate given T and the sample values 

 /), t, and u. The usual basis of procedure is to 



accept 7p=-r intuitively and to estimate A^^by the 



equation 



N-- 



nT 



(1) 



If, for example, we know there are 100 marked 

 members in the population, and a samph? of 500 



contains 50 marked members, we would estimate 

 the population by this equation to be 



N-- 



500X100 

 50 



1,000 



968620°— 51 



This method has been employed by a consider- 

 able number of mvostigators during the last two 

 decades to estimate the populations of various or- 

 ganisms. The method is much older than this, 

 however, having been employed as early as 178.3 

 by the famous French mathematician and scientist 

 Laplace in estunating the human population of 

 France. Laplace gave considerable attention to 

 the theoretical problem of the error involved in 

 employing tliis method. This problem attracted 

 the attention of another famous statistician. 

 Karl Pearson, who published an analysis of it in 

 1928. Later workers in various branches of 

 zoology seem to have overlooked Pearson's work 

 and also that of their zoological contemporaries. 

 They have apparently often "rediscovered" the 

 same method, but have in the main given little or 

 no attention to the problem of the accuracy of the 

 resulting estimate. 



Laplace determined from a sample the ratio of 



births in a year to the population producing those 



births, and then ascertained the number of births 



in a year in each urban and rural district of 



France; by multiplying the number of births by 



th{> ratio of pojiulation to births determined from 



the sample, he arrived at an estunate of the total 



population. Laplace was led to consider also the 



error inherent in his estunate. This problem, as 



restated by Pearson (1928), but using my notation, 



is as follows: "A population of unknown size A^is 



known to contain T affected or marked niembei-s. 



It is desired to ascertain — on the hypothesis of 



inverse probabilities — a measure of the error 



T 

 introduced by estimating N to ha n —> where t is 



191 



