Food: Capture-Recapture with Goldfish

Summary: This article describes the capture-recapture method of estimating the size of a population of fish in a pond and illustrates it with both a “hands-on” classroom activity using Pepperidge Farm GoldfiishTM crackers and a computer simulation that investigates two different estimators of the population size.  The activity was described in R. W. Johnson, “How many fish are in the pond?,”Teaching Statistics, 18 (1) (1996), 2-5

Specifics: To illustrate the capture-recapture method in the classroom, two different varieties of Pepperidge Farm GoldfishTM crackers are used. The instructor places all of the Goldfish from a full bag of the original variety in a bowl to correspond to the initial state of the pond (the instructor should have previously counted the true number from the bag, which turned out to be 323 in the paper’s example). Students then captured c = 50 of these fish and replaced them with 50 Goldfish of a flavored variety of different color. After mixing the contents of the bowl, t=6 ‘tagged’ fish - fish of the flavored variety were found in a recaptured sample size of r = 41, giving the estimate cr/t= 341. This used the maximum likelihood (ML method. To examine the behavior of the MLE the capture-recapture ML  method is repeated 1000 times using a computer simulation. The distribution of the results will be heavily skewed since the MLE is quite biased (in fact, since there is positive probability that t = 0, the MLE has an infinite expectation). The simulation is then redone using Seber’s biased-corrected estimate = [(c+1)(r+1)/(t+1)] – 1.  After the true value of the population size is revealed by the instructor, students see that the average of the 1000 new simulations show that the biased-corrected version is indeed closer to the truth (and also that the new estimate has less variability).

(Resource photo illustration by Barbara Cohen, 2020; this summary compiled by Bibek Aryal)

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Roger Johnson
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