Resource Library

A hands-on activity using the capture-recapture method to estimate the number of M&M’s^TM in a population The activity was described in G. D. Bisbee and D. M. Conway, “Studying proportions using the capture-recapture method”, Mathematics Teacher, 92 (3) (1999), 215-218.

Summary: Scientists use the capture-recapture method as a tool to estimate population size. Animals are captured, tagged, and then released back into the population. Later, a sample is captured and a proportion used to estimate population size.

Specifics: Let us say that we sample a beetle population of unknown size. We capture and mark ten of those beetles with a spot of India ink, then return them to the population and give them time to mix in with the population. We then recapture another sample consisting of eight beetles, one of which was previously marked. We substitute the numbers into the foregoing proportion to estimate the population size, getting 1/8 = 10/(Pop size). Solving for the Pop size gives us an estimated population of eighty beetles. Students are, predictably, less than enthusiastic about having to handle the creepy-crawly critters so this activity uses a population of M&M’s of unknown size to estimate. Each team of two to four students receives some M&M in a paper cup, which is covered on top with crumpled paper towels. The students “tag” the M&M’s from a random sample and then, after mixing them back in, sample again to estimate the number in the cup (they can later check how far off their estimates  were and compare to other teams).

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

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 Goldfiish^TM 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

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9639.1996.tb00882.x

Specifics: To illustrate the capture-recapture method in the classroom, two different varieties of Pepperidge Farm Goldfish^TM 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)

A cartoon suitable for use in teaching about scatterplots and correlation. The cartoon is number 388 (Feb, 2008) from the webcomic series at xkcd.com created by Randall Munroe. Free to use in the classroom and on course web sites under a creative commons attribution-non-commercial 2.5 license.

A joke to use in presentations about the importance of control and replication in experimentation.  The joke was written by Larry Lesser (The University of Texas at El Paso) and Dennis Pearl (Penn State University) in March 2020.

A cartoon to initiate discussions about how the correlation is a unitless number that does not change with changes in the units of the variables involved.  The cartoon was created in February 2020 by British caetoonist John Landers based on an idea by Dennis Pearl (Penn State) and Larry Lesser (Univ of Texas at El Paso). An outline of a lesson plan for the use of the cartoon is given in a 2020 Teaching Statistics article by Dennis Pearl and Larry Lesser.

A joke about the meaning of an inequality symbol like ≤ written in February 2020 by Larry Lesser from The University of Texas at El Paso and Dennis Pearl from Penn State University.

A joke to help in discussions of the value of random assignment in experiments and in discussing pedagogical options.  The joke was written by Larry Lesser from The University of Texas at El Paso in February 2020.

A joke to help discuss how random assignment is an unbiased experimental method.  The joke was written by Dennis Pearl from Penn State University in February 2020.

A poem written in 2019 by Larry Lesser from The University of Texas at El Paso to discuss systematic sampling.  Students should be familiar with the lyric being sampled from (though you could provide it to make sure) and verify that the systematic sample involved sampling every third word and starting with the lyric’s first word. Students could create their own poems with different systematic samples (or different text to sample from).  The poem is part of a collection of 8 poems published with commentary in the January 2020 issue of Journal of Humanistic Mathematics.

A poem written in 2019 by Larry Lesser from The University of Texas at El Paso to discuss the normal distribution and its percentiles.  Students could first be shown a copy of the National Center for Health Statistics growth chart graph paper so they will appreciate the details of the poem. And after reading or hearing the poem, students could verify the detail that the 40th and 60th percentiles are half a standard deviation apart. The poem is part of a collection of 8 poems published with commentary in the January 2020 issue of Journal of Humanistic Mathematics.