Estimation Principles

A cartoon suitable for use in teaching about confidence intervals and the quality of estimates made by a model. The cartoon is number 2311 (May, 2020) 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.

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)

Summary: A classroom activity using dried split peas exploring the reliability of a basic capture-mark-recapture method of population estimation is described using great whale conservation as a motivating example. The activity was described in C. du Feu, “Having a whale of a time,” Teaching Statistics, 31 (3) (2009), 66-71.

Specifics: The hands-on activity uses dried split peas and involves much larger populations and has two advantages. Firstly, the split-pea populations are too large for any sensible student to contemplate counting the full population. Secondly, unlike Smarties^TM, or M&M’s^TM dried split peas will not suffer loss through eating (so there is a fixed population size to be estimated). Beforehand, soak some split peas in colored food dye or simply buy both green peas and yellow peas. Students add exactly 50 of the differently colored peas to each population of unmarked split peas. We now have hundreds, if not thousands, of members in each population of which 50 were ‘captured’ and marked in the first sampling event. The sampling can be done using a teaspoon of about 5mL capacity, which gives samples of about 50 individuals. The number of marked and unmarked split peas in the spoon are counted and a population estimate is made. The peas are replaced, the populations is mixed (stirring or shaking with the lid on) and the next sample is taken. This is repeated as long as required. Once there are sufficient estimates, the sampling can be drawn to a close and discussion of the estimates can take place. 

Supplementary materials include expository material on the motivating example and student worksheets.

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

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)