second sample. This count gives you the value of (MS2).
Record this value on the worksheet (see student worksheet
and example worksheet).
14. Students will use their counts from the samples and complete
a Lincoln-Petersen estimation of the M&Ms by utilizing the
formula given in the worksheet (found in the Appendix).
15. Students should be reminded (if necessary) to add 1 to
their counts and subtract 1 from the obtained population
estimate to account for sampling bias.
Obtaining a Second Population Estimate
16. Next, have students utilize the color of M&Ms occurring in
the lowest number (green, in our example found in the
Appendix) as the number of initially captured marked
and released individuals in an initial sample (MS1).
17. Have the student return all M&Ms into their container and
gently shake the container to randomly distribute the M&Ms.
18. Students (with their eyes closed and drawing M&Ms one at
a time) will then draw out the same number of M&Ms as in
Step 8. For instance, if students sampled 10 M&Ms in their
first MS1 calculation, they will once more randomly draw
out 10 M&Ms for the second initial sampling event and
population estimate for the color of M&Ms in fewest number (green, in our example). If any of the randomly selected
M&Ms are of the least common color, students will scratch
them for marking purposes, then return all M&Ms to the
container. Record the number of least common M&Ms
obtained for MS1 on the provided worksheet.
19. Utilizing this sampling methodology assures that sample
sizes are equivalent for the two population estimates. This
will allow for comparison and discussion of sample size
effects upon the Lincoln-Petersen population estimate.
20. Students will again gently shake or stir the candies in the
container, and then randomly draw out the designated
number as described in Step 8. Count the number of
marked M&Ms (MS2) of the second, random sample nS2
(as described in Step 11, above).
21. As before, the number of marked green M&Ms in the second sample represents MS2. Record this value on the
22. Using the provided formula on the worksheet, calculate a
population estimate for the green M&M species in the
community (as in Step 14).
23. Students will then compare their population estimates to
the actual number of M&M species of both colors (those
in highest [blue] and in lowest [green] numbers) in their
simulated community, and discuss how accurate their estimates are compared to the actual counts of M&M candies.
A few questions for students to consider are:
A. How close were your population estimates to the actual
number of M&M species in the community?
B. Were your estimates of the populations higher or lower than
the actual population number in either or both estimates?
C. If your population estimates were not very close to the actual
numbers of the M&M species, what might be some factors
that would account for the difference?
D. Assume that all the conditions required for the Lincoln-Petersen index are demonstrated for a population of animals
that you are studying. What could you infer about the population numbers of a particular species if you captured,
marked, and returned several individuals during your first
sampling period, but obtained no marked individuals after
the second sampling session?
E. Could you suggest a way (or ways) to obtain a more accurate
After the activity is completed, instructors should discuss the
results and students’ conclusions concerning their obtained population estimates. It is reasonable to assume—because of the relatively
small total number of M&M candies in the sample communities—
that almost all of the population size estimates obtained would be
One way to improve our estimates would be to combine more
student groups’ M&Ms and thereby obtain the larger sample sizes
necessary for more accurate estimates. Also, as a possible extension
of this activity, the instructor might point out that a repeated sampling method, such as the Schnabel method, will likely give more
accurate results than the more common Lincoln-Petersen method.
However, in a real-world application, it may not be possible for a
researcher to obtain multiple samples in some locations or ecologi-cal/environmental settings. For instance, it might be undesirable to
disturb the environment or species of concern on more than two
occasions if the researcher is working within a sensitive environment or with a threatened or endangered species.
Although perhaps not the most ideal method to reinforce or
teach ecological principles and/or techniques to students, we have
found that our students have responded favorably to this technique
and seem to obtain better conceptualization and understanding of
the underlying mathematical and ecological concepts related to
estimation of biological populations.
Because students have already counted the differently colored
M&M species within their communities, a natural extension of
our activity is to have students calculate a Shannon-Weaver diversity index number for their M&M communities and discuss the
resulting value in terms of species richness and evenness. Because
the numbers of the differently colored M&M species have already
been determined, it is relatively easy (with calculators having a
“natural log” function key).
Procedure: Shannon-Weaver Diversity Index
The formula for calculation of the Shannon-Weaver diversity
H ¼ ΣðpiÞð−lnpiÞ
Wherein: H = Calculated Shannon-Weaver index value
Σ = total sum
pi = proportion or average of a particular species
obtained in a sample
-lnpi = inverse (opposite) of the natural log of the proportion for a particular species in the sample