Maths IA · Statistics and probability · topic 40 of 97
How many fish are in the pond? Capture-recapture · IB Maths IA idea: guide, sample graph and step-by-step plan
one estimate means little: look at the spreadHow do ecologists count fish they cannot see all at once? Catch some, mark them, release them, then catch again and see what fraction are already marked.
- ✓ you like an experiment you can do at a desk
- ✓ you want to count something you cannot count directly
- ✓ you enjoy checking a formula by simulating it thousands of times
The big idea
If M animals are marked and later a second sample of size n contains m marked ones, the proportion m/n in the sample should be close to the proportion M/N in the whole population. That gives the estimate N ≈ Mn/m. The estimate is a ratio of random quantities, so it is biased and uneven, and the number of marked animals in the sample follows a known distribution you can write down. You derive the estimate, find how likely each m is, and use simulation to measure its bias and spread for different sample sizes. A classroom experiment with marked beans gives you real data where you know the true N. The depth comes from judging the method: a better estimator, an interval for N, and what happens when the assumptions are broken.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Use the idea that the marked fraction in the sample matches the marked fraction in the population to get N ≈ Mn/m.
→ A proportion argument you can state precisely. Say what assumptions it needs: a closed population, random sampling, marks that stay.Count the ways to choose a sample with exactly m marked animals and divide by all samples.
→ A probability formula with combinations. Compare it with a binomial approximation and say when the two are close.Use a bag of beans with known N, mark some, mix, draw a sample and estimate N. Repeat many times.
→ Real results to compare with theory. Because you know N, you can measure the error directly.Simulate many samples for different sizes n and plot the estimates.
→ A picture of the method's bias and variability, and a clear message about how large the second sample should be.Compare the simple estimate with the Chapman version (M+1)(n+1)/(m+1) − 1.
→ A second estimator and a fair comparison of bias and spread. Say which one you would use and why.Use simulation, or an estimate of the variance, to give a range of likely values of N.
→ A confidence-interval idea for a ratio estimator. Check by simulation how often your interval contains the true N.Your Maths IA research question: run the check
which population? which assumptions?Write your own question first. Then make sure it ticks every box:
- Names the population (real or a controlled experiment with beans) and what is marked.
- States the assumptions: closed population, equal chance of capture, marks not lost, no effect of marking.
- Says what you will measure: bias, spread or the chance an interval is correct, and for which sample sizes.
- Says how many repeats you will use and why that is enough.
- Goes beyond a single estimate: there is a distribution, a comparison or an interval.
The model
Mark M animals, release them, take a second sample of n, and count the m that are marked. The proportion argument gives N. Draw your own pond or bag.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The population, why direct counting is hard, and what you expect before calculating.
The proportion argument, the formula, the assumptions it needs and a worked example.
The probability of each m, built from combinations, with a check that the probabilities sum to 1.
The bean method, a table of repeated estimates, and the error against the true N.
The simulation, the number of repeats, and graphs of the estimates for several sample sizes.
The Chapman version compared with the simple one, and a way of giving a range for N.
Back to your aim: how good the method is and how big the samples should be, with numbers.
Which assumptions failed in your experiment, how a real animal population differs, and which limitation matters most.
Where every formula and result came from.
What your Maths IA graph could look like
show the true value!Made-up numbers to show the form of a strong graph, not the answer: the estimate of the population against the size of the second sample, with the true value marked. example only
- The average estimate and the true value on one set of axes, so the bias is visible.
- Axis labels say what is plotted: sample size on x, estimated population on y.
- State how many repeats each point is based on; one run is not a result.
- Plot your experiment's repeated estimates as points around the average.
- Another graph to consider: the whole distribution of estimates for one sample size.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out how capture-recapture is used. Before any calculation, guess how the estimate behaves when the second sample is small.
- Fix the assumptionsList what must hold: closed population, random sampling, marks that stay and do not change behaviour.
- Derive the estimateUse the proportion argument to get the formula and apply it to a small made-up example by hand.
- Trial run (1 lesson)Mark 20 beans in a bag of 100, mix, draw 20 and estimate N. Do it five times and note the spread. Fix your procedure now.
- Find the distributionWrite the probability of m marked in the sample using combinations, and check it against your repeated draws.
- SimulateSimulate thousands of experiments for several sample sizes. State the number of repeats and plot the estimates.
- Compare estimatorsCompare the simple estimate with a corrected one for bias and spread, and decide which to recommend.
- Go further, with a reasonBuild an interval for N, or break an assumption and see what happens. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a real survey and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Proportion and ratio – Equate the marked fraction in the sample with in the population to get .
- Combinations and hypergeometric probability – Write and check it sums to 1.
- Expected value and bias – Compute over the distribution of to show that the simple estimate overshoots, and compare with the Chapman version.
- Binomial approximation – Approximate by and measure the error against the exact probabilities.
- Simulation – Simulate thousands of samples, plot the distribution of estimates, and estimate the bias, standard deviation and the coverage of an interval.
Key equations
Syllabus links
- 4.1Populations, samples and sampling
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 1.10Counting principles; binomial theorem for any indexHL
- 4.1Populations, samples, bias and sampling
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 4.14Linear combinations of random variables; unbiased estimatesHL
- 4.16Confidence intervals for a meanHL
Read up first
- Probability (AA HL) →
- Counting principles (AA HL) →
- Probability distributions (AI HL) →
- Confidence intervals (AI HL) →
Data and sources to start from
A bag of beans or counters with known N, marked and resampled many times; extended by a computer simulation of the same process.
References to cite
- Chapman, D. G. (1951). Some properties of the hypergeometric distribution with applications to zoological sample censuses. University of California Publications in Statistics, 1(7), 131–160.
- Seber, G. A. F. (1982). The estimation of animal abundance and related parameters (2nd ed.). Griffin.
- Schwarz, C. J., & Seber, G. A. F. (1999). Estimating animal abundance: Review III. Statistical Science, 14(4), 427–456.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
no recaptures? no estimate- Treating one estimate as the answer. A single run has a large error; the spread is the result.
- Forgetting that the estimate is undefined when no marked animal is recaptured.
- Ignoring that marked animals may be easier or harder to catch, which breaks the model.
- Using the binomial for the number of marked animals without saying the sample is drawn without replacement.
- Too few repeats in the simulation, so the bias and spread are themselves noisy.
- Stating a confidence interval without checking, by simulation, how often it contains the true value.
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.