mathematics

Lincoln index

Lincoln index is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lincoln index rather than just read about it. In short: The Lincoln index is a statistical measure used in several fields to estimate the population size of an animal species. Described by Frederick Charles Lincoln in 1930, it is also sometimes known as the Lincoln-Petersen method after C.G.

Key takeaways

Lincoln index belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lincoln index to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lincoln index from memory before moving on to harder problems.

Reference excerpt

The Lincoln index is a statistical measure used in several fields to estimate the population size of an animal species. Described by Frederick Charles Lincoln in 1930, it is also sometimes known as the Lincoln-Petersen method after C.G. Johannes Petersen who was the first to use the related mark and recapture method.

Applications Consider two observers who separately count the different species of plants or animals in a given area. If they each come back having found 100 species but only 5 particular species are found by both observers, then each observer clearly missed at least 95 species (that is, the 95 that only the other observer found). Thus, we know that both observers miss a lot. On the other hand, if 99 of the 100 species each observer found had been found by both, it is fair to expect that they have found a far higher percentage of the total species that are there to find. The same reasoning applies to mark and recapture. If some animals in a given area are caught and marked, and later a second round of captures is done: the number of marked animals found in the second round can be used to generate an estimate of the total population. Another example arises in computational linguistics for estimating the total vocabulary of a language. Given two independent samples, the overlap between their vocabularies enables a useful estimate of how many more vocabulary items exist but did not happen to show up in either sample. A similar example involves estimating the number of typographical errors remaining in a text, from two proofreaders' counts.

Formulation The Lincoln Index formalizes this phenomenon. If E1 and E2 are the number of species (or words, or other phenomena) observed by two independent methods, and S is the number of observations in common, then the Lincoln Index is simply

L = E 1 E 2 S . {\displaystyle L={E_{1}E_{2} \over S}.}

For values of S < 10, this estimate is rough, and becomes extremely rough for values of S < 5. In the case where S = 0 (that is, there is no overlap at all) the Lincoln Index is formally undefined. This can arise if the observers only find a small percentage of the actual species (perhaps by not looking hard enough or long enough), if the observers are using methods that are not statistically independent (for example if one looks only for large creatures and the other only for small), or in other circumstances.

Limitations The Lincoln Index is merely an estimate. For example, the species in a given area could tend to be either very common or very rare, or tend to be either very hard or very easy to see. Then it would be likely that both observers would find a large share of the common species, and that both observers would miss a large share of the rare ones. Such distributions would throw off the consequent estimate. However, such distributions are unusual for natural phenomena, as suggested by Zipf's law. T. J. Gaskell and B. J. George propose an enhancement of the Lincoln Index that claims to reduce bias.

See also Sampling Theory Drake equation German tank problem Inter-rater reliability

Further reading Lincoln, Frederick C. (May 1930). Calculating Waterfowl Abundance on the Basis of Banding Returns. Circular. Vol. 118. Washington, DC: United States Department of Agriculture. Retrieved 21 May 2013. Petersen, C. G. J. (1896). "The Yearly Immigration of Young Plaice Into the Limfjord From the German Sea", Report of the Danish Biological Station (1895), 6, 5–84. T. J. Gaskell; B. J. George (1972). "A Bayesian Modification of the Lincoln Index". Journal of Applied Ecology. 9 (2): 377–384. doi:10.2307/2402438.

Notes

Worked examples

Example 1 — a first encounter with Lincoln index

Start with the simplest possible case. Write down what Lincoln index claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lincoln index before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lincoln index ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lincoln index

In research: Lincoln index appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lincoln index in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lincoln index is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ecological techniques, Environmental statistics, Estimation methods, so understanding it makes those chapters shorter.
In everyday life: Look for Lincoln index outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lincoln index in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lincoln index means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lincoln index out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lincoln index in simple terms?

Why does Lincoln index matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lincoln index?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lincoln index.